i\~

Linear and M

ultilinear Algebra, 1990, V

ol. 28, pp. 3-33R

eprints available directly from the publisher

Photocopying perm

itted by license onlyC

9 199 Gordon and B

reach Science Publishers S.A,

Printed in the United States of A

merica

The Largest E

igenvalue of a Graph: A

Survey

D. C

VE

TK

OV

ICD

epartment of M

athematics, F

aculty of Electrical E

ngineering, University of B

elgrade, PO

Box 816,

11001 Beograd, Y

ugoslavia

P. R

OW

LINS

ON

Departm

ent of Mathem

atics, University of S

tirling, Stirling F

K9 4LA

, Scotland

(Received M

arch 1, 1990)

Ths article is a survey of results concerning the largest eigenvalue (or index) of a graph, categorized

as follows: (i) inequalities for the index, (2) graphs w

ith bounded index, (3) ordering graphs by theirindices, (4) graph operations and m

odifications, (5) random graphs, (6) applications,

INT

RO

DU

CT

ION

Alm

ost all results related to the theory of graph spectra and published before1984 are sum

marized in the m

onographs (27) and (28). In view of the rapid grow

thof the subject in subsequent years it is no longer reasonable to expect a single bookto provide a com

prehensive survey of the latest results. Instead it seems m

ore ap-propriate that expository articles should be devoted to specific topics. F

or example

the paper (68) reflects the recent realization that many results from

analytic proba-bility theory have im

plications for the spectra of infinite graphs.H

ere we survey w

hat is known about the largest eigenvalue of a finite graph. T

histopic em

braces early results which go back to the very beginnings of the theory of

graph spectra, together with recent developm

ents concerning ordering and pertur-bations of graphs. Proofs w

hich appear in (27) and (28) are not repeated here. We

discuss only finite undirected graphs without loops or m

ultiple edges, and we start

with som

e basic definitions. Let G be a graph w

ith n vertices, and let A be a (0,1)-

adjacency matrix of G

, regarded as a matrix w

ith real entries. Since A

is symm

etric,its eigenvalues À

i, Ài,. .., À

n are real, and we assum

e that Ài ~ À

i ~ .. . ~ Àn. T

heseeigenvalues are independent of the ordering of the vertices of G, and accordingly

we w

rite Ài(G

) = À

i(A) =

Ài (i =

i,...,n) and refer to Ài,....,À

Ii as the spectrum of

G. T

he largest eigenvalue Ài is called the index of G

(orl spectral radius of A). W

ecall det(xI - A) the characteristic polynomial of G, denoted by ltG(x). The distinct

eigenvalues of G w

il be denoted by lli,..., llm, ordered as required. Since A

is anon-negative m

atrix, some general inform

ation on the spectrum of G

is providedby the Perron-Frobenius theory of m

atrices (45, 49, 57, 65, 67). In particular, if Gis connected then A

is irreducible and so there exists a unique positive unit eigen-vector corresponding to the index À

i. This vector w

e call the principal eigenvectorof G

: note that entries corresponding to vertices in the same orbit of A

ut( G) are

3

4D

. CV

ET

KO

VlC

AN

D P. R

OW

LIN

SON

equal. We shall also use im

plicitly the fact that if G' is obtained from

G by adding

an edge then À1(G

') ~ À1(G

), with strict inequality w

hen G is connected: this is an

imm

ediate consequence of the fact that the spectral radius of a non-negative matrix

increases with each entry. Further fundam

ental results in matrix theory w

hich serveas a background to problem

s concerning the largest eigenvalue may be found in

(19, 35). We note in passing that although the eigenvalues of a directed m

ultigraphneed not be real, such a digraph has a positive eigenvalue À

1 such that IÀI ~

À1 for

all eigenvalues À. Som

e results on the spectral radius of digraphs may be found in

(6, 83, 93, 98). For differing approaches to the spectra of infinite graphs, with som

eresults on the largest eigenvalue, the reader is referred to C

hapter VI of the m

ono-graph (27), the expository article (68) and the paper (17). For the index in particular,see (8). T

he paper (96) extends to infinite graphs a result on finite graphs whose in-

dex does not exceed V2 +

v' (Theorem

2.4). Some discussion of the index of a

graph may be found in the expository papers (84, 91, 101).

Finally we point out that several of the authors' results m

entioned in this articlew

ere conjectured on the basis of numerical evidence furnished by the expert system

"Graph" (22, 23, 31, 90). In som

e instances, proofs were com

pleted by using thesystem

to check outstanding cases.

1. BOUNDS FOR THE INDEX OF A GRAPH

Here w

e give upper and lower bounds for the index of a graph w

hich are ex-pressed in term

s of various graph invariants. These bounds are interpreted and used

from different view

points in other sections of this article: in Section 2 for example

we survey classes of graphs defined by som

e bounds on the index. The discussion

of graph ordering in Section 3 and of graph perturbations in Section 4 is often con-cerned w

ith bounds on the index. In §6.1 we return to the question of bounding

other graph invariants in terms of the index.

In the fundamental paper (18) on the theory of graph spectra, C

ollatz and Sino-gow

itz observed that the index À1 of a connected graph on n vertices satisfies the

inequality11

2cos- .: À1'': n - 1.

n+1 - -

The lower bound is attained by the path Pn while the upper bound is attained by

the

complete graph K

n. If we om

it the assumption of connectedness, then for a graph

without edges w

e have À1 =

0 and otherwise À

1 ~ 1.A reformulation of inequalities from the theory of non-negative matrices (67,

Chapter 2) yields the follow

ing theorem.

TH

EO

RE

M 1.1 Let dm

im d, dm

ax respectively be the minim

al, mean and m

axmal val-

ues of the vertex degrees in a connected graph G. If À

1 is the index of G then

dmin ~ d ~ À1 ~ dma.

Equality in one place im

plies equality throughout; and this occurs if and only if G is

reguar.

LAR

GE

ST

EIG

EN

VA

LUE

OF

A G

RA

PH

5

,

Let A denote the adjacency matrix of an n-vertex graph G. For any non-zero

vector vE W

the Rayleigh quotient vT

Av/vT

v is a lower bound for À

1(G), as can

be seen by diagonalizing A. If w

e take v to be the all-1 vector j then we obtain the

inequality d ~ À1 of T

heorem 1.1. Sim

ilarly, IAvl/lvl is a low

er bound for À1, and

the next two results m

ay be obtained by setting v = j, v =

ei respectively, where the

i-th vertex of G has m

axmal degree and ei =

(6ii,...,6inl.

TH

EO

RE

M 1.2 (H

ofmeister (54)) If the vertices of G

have degrees d1,d2,.. .,dn then

À1(G) ~ l(lln)

,,7=1 dr.

TH

EO

RE

M 1. (N

osal (70); Lovász &

Pelikán (62)) If dmax is the m

axmal degree of

a vertex in G then À

1(G) ~ ldm

ax.

Hofm

eister went on to show

that for any graph G, there exists a real num

ber p,

unique if G is non-regular, such that À

1 (G) =

\! (11 n) ,,7=1 dl.

Theorem

s 1.1 and 1.3 show that the index of a graph is controlled by the m

axmal

degree in the sense that À1 is bounded above and below

by a function of dmax. H

off-m

an (51) observes that Ram

sey-type arguments m

ay be used to prove that À1(G

) iscontrolled by the least num

ber t such that neither Kt nor the star K

1,t is an inducedsubgraph of G

.W

e state one more result w

hich demonstrates the interplay betw

een index andvertex degrees. In a regular graph G

of degree r on n vertices the number N

kof w

alks of length k is given by Nk =

nrk; thus N N

k I n = r and this suggests that

N N

k I n in the general case might be regarded as a certain kind of m

ean valueof vertex degrees. A

ccordingly d = lim

k~+

oo NN

kln is defined to be the dynamic

mean of the vertex degrees of an arbitrary graph G

.

THEOREM 1.4 (D. Cvetkovic (20)) For any graph G, the dynamic m~an of

the vertex

degrees is equal to the index of G.

We give next som

e inequalities for the index of a graph involving the number n

of vertices and the number e of edges. A

part of Theorem

1.1 may be stated in

the form À

1 ~ 2e I n, since 2e I n is the m

ean degree. An upper bound in term

s of eand n m

ay be obtained by maxm

izing À1 subject only to the constraints ,,7=

1 Ài =

o and ,,7=1 À

T =

2e (that is, tr(A) =

0 and tr(A2) =

,,7=1 di). T

hen we obtain the

following result of W

ilf.

THEOREM 1.5 (100) For any graph G, À1(G) ~ V2e(1-1In).

The inequality in the next result w

as announced by Schwenk in (82), and a proof

by Yuan appeared som

e thirteen years later.

TH

EO

RE

M 1.6 (103) F

or any connected graph G, À

1(G) ~

v2e - n + 1. E

qualityholds if and only if G

is the star K1,n-1 or the com

plete graph Kn.

Proof Suppose that G has vertices 1,2,..., n and adjacency m

atrix A. L

et di de-note the degree of vertex i and let C

i be the i-th column of A

. Let x be the principal

eigenvector of A, say x =

(xi,...,Xn)T

, and let Vi be the vector w

hose j-th entry is

6D

. CV

ET

KO

viC A

ND

P. RO

WL

INSO

NLA

RG

ES

T E

IGE

NV

ALU

E O

F A

GR

AP

H7

Summ

ing over i, we obtain

A low

er bound for the chromatic num

ber is given in the next theorem, the proof

of which is outlned in (28, C

hapter 3).

TH

EO

RE

M 1.8 (A

. J. Hoffm

an (52)) Let),1 and ),n be the largest and the least eigen-

value of a graph G. T

hen the chromatic num

ber ¡(G) satisfies

),1¡(G

) 2' 1 + I),n I.

Xj if j '" i ("j is adjacent to i") and 0 otherw

ise. We have c7V

¡ = c7x =

),1Xi, and so

by the Cauchy-S

chwarz inequality,

),r xl ~ lei IZlv¡\Z = di (1 - L xi) .

j-fi

),r ~ 2e - tdi (LX

Y) .

i=1 j-fi

Subsequently D. C

vetkovic (21) proved thatn¡(G

) 2' n - ),1'(1.1)

But

t.d¡ (j;xi) ~ t.diX

¡ + t.d¡ (1;/1)

n n ( ) n~LdiXr+L L xy =LLxy=n-1

i=1 i=1 i'f-fi i=1 di

a result which w

as rediscovered in (38).L

et ~(G) be the size of the largest clique in G

: ~(G) is called the clique num

berof G

. Since ¡(G) ~ ~(G

) one might ask w

hether nj(n - ),1) is also a lower bound

for ~(G). T

his was proved in (39) for planar graphs, w

hile the authors of (38) offerthe inequality

-1 n

~(G):?-+

-.- 3 n - ),1

Both (1.1) and (1.2) can be im

proved however because W

ilf has shown:

TH

EO

RE

M 1.9 (W

ilf (102)) For any graph G

,

(1.2)

and so ),t ~ 2e - n + 1. E

quality holds if and only if for each i, either di = 1 or

di = n - 1, and the result follow

s. ·T

here are several inequalities involving the index ),1 and the chromatic num

ber

¡( G) of a graph.

TH

EO

RE

M 1.7 (H

. S. W

ilf (100)) For the chrom

atic number ¡(G

) of a graph G w

ehave

n):?-.

~(G . - n - ),1

¡(G) ~

1 + ),1'

For a connected graph G, equality holds if and only if G

is complete or a cycle of odd

length.

Proof (101) Delete edges from

G until a critical graph G

is obtained: thus thedeletion of any further edge w

ould reduce the chromatic num

ber. In G all vertex

degrees are at least ¡(G) - 1. If dm

in(G) denotes the m

inimal degree of G

then we

have¡(G

) -1 ~ dmin(G

) ~ ),1(G) ~ ),1(G

).

If G is connected and equality holds then G

is regular of degree ¡(G) - 1. T

hesecond assertion of the T

heorem now

follows from

the classical result of Brooks

(10), which for a connected graph G

states that ¡(G) ~

1 + dm

ax, with equality if

and only if G is com

plete or a cycle of odd length. ·A

nalogous results for point-arboricity and related invariants (61) have been ob-tained by L

ick (60): see (28, pp. 90-91). A com

putational comparison of several

bounds for the chromatic num

ber of a graph appears in (37): the spectral boundfrom

Theorem

1.7 is reported to be in the middle of the list.

In addition he was able to derive a better lower bound for K,( G) which involves

),1 and a corresponding eigenvector.T

he following condition for K

,( G) ~ 3 is established in (28, p. 86):

TH

EO

RE

M 1.10 (N

osal (70)) Let),1 be the index and e the number of edges of a

graph G. If ),1 ? ve, then G

contains a triangle.

This result is extended in (9) to provide conditions on ),1 w

hich ensure a girthno greater than 2k +

1 for k a positive integer. Also a condition is presented w

hichguarantees that the girth is no m

ore than four.F

inally we m

ention two results involving the index of the com

plement G

of agraph G

. The first m

ay be proved by applying Theorem

s 1.1 and 1.5 to G and G

.

TH

EO

RE

M 1.11 (E

. Nosal (70), A

. T. A

min and S

. L. Hai6~

i (1)) Let G be a graph

on n vertices. We have

n -1 ~ ),1(G) +

),1(G) ~ V

i(n -1).

TH

EO

RE

M 1.12 (D

. C. Fisher (40)) L

et fez) = 1- C

1Z +

czzz - C3Z

3 + ... w

here Ck

is the number of com

plete subgraphs on k vertices in G. If r (G

) is the reciprocal ofthe sm

allest real root of fez) then ),1(G) ~ reG

) - 1.

8D

. CV

ET

KO

VIC

AN

D P. R

OW

LIN

SON

LAR

GE

ST

EIG

EN

VA

LUE

OF

A G

RA

PH

9

Proof We give only a brief outline of the proof. L

et V(G

) denote the set ofvertices of G

, and let M (G

) be the monoid generated by V

(G) subject to the re-

lations uv = vu precisely w

hen uv is an edge of G. For n =

0 let Pn be the num-

ber of n-Ietter words in M(G). An inclusion-exclusion argument (41) shows that

for n)o 0, Pn = cipn-i - CZPn-Z +... + (-It+lcnPo, with po = 1. Thus f(z)(po +

piz + P

2ZZ

+ ...) =

1, and it follows that lim

n--oop~/n =

reG). N

ext let qn be thenum

ber of n-Ietter words w

on V(G

) as alphabet, subject to the restriction thatadjacency in G

precludes adjacency in w. W

e have Pn;: qn = jT

(A +

I)j, where A

- i/n i/n i/n -

denotes the adjacency matrix of G. Hence pn ;: qn , and qn -+ Ài(G) + 1 as

n -+ 00. T

he result follows. ·

. ,"..:::,-~:n

\"-,I/./

---en In edges'

2. GRAPHS WITH BOUNDED INDEX

'ij~\1!;~!""m

i

¡ID-

f1i!:il1I,li

"

f~ .;¡,:,

m~.'1l~~

00:;.1',

')1

~!~j

~jWl, :

Some interesting classes of graphs can be obtained by prescribing an upper bound

for the index. The graphs for w

hich .\i :: 2 can be determined essentially because

their vertices have mean degree:: 2 and m

axmum

degree:: 4 (cf. Theorem

s 1.1and 1.3): the classification of such graphs is given in Section 2.1. In Section 2.2 w

ediscuss the graphs for w

hich ..i :: ";2 + 0: note that if 7 denotes the golden ratio

lci + 0) then ";2 + 0 = 73/2 = 7i/2 + 7-i/2 ~ 2.058171. The significance of this

number as an upper bound is explained (in part) as follow

s. If G is a connected

graph which is neither a tree nor a cycle then for som

e m ?: 3 it has a subgraph H

mconsisting of an m

-cyc1e and a pendant edge. It follows from

a result of Hoffm

an

(50) that Ài(Hm) approaches ";2 +.. from above as m -+ 00, and consequently

..i(G))o ";2 + ... Thus apart from cycles, the connected graphs with index at most

";2 + 0 are trees. T

heorem 2.4 provides a classification of the trees G

for which

20( ..i(G):: ";2 + ... Since ";2 +.. = limm--oo..i(Hm), the number ";2 +.. is

said to be a limit point of graph indices: such limits are the topic of Section 2.3,

where ";2 +

.. wil assum

e further significance.

iG

2

3

45

64

2

G)

FIGU

RE

1.

2.1. Graphs w

hose largest eigenvalue does not exceed 2W

e start with the follow

ing result.

TH

EO

RE

M 2.1 (91) T

he connected graphs whose largest eigenvalue does not exceed 2

are precisely the induced subgraphs of the graphs shown in F

igure 1.

then either G =

G2 or a second path has length less than 3. In the latter case G

isan induced subgraph of G

3 or some W

n. Finally, if the maxm

al degree of a vertexin G is 2 then G is a path and hence an induced subgraph of some en' .

Note In Figure 1, W

o = K

i,4 and each graph has index 2: the numbers attached

to vertices are components of a corresponding (positive) eigenvector.

Proof Let G

be a connected graph with ..i(G

):: 2. Since ..i increases strictly

monotonically w

ith the addition of vertices, provided the graph remains connected,

G is either a cycle en or a tree; m

oreover Wo is the only possible tree w

ith a vertexof degree greater than 3. If the m

axmal degree is 3, then either G

= W

n for some

n )0 0 or G has a unique vertex of degree 3 w

ith three paths attached. In the secondcase either G

= G

i or one of the three paths has length 1. If one path has length 1

Theorem

2.1 and its proof are due to J. H. S

mith (91), and accordingly graphs

from Figure 1 are often called Sm

ith graphs in the literature (see, for example, (29)).

Seidel (84) proposed the nam

e 'Coxeter graphs' because the graphs inqueestion ap-

pear implicitly in C

oxeter's work on discrete groups generated by reflections in hy-

perplanes. Since the topic of this article is a part of graph theory rather than grouptheory, w

e prefer the elegant graph-theoretic proof by Sm

ith. For another proof

of Theorem

2.1 see (69). For the rôle of these graphs in algebra see, for example,

(47), where they appear as D

ynkin diagrams. T

he Smith graphs are very im

portantfor another part of the theory of graph spectra, namely the study of graphs with

least eigenvalue bounded below by -2. T

hese graphs have been characterized byC

ameron, G

oethals, Seidel and Shult in terms of root system

s (14). On the other

hand, root systems can be generated starting from

the Sm

ith graphs (14, 25) and

10D

. CV

ET

KO

VlC

AN

D P. R

OW

LIN

SON

LAR

GE

ST

EIG

EN

VA

LUE

OF

A G

RA

PH

11

ra A2.2. Graphs whose largest eigenvalue does not exceed V2 + V5

In view of T

heorem 2.1, the graphs of the title w

il be determined once w

e havefound the connected graphs w

ith index in the interval (2, V2 +

VS); and w

e havealready noted that such graphs are trees. In order to describe the trees w

hich arise,let T

( a, b, c) denote the graph with a vertex v of degree 3 such that T

( a, b, c) - v =Pa U

Pb U Pc. For a ;: 2, b;: 1, c;: 2 let Q

( a, b, c) be the tree obtained from the path

with vertices 1,2,.. .,a +

b + c - 1 (in order) by adding pendant edges at vertices a

and a + b. If A

and B are rooted graphs, Pn(A

,B) denotes the graph obtained by

joining an endvertex of Pn to the root of A and the other endvertex of Pn to the

root of B. All of these graphs are ilustrated in Figure 4.

The follow

ing classification theorem com

bines the results of several authors, andincludes im

plicitly the fact that no graph has index equal to V2 +

VS

.

TH

EO

RE

M 2.4 (l1, 26) T

he connected graphs with index in the interval (2, V

2 + V

S)

are precisely the trees of the following types.

(a) T(a,b,c)for

Note that am

ong the graphs in Figure 3, the graph G

9 has smallest index, approx-

imately 2.007. T

hus there is no graph with index in the interval (2,À

i(G9)).

FIGURE 2

this makes it possible (25) to give elem

entary alternative proofs of some im

portanttheorem

s from (14).

Let S be the set of all graphs (not necessarily connected) w

hose index does notexceed 2. C

vetkovic and Gutm

an (29) determined explicitly the spectra of all graphs

in S. One of their observations w

as that any eigenvalue is of the form 2cos(pjq)7r

for some integers p,q (q =

f 0). It is interesting that the same conclusion follow

s froman early result of L

. Kronecker (56), as indicated in a review

of (29) by J. H. Sm

ith(cf. M

R57, #5079). K

ronecker's Theorem

reads: let À be a non-zero real num

berw

hich is a root of a monic polynom

ial p with integer coeffcients. If all roots of p

are real and in (-2,2), then À =

2cos27rr for some rational num

ber r.T

he index of a graph in S is either equal to 2 or is of the form 2 cos 7r j q for a

positive integer q.C

o spectral graphs from

the set S have been studied in (29), too. F

or example,

the graph Wn from Figure 1 is co

spectral with the disjoint union of C

4 and Pn+1'

Note that for n = 0 we obtain the smallest pair of non-isomorphic cospectral graphs,

consisting of Ki,4 and C4 U Pi.

The problem

of deciding whether there exists a graph w

ith given spectrum seem

sto be intractable in the general case. E

ven more diffcult is the problem

of construct-ing all graphs w

ith a given spectrum. H

owever, if the eigenvalues (given) belong to

the segment (-2,2) both problem

s are easily solved and an appropriate algorithm is

formulated in (29).

The difference À

i - Àn betw

een the largest and smallest eigenvalues is called the

spectral spread of a graph. In graphs from the set S the spectral spread is bounded

above by 4. Since Àn =

-Ài in bipartite graphs, bipartite graphs outside S certainly

have spectral spread greater than 4. How

eyer, there are only finitely many non-

bipartite graphs outside S having spectral spread bounded by 4, as the follow

ingresult of Petrovic show

s.

TH

EO

RE

M 2.2 (71) A

connected graph has spectral spread bounded above by 4 if andonly if it is an induced subgraph of one of the Sm

ith graphs or of one of the graphs inFigure 2.

This theorem

may be regarded as a generalization of T

heorem 2.1.

For the next result, due to C

vetkovic, Doob and G

utman, recall that a graph is

minim

al with respect to a property P

if it has property P but none of its vertex-

deleted subgraphs has property P.

TH

EO

RE

M 2.3 (26) T

here are exactly 18 graphs which are m

inimal w

ith respect tothe property of having index greater than 2. T

hese graphs, together with their indices

are shown in F

igure 3.

a = 1,

b = 2,

c)- 5

or

a = 1,

b)-'2,c)- 3

or

a = 2,

b = 2,

c)- 2

or

a =2,

b = 3,

c = 3.

(b) Q(a,b,c)for

(a,b,c) E H2, 1,3),(3,4,3),(3,5,4),(4, 7,

4),(4,8, 5)J or

(a,c) =f (2,2) and

a)- 1,

b;: b*(a,c),

c)- 1

where

r+'

fora)- 3,

b*(a,c) = 2 +

cfor

a = 3,

-1+ c

fora =

2./

A slightly w

eaker form of T

heorem 2.4, w

ithout specification of the functionb*(a,c), was proved by Cvetkovic, Doob and Gutman (26).

The function b*(a,c)

was determ

ined by Brouw

er and Neum

aier (11). The proof is sim

ilar in spirit tothat of T

heorem 2.1, but first it is necessary to verify that lim

n_oo.Ài(T

(l,n,n)) =lim

n_ooÀi(T

(2,2,n)) = V

2+V

S; and to determine the conditions under w

hich/À

i(Pn(A,B

)) is greater or less than Ài(Pn+

i(A,B

)). The relevant results on the sub-

division of an edge are given in Theorem

4.5.

;":î~~ '~:_~sk-:)~~~:-*~~ß~~;~¿$j€f~~i~!;~#~~-'~:!~~:-1í:~!:~!!!!~~!!~li~#i'~¡m5fil2$~J~~~;:;æ~~1~r_3l=:;;:;;~;:;,::'t,~-\~::~:_::§ê'gE'~:E~:~1~;:!:~~!~::tli~Jt¡:._. -~- ~ - f . .

/:

~Gi

G6Gs

~o- 2.007

GB

FIGURE 3. The eighteen minimal graphs with index greater than 2.

Gg

'.

"10 Gii GI2 GI3

Gis GI6 GI7

FIGURE 3. (Continued)

-N

~,tl()..

~o::(),~a

oG7 ~

:;

~zvioz

-0

~

~:;QtTvi--t:QtT

~5tIo'I;iQ:;~'i::

GI4

Gis

..~

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OF

A G

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PH

92.3. limit points of graph indices

We say that the real num

ber À is a lim

it point of graph indices if there is aninfinite sequence of graphs G

n such that À =

limn__coÀ

i(Gn) and the À

i(Gn) are

distinct. The study of lim

it points of graph eigenvalues was initiated by H

offman in

the paper (50), where he determined all limit points less than ';2 + 0. It is clear

from Section 2.2 that no num

ber less than 2 is a limit point of graph indices; and

we have noted that ';2 +

vi is itself a limit point. H

offman's result is as follow

s.

TH

EO

RE

M 2.5 (50) F

or n E N

, let ßn be the unique positive solution of the equation

xn+l =

1 + x +

x2 + ... +

xn-i, and let an = ß~/2 +

ß;;il2. The num

bers an (n E N

)are the numbers less than ';2 + vi which are limit points of graph indices. Moreover

2 = ai -: a2 -: .,. and lim

n_co an = ';2 +

VI.

Each an is realized as the lim

it point of indices of trees which consist of a path

with a pendant edge attached. In fact, the num

bers an (n E N

) are all the limit

points in (2,';2 + vi) of spectral radii of symmetric matrices whose entries are

non-negative integers.In (50), H

offman suggested that possibly there exists a real num

ber À such that

every number at least À

is a limit point of graph indices. T

his turned out to be truew

ith À =

';2 + V

I, the value which T

heorem 2.5 show

s to be the smallest possible

candidate. Indeed, Shearer (85) proved by direct construction that if ¡i :2 ';2 + vi

then JL is a limit point of indices of trees. Limit points of

eigenvalues other than the

largest are studied in (36).

Û.t

NN0=.5'0~::U'":a1l0.(0..co

~

.!()

3. ORDERING OF GRAPHS

'"

J

iT

he ordering of graphs by index was proposed by C

ollatz and Sinogow

itz (18)follow

ing their investigation of indices of trees. Lovász and Pelikán (62) suggested

that among trees w

ith a prescribed number of vertices, the higher the index the

more 'dense' the tree. In support of this view

they proved that among trees w

ithn vertices the star K

i,n-i has largest index (vn -1) and the path Pn has smallest

index (2cos7rj(n + 1)). In fact the first assertion is im

mediate from

the observation(cf. W

ang (97)) that the spectrum of a tree is sym

metric about zero and satisfies the

relation Ài +

... + À

~ = 2(n -1). W

e can go on to show that K

i,n-i is the only n-vertex tree with index vn -1, for then the spectrum is (.J,o,O,.. .,O,-.J);

the adjacency matrix therefore has rank tw

o, and so with appropriate ordering of

columns has the form

(i, g) where each entry of B

is /1. The tree is therefore a

complete bipartite graph and hence a star.

To show

that Pn alone has sm

allest index, Lovász and Pelikán exploit the follow

-ing expression (28, T

heorem 2.12) for the characteristic polynom

ial of a graph Gwith a bridge uv:

J

"t '11 "U ~

-.¡i~;:òtiqiG(x) = qiG-uv(X) - qiG-u-v(X).

(3.1)

If G is a tree other than a path then w

e can construct a tree G' w

ith index smaller

than G as follow

s. Choose vertices u, w

in G such that deg( u) ? 2, deg( w

) = 1

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H17

and d(u, w) is minimaL. Then u is adjacent to a vertex v not on the u - w path

in G and w

e obtain G' from

G by replacing the edge vu w

ith vw. Since the

graphs G - uv,G

' - vw are isom

orphic we have ÍJG

(x) - ÍJG'(x) =

ÍJG'-v-w

(x)-ÍJG-u-v(x) from (3.1). Now G - u - v is isomorphic to a spanning subgraph of

G' - v - w

, while repeated application of (3.1) show

s that for any spanning subgraphH

' of a tree H w

e have ÍJH(X

) -( ÍJH'(X

) for all x:; Ài(H

). It follows that ÍJG

(x)-(ÍJG

'(x) for all x:; Ài(G

' - v - w), in particular for x =

Ài(G

'); hence Ài(G

')-(À

i(G). A

ccordingly we have the follow

ing result.

TH

EO

RE

M 3.1 (18, 62, 97) A

mong the trees w

ith n vertices (n:; 1), the star Ki,n-i

alone has largest index and the path Pn alone has sm

allest index.

Sim

ilar arguments concerning characteristic polynom

ials were used by S

imic in

an analogous investigation of unicyclic graphs. The argum

ents make use of som

egeneral theorem

s on the change in index of a graph resulting from various m

odifi-cations described in S

ection 4.

of Cs and a vertex of C

d-s then the index of the resulting graph decreases as sincreases (3 ~

s ~ (d 12)).

So far w

e have been concerned primarily w

ith trees and unicyclic graphs. The

impetus for investigating the À

i-ordering of other classes of graphs came from

two

quarters: Brualdi and H

offman (7, p. 438) posed the problem

of finding the maxm

alspectral radius of a (O

,I)-matrix w

ith a prescribed number of ones; and C

vetkovic(81, p. 211) asked how

the index of a graph consisting of a fixed cycle and a chordvaries w

ith the position of the chord. This second question w

as answered inde-

pendently by Sim

ic and Kocic (89) and (for a cycle of even length) by R

owlinson

(75), using entirely different methods. Sim

ic and Kocic consider the m

ore generalclass of n-vertex graphs consisting of k disjoint paths (k :; 2) betw

een two vertices

u and v. If the components of the principal eigenvector corresponding to vertices

in the i-th path are x~,xL...,X

~I¡' then we have m

i+m

i+...+

mk=

n+k-2,

xij = x5 =... = x~, xj = X~i¡_j (j = O,...,mi) and

!-;." "

Ldi,i

~' i

I.!

" "c;"

i~."'~ij,.'...,...

, ¡'

~ it.~

I.,"..

' r¡Ii

W~, ,

~"."..'

'h'

di~tt.,'",m:~;

ij'.,.l'"

THEOREM 3.2 (13, 86) Let Ktn-i denote the graph obtained from Ki.n-i by

addingan edge. A

mong the uiiicyclic gráphs w

ith n vertices, the graph Ktn-i alone has largest

index and the cycle Cn alone has smallest index. '

The second statem

ent here is imm

ediate from the fact that a unicyclic graph

has mean degree 2 and this low

er bound for the index is attained precisely when

the graph is regular. Further results concerning the Ài-ordering of unicyclic graphs

are derived by Cvetkovic and R

owlinson in (32). For exam

ple, let Gm

,n,r,s(n ~ m ~

1, m + n ~ 3, r ~ 1, s ~ 1) denote the graph obtained from Cm+n by attaching

paths Pr+

i,Ps+

i at vertices distance m apart in C

m+

n. (Attachm

ent of a path istaken to m

ean attachment by an end-vertex.) T

hen Ài(G

m+

i,n-i,r,s) -( Ài(G

m,n,r,s)

for 1 ~ m ~ n - 2. Let Eel (e ~ 3, f ~ 1) denote the graph obtained from Ce by

attaching a path PI+l by

an end-vertex. Then À

i(En+

i,d) -( Ài(G

i,n.i,d-i). The fol-

lowing results of Li and Feng (59) may be applied to show that if r + s = d and 1 ~

r~(dI2)-1 then À

i(Gi,n,r,s)-(À

i(Gi,n,r+

i,s-i); and if r-l~r-s~

m:;1 then

Ài(G

m,n,r,s):; À

i(Gm

,li,r+i,S-i). M

oreover one can compare the indices of graphs ob-

tained by attaching two paths at the sam

e vertex of a fixed cycle (cf. equation (4.2)).

TH

EO

RE

M 3.3 (59) Let u, v be vertices of G

such that d(u, v) = m

. Let Gr,s denote

the graph obtained from G

by attaching a path of length r at u and a path of length sat v. Then Ài(Gr,S):; Ài(Gr+i,S-i) under any of the following conditions

(i) m =

0, deg(u) ~ 1, and r ~ s ~ 1;(ii) m

= 1, deg(u) ~ 2, deg(v) ~ 2 and r ~ s ~ 1;

(iii) m:; 1, deg(u) ~ 2, deg(v) ~ 2, r - s ~ m

and s ~ 1.

xi. i - Àixi, i + xi. = 0

J+ J+

J(j=

0,...,mi-2).

(3.2)

These results too are proved by com

paring characteristic polynomials (and using

equation (4.6)). In the situation (i) of Theorem

3.3, we m

ay regard Gr,s as obtained

from G

by amalgam

ating u with an interm

ediate vertex of a path of fixed lengthd = r + s: for 1 ~ r ~ (dI2), Ài(Gd-s,S) increases with s. Using different methods

(d. Theorem

4.6), Sim

ic (87) proved that if instead we am

algamate u w

ith a vertex

The recurrence relations (3.2) may be solved to give x~ as a function of mi,...,mk

and Ài. From

the relation Ax =

Àix w

e know that À

ix~ = xi +

... + xt, and this

equation defines Ài as an im

plicit function of mi,..., m

k. It is now a m

atter of cal-culus to show

that if (say) m3,...,m

k are held fixed then Ài is a strictly increasing

function of Imi - mil. (Entirely analogous results hold for the class of graphs ob-

tained by amalgam

ating the vertices u and v.) Setting k = 3 w

e can now answ

erC

vetkovic's question as follows:

TH

EO

RE

M 3.4 (89) Let G

n,k denote the graph consisting of an n-cycle 123... nl to-gether w

ith the chord lk (3 ~ k ~ n - 1). Then À

i(GIi,3):; À

i(GIi,4):; ... :; À

i(GIi,s)

where s =

1 + (nI2).

Thus the bicyclic H

amiltonian graphs w

ith n vertices are distinguished by theirindices; in particular G

Ii,3 alone has largest index and Gii,s alone has sm

allest index.S

imic (87) subsequently extended T

heorem 3.4 by show

ing that the same conclusions

hold if instead of adding to the cycle the chord lk we add an arbitrary connected

graph G w

ith similar vertices ll, v identified w

ith l,k respectively. (Vertices are sim

-ilar if they lie in the sam

e orbit of the automorphism

group of G.)

Row

linson's approach to Theorem

3.4 was to introduce an algorithm

which en-

ables the characteristic polynomial of a m

ultigraph G to be com

puted recursivelyin term

s of characteristic polynomials of local m

odifications of G. Suppose that G

has at least three vertices, let ll, v be distinct vertices of G and let m

be the number

of edges between II and v. L

et G - (uv) denote the m

urÙgraph obtained from

G by

deleting all edges between u and v, and let G

* be the multigraph obtained from

G - (uv) by am

algamating u and v. A

ppropriate determinantal expansions yield the

relation

ÍJG(X

) = ÍJG

-iuv)(x) + m

ÍJG-(x) +

m(x - m

)ÍJG-u-v(x) - m

ÍJG-u(x) - m

ÍJG-v(x),(3.3)

~:111,Ii'

;l~ ;.;¡

~~~.~;~1~~i'.R._.I'

~~~'~~~. i~

i

Lt'iijUJI~~r!~

L \1..I~~~h~~~N ~

Iff.* I~"~~R:iii¡J!I!.;i.iil~.~ ~

~

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Equation (3.3) is called the deletion-contraction algorithm

. Note that if G

is a graphthen G

* wil have m

ultiple edges precisely when u and v have a com

mon neighbour

in G; hence the m

ultigraph setting. If we apply (3.3) to the graph G

ii,k of Theo-

rem 3.4 w

ith u = 1 and v =

k, we can eventually express the characteristic polyno-

mial of G

ii,k in terms of characteristic polynom

ials of paths and cycles. (Here w

em

ake repeated use of (3.1) and (3.3).) Now

Pr,C

r have characteristic polynomials

Ur(lx),2T

r(lX) - 2 respectively, w

here T"U

r are Chebyshev polynom

ials of the firstand second kind (28, p. 73). A

ccordingly the characteristic polynomial of G

ii,k hasan expression in term

s of Chebyshev polynom

ials; for even n this expression canbe analyzed to yield the conclusion of T

heorem 3.4. Sim

ilar techniques were used

by Bell and R

owlinson (3) in an investigation of tricyclic H

amiltonian graphs (cycles

with tw

o chords). They show

ed first that if such a graph has n vertices and max-

mal index then the tw

o chords have a vertex in comm

on. Let lL

(h,t,k) denote theindex of the graph Gh,t,k (h ~ 1, t ~ 0, k ~ 1, h + t + k + 3 = n) consisting of an n-

cycle 123... n1 together with chords joining vertex 1 to vertices h + 2 and n - k.

The results on À

i-ordering are (i) if 1:: k :: t then lL(h,t,k) -( lL(h,k - 1,t + 1),

(ii) if k ~ t ~ 1 then lL(h,t,k) -( lL

(h,t - I,k + 1), (iii) if 2:: h:: k then lL

(h,O,k)-(

¡.(h - 1,0,k + 1). It is now

easy to identify the unique graph with m

axmal index.

TH

EO

RE

M 3.5 (3) A

mong the tricyclic H

amiltonian graphs w

ith n vertices (n ~ 5),

the graph Gi,O

,Ii-4 alone has largest index.

It should not prove too diffcult to determine the tricyclic Hamiltonian graphs

with sm

allest index. Another open question concerning indices of H

amiltonian

graphs relates to the family 1111 of m

axmal outerplanar graphs on n vertices. E

vi-dence from

"Graph" suggests that the fan K

i 'iPIi-i is the unique graph in 1111 w

ithlargest index and that P; is the unique graph in 1111 with smallest index. (The graph

Ki 'i Pii-i is obtained from

Pii-i by adding a vertex adjacent to each vertex of Pii-i;and the graph P

; is obtained from P

ii by joining vertices which are distance 2 apart

in Pii') In support of this conjecture, Raw

linson has proved the following.

TH

EO

RE

M 3.6 (80) L

et gii (n ~ 4) denote the class of maxim

al outerplanar graphsw

hich have n vertices and no internal triangles. Then K

i 'i Pii-i is the unique graphin gii with maximal index and pi; is the unique graph in 911 with minimal index.

Here an internal triangle of the m

aximal outerplanar graph G

is a 3-cycle which

has no edges in comm

on with the unique H

amiltonian cycle 2 of G

. If G E

giithen the graph G

- E(2) consists of a tree G

** and two isolated vertices; and jf

G =

Ki 'i Pii-i then G

** = K

i,Ii-3 while if G

= P; then G

** = PIi-2. N

ow in view

öf Theorem

3.1 it is natural to ask whether À

i(Gi) -( À

i(G2) w

henever Gi,G

2 E gii

and Ài(Gi*) -( Ài(Gi*). A pair of 10-vertex graphs exhibited in (80) shows that this

is not always the case; in other w

ords the Ài-orderings of the graphs G

E g10 is in-

consistent with the À

i-ordering of the trees G**. T

he proof of Theorem

3.6 extendsthe techniques used by B

rualdi and Hoffm

an in a paper (12) which provides partial

answers to the questions they posed nearly ten years earlier (7, p. 438). A

s far asgraphs are concerned, the basic problem here is to determine those graphs which

have maxim

al index when just the num

ber of edges in prescribed. (Note that T

he-orem

s 3.1 to 3.6 pertain to graphs for which both the num

ber of vertices and the

number of edges are prescribed.) Let S

ee) denote the class of all graphs havingpreciselye edges, and let fee) denote the m

axmal index of a graph in S

ee). Brualdi

and Hoffm

an showed that w

hen e = (~) and d ? 1, a graph G

in S( e) has indexfee) if and only if K

d is the only non-trivial component of G

. They conjectured

that when e =

(~) + t, 0 -( t -( d, a graph G

in See) has index fee) if and only ifthe only non-trivial com

ponent of G is the graph G

e obtained from K

d by addingone new

vertex of degree t. By applying perturbation-theoretic m

ethods to adja-cency matrices, Friedland (43) proved that the conjecture is true for t = d - 1 and

further that there exists K(t) ? 0 such that the conjecture is true for d ~ K

(t). Sub-sequently, Stanley (92) proved that f (e) :: l( - 1 +

vI + 8e), w

ith equality preciselyw

hen e = (~). Friedland (44) refined Stanley's inequality and thereby proved that the

conjecture holds when t is 1, d - 3 or d - 2. T

he conjecture was finally proved true in

general by Raw

linson (76). Since the components of the principal eigenvector of G

etake only three values, À

i(Ge) is a root of a cubic equation, and one can show

easilythat fee) =

d -1 + € w

here 0 -( € -( 1 and é + (2d -1)('2 +

(d2 - d - t)€ - t2 = O

.T

he starting point for investigations of See) is the observation that the maxm

alindex of a graph in S

( e) is attained by a graph having a stepwise adjacency m

atrix,that is an adjacency m

atrix (aij) which satisfies the condition

(*) if i -( j and aij = 1 then ahk = 1 whenever h -( k :: j and h:: i.

To see this, suppose that A

is the adjacency matrix (aij) of a graph in See) and

Ax =

Ài(A

)x where IlxlI =

1, x = (xi,...,xii) and vertices are ordered so that X

i ~X

2 ~... ~ Xii ~ O

. If for example apq =

0 and ap,q+i =

1 where p -( q then take

A' to be the m

atrix obtained from A

by interchanging the (p,q) and (p,q + 1)

entries and interchanging the (q,p) and (q + 1,p) entries. T

hen Ài(A

') - Ài(A

) ~xT

(A' - A

)x = 2xp(xq - xq+

i) ~ O. Sim

ilar arguments deal w

ith the case in which

apq = 0, ap+

i,q = 1 and p +

1 -( q; and repeating the procedure as necessary we

can obtain a stepwise m

atrix B such that À

i(B) ~ À

i(A). R

efining the arguments w

ecan show

that Ài(B

)? Ài(A

), and it follows that every graph in See) w

ith maxm

alindex has a stepw

ise adjacency matrix. In the special case that e =

(Ü, B

rualdi andH

offman are able to construct from

B in four stages a m

atrix E such that À

i(B) ::

Ài(E):: d - 1. Thus Kd has maxmal index in See); moreover it can be shown that

when À

i(B) =

d - 1, B has the required form

, namely (~-I ~), w

here each entry ofJ is 1. (A short proof of this result is given below.) The general case e = (~) + t

(0 -( t -( d) proved less tractable (see (42)): here a graph with m

aximal index has an

adjacency matrix of the form

/¡J - I C OJ

A' =

cT 0 0

o 0 0

where J - I has size d x d and c T

= (1,..., 1,0, . ..,0) w

ith t non-zero entries. To

prove this, Row

linson (76) first shows that the condition (*) m

ay be strengthened by

d;Mi

,'I¡l~',fi,¡Ii '.-a,~~\;~¡":i\!i!!1

~li

l,ij"i,i

Wi

J¡~:t!:,",,~(

¡~¡

l~~1~"i,,',L

4 :¡

~l .'if:

;\'~iII ~,~

',', ",'1,''",

¡'i"

~i,,!.

~".,';" ",,':. :¡j

"!J1m~, '~.""1r:

~~¡'rm

~,I.:ii ..,.~~ ';

~'~~

i¡ , ':: t~it~,:,J

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OF

A G

RA

PH

adding the requirement

( ** ) if (i) h -( p -( q -( k,

and (ii) ahk = 1, alij =

0 whenever j ? k; aik =

0 whenever i ? h,

and (ii) apq = 0; apj =

i whenever p -( j -( q; aiq =

i whenever i -( p,

r~~'~ p edges

\r~~.then p +

q :: h + k +

L

For a matrix A

other than Æ satisfying (*) and (**) he com

pares indices by consid-ering the relation

PC,., p. ql

()'i(A') - À

i(A))xT

xl = xT

(AI - A

)x' (3.4)w

here A'xl =

Ài(A

')x', x':: 0 and IIx'll = 1. Since xT

x'? 0 the sign of Ài(A

')-À

i(A) is determ

ined by the sign of the biquadratic form on the right hand side of

(3.4). This is expressible as a - ß w

here each of a and ß is a sum of r term

s of theform

xixj + X

iXj, and 4r is the num

ber of non-zero entries in AI - A

. The proof of

the Brualdi-H

offman conjecture requires a delicate analysis of these term

s which

exploits the condition (**).W

e note in passing that equation (3.4) enables us to deal imm

ediately with the

special case in which e =

(~). If A' =

(~-I ~) and A is any other stepw

ise adjacencym

atrix of the same size then a is the sum

of r terms X

i xj + xi X

j for which i -( j :: d

and ß is the sum

of r terms xixj +

XiX

j for which i -( j and j :: d +

1. Since X

l =... = xd, xi = 0 for i? d, and Xi :: X2 :: ... :: Xd :: Xd+i ? 0 we have

Bl,., p. q)

FIGU

RE

5.

(Ài(A') - Ài(A))xTxl = a - ß:: rXi(2Xd - Xd+i)? O.

It follows that, to w

ithin isolated vertices, Kd is the unique graph w

ith (~) edgesand m

axmal index. T

he general result is the following.

TH

EO

RE

M 3.7 (12, 76) Let e =

(~) +

t where d? 1 and 0:: t.. d. F

or t ? 0 let Ge

be the graph obtained from K

d by adding one new vertex of degree t. If G

is a graphw

ith maxim

al index among the graphs w

ith e edges then G has a unique non-trivial

component H

; H =

Kd w

hen t = 0 and H

= G

e when t? O

.

Note that K

d is Ham

iltonian when d :: 3 and G

e is Ham

iltonian when d:: 3 and

1.. t.. d. Accordingly to find the H

amiltonian graphs in S

ee) with m

axmal index

it suffces to consider the case e = (~) +

1. For d? 1 let Kd denote the graph ob-

tained from K

d by deleting an edge; and for d :: 4 let Hd denote the graph obtained

from K

d by adding one new vertex adjacent to precisely tw

o vertices of degree d - 1in K

i. Then H

d has (~) + 1 edges and is H

amiltonian for d? 4. T

he techniques of(76) m

ay be extended to show that (for d:: 4) H

d has the second largest indexof any connected graph w

ith (~) + 1 edges, and that H

d is unique in this respect.Indeed, R

owlinson (77) show

s that for d ? 4 the only graphs with e =

(~) + 1 and

index greater than Ài(H

d) are, to within isolated vertices, the graphs K

d U K

2 andG

e. Sm

all values of e are treated separately and the complete picture is as follow

s.

TH

EO

RE

M 3.8 (77) Let G

be a Ham

iltonian graph with e edges, e :: 3. If the index

of G is m

axmal then one of the follow

ing holds:

(a) e = 4 and G = C4,

(b) e = 7 and G is the unique maxmal outer

planar graph on 5 vertices,

(c) e = (~), d:: 4, and G

= K

d,(d) e =

(~) +

1, d :: 5, and G =

Hd,

(e) e = (~) +

t, 1.. t.. d, d:: 3 and G =

Ge.

We now

return to the situation in which both the num

ber of edges and the num-

ber of vertices are prescribed. Let 'H

( n, e) denote the class of all connected graphsw

ith n vertices and e edges. In seeking the graphs with m

axmal and m

inimal index

in 'H( n, e), w

e may suppose, in view

of Theorem

s 3.1 and 3.2, that e = n +

k where

k:: 1.

In complete generality, this problem

is harder than the corresponding problemfor S

ee). As far as m

inimal index is concerned w

e have just the following result

of Sim

ic, proved by extending to 'H ( n, n +

1) the techniques he used in provingT

heorem 3.4.

TH

EO

RE

M 3.9 (88) A

mong the bicyclic graphs w

ith n vertices, n:: 7, there are pre-cisely tw

o graphs with m

inimal index. In the notation ot,F

igure 5, one is the graphP(k,n + 1 - 2k,k) and the otheris the graph B(k,n + 1 - 2k,k), where k = rnj3l-

If À(m,p,q) denotes the index of P(m,p,q) then (as we noted in the preamble to

Theorem

3.4) for fixed m,À

(m,p,q) is a strictly increasing function of Ip - ql (87).

If p(m,p,q) denotes the index of B

(m,p,q) then for m

.. qand fixed p,p(m,p,q)

is a strictly increasing function of q (88).T

he remaining results in this section are concerned w

ith graphs in 'H(n,n +

k)w

hich have maxm

al index. Brualdi and Solheid (13) show

that again such a graph

'~

,L.

22D

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KO

viC A

ND

P. RO

WL

INSO

NLA

RG

ES

T E

IGE

NV

ALU

E O

F A

GR

AP

H23

l~i~~~

,,,

I.~~..'

ili

m.~~~I

~ ,~

.'Ii~;m~~l.ai!

I~l~:~~

G has a stepw

ise adjacency matrix (aij)' T

hus aiz = an'" =

ain = 1 and G

has aspanning star w

ith central vertex 1. Let G

* denote the graph induced on vertices2, .. ., n by the rem

aining k + 1 edges of G

. In all known cases, a graph G

with

maxm

al index is one of two types. T

he first type, denoted by Hn,k (0:: k :: n - 4),

is the graph G for w

hich G* consists of a star and isolated vertices. T

o describethe second tye, denoted by G

n,ki let k + 1 =

(di-i) + t (0:: t :: d - 2): if t =

0 thenG

* consists of Kd-i and isolated vertices, w

hile if t ? 0 then the only non-trivialcom

ponent of G* is the graph obtained from

Kd-i by adding one new

vertex ofdegree t. The graphs Hn,k, Gn,k may also be described by assigning k + 1 ones to the

triangle of positions (i,j), 1, i , j, in a stepwise adjacency matrix. In the first case,

they are assigned to the first available row; in the second case they are assigned

column by colum

n. Note that H

n,i = G

n,i. We shall see later that for certain other

values of nand k we have ),i(H

Il,k) = ),i(G

n,ÙF

or a specific value of k there are only finitely many possibilities for G

*, and thisfact enabled B

rualdi and Solheid to undertake an exhaustive analysis of the cases inw

hich k :: 5. Then the characteristic polynom

ial of G has the form

xn-r fr(x) where

deg(fr) = r :: 6 and in fr the coefficients are linear functions of n. For sufficiently

large n, the graphs in 'H( n, n +

k) with m

aximal index can then be identified. W

henk :: 2, these graphs are know

n for all n and we record the result as follow

s.

TH

EO

RE

M 3.10 (13) A

mong the bicyclic graphs w

ith n vertices the graph Gn,l alone

has maxm

al index; and among the tricyclic graphs w

ith n vertices, the graph Gn,z

alone has maxm

al index.

Brualdi and Solheid prove also that for k E P,4,5) there exists N(k) such that

for n? N(k), Hn,k is the unique graph in 'H(n,n + k) with maxmal index; and they

conjecture that the same conclusion holds for all k ~ 3. This conjecture was con-

firmed by C

vetkovic and Row

linson (33), and the essential ingredient for their proofis the biquadratic form

given in equation (3.4).

TH

EO

RE

M 3.11 (33) L

et 'H(n,n +

k) denote the class of all connected graphs with n

vertices and n + k edges. For k ? 2 there exists N

(k) such that for n ? N(k), H

n,k isthe unique graph in 'H

(n,n + k) w

ith maxm

al index.

Some isolated results on the ),i -ordering of 'H

( n, n + k) (k :: 5) are given in (13)

and (33) with a view

to estimating N

(k). Bell (2) has pursued this question in the

case that k + 1 = (di-i) (5:: d :: n -1). (This case corresponds to the special case

e = (~) for See) considered by B

rualdi and Hoffm

an (12).) Bell not only gives N

(k)as an explicit function g(d) but also show

s that when n, g(d) the graph G

n,k isthe unique graph w

ith maxim

al index in 'H(n,n +

k). Moreover if g(d) is an integer

then ),i(GIi,k) =

),i(lfii,k) when n =

g(d). In fact,

1 32 16

g(d) = Zd(d + 5) + 7 + d _ 4 + (d _ 4)Z (d ~ 5) (3.5)

and so n = g(d) if and only if (n,n + k) E t(60,69),

(68, 88),

(80, 85)). T

he complete

result is the following.

TH

EO

RE

M 3.12 (2) Let k +

1 = (di-i), w

here 5:: d :: n -1, and let G be a graph

with m

aximal index in 'H

(n,n + k). T

hen

(i) G =

Gii,k if n ,g(d),

(ii) G = Hii,k or Gn,k if n = g(d),

(iii) G =

Hii,k if n? g(d),

where g(d) is defined by equation (3.5).

It follows that for e =

n + k =

n -1 + (di-i), G

ii,k is the unique graph with m

ax-m

al index in 'H(n,e) w

henever n, 60 or whenever e ~ 2n - 47; m

oreover Theorem

3.12 provides an improvem

ent on Yuan's upper bound (T

heorem 1.6) for the index

of a graph in 'H(n,e), e =

n -1 + (di-i). B

ell's methods again m

ake use of equation(3.4) and represent a further refinem

ent of the arguments in (76), but in this special

case an analogue of condition (**) is not needed.

4. GRAPH OPERATIONS AND MODIFICATIONS

We begin by considering the indices of graphs constructed in various w

ays fromtw

o graphs Hand K

. Let IV

(H)I =

s, IV(K

)I = t. First, the disjoint union H

ÜK

clearly has index equal to maxpi(H

),),i(K)). It is also straightforw

ard to dealw

ith the sum H

+ K

, product H x K

and strong product H *K

, each of which has

vertex set V(H

) x V(K

). Vertices (U

b vi),(uz, vz) are adjacent in H +

K if and only

if either Ui =

Uz and V

i rv Vz or U

i rv Uz and V

i = vz; adjacent in H

x K if and only

if Ui '" V

i and Uz rv vz; and adjacent in H

*K if and only if they are adjacent in H

+K or H x K. If H,K have adjacency matrices A,B respectively then H + K,H x

K,H

*K have adjacency m

atrices (A (9li) +

(Is (9 B), A

(9 B, (A

(9 Ii) + (Is (9 B

) +(A

(9B) respectively, and it follow

s that ),i(H +

K) =

),i(H) +

),i(K), ),i(H

x K)

= ),i(H

)),i(K), ),i(H

*K) =

),i(H)),i(K

) + ),i(H

) + ),i(K

). For a general setting forthese results, see (28, section 2.5J.

Now

let u be a vertex of H, va vertex of K

. If HuvK

denotes the graph obtainedfrom

HÜ

K by adding the edge uv then

aiHuvK(X) = ØH(X)(/J(x) - aiH-u(X)aiK-v(X).

If G is obtained from

HÜ

K by am

algamating u and v then

aiG(x) = aiH(X)ØK-v(X) + ØK(X)ØH-u(X) - XaiH-u(X)aiK-v(X), (4.2)

(4.1)

The relations (4.1) and (4.2) m

ay be established by using appropriate determInantal

expansions: see (28, Theorem

2.12) and (75, Rem

ark 1.6). In either case, the spec-trum

(and hence the index) of the graph concerned is determined by the spectra

of H,K

,H - u and K

- v. Accordingly it is of interest.to investigate further the

characteristic polynomial of a graph w

hich is modified by the rem

oval of a vertex.Let G

be a graph with vertices 1,2,..., n and adjacency m

atrix A. Let A

havespectral decom

position ¡.iPi +

... + ¡.niP

ni, and let ei,.. .,en comprise the standard

orthonormal basis of R

" .G

odsil and McK

ay (46) pointed out that an expression for øG-u(x), u E

V(G

),m

ay be obtained by expressing in two w

ays the (u,u)-entry of the matrix generat-

ing function Lr:o x-k A

k. On one hand, L

r:o x-k Ak =

(I - x-i A)-i w

ith (u, u)-entry xaiG

-u(x)/lG(x), because aiG

-u(x) is the (u,u)-cofactor of xI - A. O

n the

24D

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KO

viC A

ND

P. RO

WL

INSO

NLA

RG

ES

T E

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NV

ALU

E O

F A

GR

AP

H25

~~~¡,1i,~F11t

'...1' ~'l" , ,II

1"-'

W('

~iIt:1

~.ßj~~"I;~i.1'.~..'..1,...'.

00 ;,:¡ìl,¥' i'~'i,

I".".

~.,iö':

.1~.,.........I..~,;I

I' , '. ',',i.),

. i~:!

~~II.....'.'~-1:

l...,c.S!.I.. ..-.:.,.,Ii...,.il..,..

'~~;1;: -ii;

m+

.'~;~~~~ 'Ii!

~.~li.it~

,:'.i~~¡F.i'.IlI~.

..- ....i~:

~ :

¡~.11'

WI

.,6.,..;

¡".I...~

n.¡~J~l!!~

other hand, Ak = 'L':i p.f Pi and Pi has (u, u)-entry afu where aiu = IP¡eu I; hence

the (u, u)-entry of 'L~o x-k A

k is 'L':i 'L

~o x-k p.f afu, which is expressible as

'L':i afu/(I- X-iP.i). Therefore,

of P.i then sum(Pi) = nß¡, k(t) = 'L7~l'L'lonßfp.ftk = 'L':i nßfI(l- tP.i) and

m 2

CPG-u(X) = CPG(X) L ~.

¡=i x- Pi

f m nß¡ ì.

CPG( X) = (-It cpo( - x-I) ì. 1 - n f; x + 1 + p¡ f .

( 4.5)

( 4.3)T

he numbers ß

i,...,ßm

are called the main angles of G

. In view of equations (4.4)

and (4.5) we have the follow

ing result.

TH

EO

RE

M 4.2 T

he spectrum of H

'V K

is determined by the spectra and m

ain anglesofH

andK.

Note that cos-i(aiu) is the angle betw

een eu and the i-th eigenspace of A. T

henum

bers aiu,...,amu are com

monly called the angles of G

at u, abusing terminol-

ogy. In view of (4.1), (4.2) and (4.3) w

e conclude the following.

TH

EO

RE

M 4.1 Let u be a vertex of the graph H

, v a vertex of the graph K. If the

graph G is obtained from

H U

K either by am

algamating u and v or by adding the

bridge uv then the spectrum of G

is determined by the spectra of H

and K, the angles

of H at u and the angles of K

at v.

Let us w

rite Hu for the graph obtained from

H by adding a pendant edge at ver-

tex u. We note in passing that Z

hang, Zhang and Z

hang (104) extend Theorem

3.3by show

ing that if CPH

u(X) -( C

PHv(X

) for all x? Ài(H

v) then Ài(H

uwK

)? Ài(H

vwK

)for all vertices w

of K.

We now

turn to another means of com

bining two graphs H

and K: the join (or

complete product) H'V K is obtained from H U K by joining every vertex of H to

every vertex of K. In other w

ords, H'V

K is the com

plement of H

UK

, and thism

akes it possible after a little work (28, T

heorem 2.7) to express the characteristic

polynomial of H

'V K

in the following w

ay:

For the remainder of this section w

e discuss various modifications of a graph G

,using the above notation for angles and orthogonal projections. W

e have alreadyseen from

equation (4.3) that for any vertex u of G, À

i(G - u) is determ

ined asthe largest root of C

PG

(x) 'L7~i afu/(x - P

i). If again we denote by G

u the graphobtained from

G by adding a pendant edge at u then as a special case of (4.1) w

ehave

iPGu(x) = xiPG(x) - CPG-u(x),

(4.6)

a relation which is easy to prove directly. A

ccordingly

f m 2 ì.

CPG,(X) = iPG(X) ì. x - f; X ~upi f

(4.7)

ar(x) = (-I)ncpG

(-x -1) - (-It(x + 1)~i k (~) C

PG( -x -1)

G X

+ 1

and so Ài(G

u) is determined by the spectrum

of G and the angles aiu,...,am

u'More generally we have the following

observation.

TH

EO

RE

M 4.3 F

or any vertex u of G, the spectra of both G

- u and Gu are deter-

mined by the spectrum

of G and the angles of G

at u.

If, in the construction of Gu from

G, the new

pendant vertex is labelled 0 thenCPG, (x) = det(xl - Ao - B) where Ao is the (n + 1) x (n + 1) matrix (0 A) and the

only non-zero entries of B are ones in positions (0, u) and (u, 0). In the case that G

is connected and not a complete m

ultipartite graph, Bell and R

owlinson (4) sought

an explicit expression for Ài(G

u) in terms of the spectrum

and appropriate anglesof G by expressing the largest root of det(xl - Ao - (B) ((? 0) as a power series

in (. For this to be of any use we require convergence at ( =

1, and it turns out thatthe radius of convergence of the series does exceed 1 for large enough Pi - P2. Iffor example Pi - P2 ? 4 then Ài( Gu) = pi + 'L~i Ck where the Ck are recursively

defined functions of the 2m invariants Pi, a¡u of G

(l;= 1,...,m

). In fact the Ck

are just the so-called perturbation coefficients which arise in the analytical theory

of matrix perturbations (57, §§11.5, 11.6) applied to the linear perturbation A

o +(B(( E C). Here, if xo denotes an eigenvector of Ao corresponding to pi, one finds

sufficient conditions for the existence of analytic functions pi(() = pi +

'L~i C

k(k,x(() = Xo + 'L~i Xf(k such that (Ao + (B)x(() = pi(()x((). Rowlinson (78) applied

this theory in the case of a graph G +

uv obtained from a connected graph G

byjoining tw

o non-adjacent vertices u and v. Not surprisingly, m

ore invariants of G are

required to determine À

i(G +

uv) in this general case: in addition to the spectrum

CPW\lK(X) = (-iyCPH(X)~( -x -1) + (-iyCPK(X)CPii -x -1)

+ (-ly+t+icpii(-x -1)cpï((-x -1). (4.4)

If H is regular of degree d and K

is regular of degree e then it follows that

Ài(H

'VK

) = itd +

e + vI(d - e)2 +

4st1 because then the spectrum of H

consistsof s - 1- d, -À2(H) - 1,..., -Às(H) - 1 and the spectrum of K consists of t -1- e,

-À2(K) - 1,..., -Àt(K) - 1. In general however we need to investigate further the

characteristic polynomial of the com

plement of a graph (the result of a unary graph

operation).If as before G has adjacency matrix A then G has adjacency matrix J - I - A

and characteristic polynomial det((x + 1)1 - J + A), where each entry of J is 1.

Thus if(x) = det((x + 1)1 + A) - sumadj((x + 1)/ + A), where sumadj( ) denotes

the sum of all entries of the adjoint m

atrix. It follows (cf (34, T

heorem 5)) that

where fG(t) = sumadj(1 - tA)/det(1 - tA) = sum

(I - tA)-i =

sum'L

~oAktk.N

owsum

(Ak) =

'L7~ipfsum

(Pi) and sum(Pi) =

'LuvPieu' Piev =

IP¡jI2 where j denotes

the all-l vector. Thus if cos-1(ß¡) is the angle between j and the eigenspace

26D

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AN

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OW

LIN

SON

LAR

GE

ST

EIG

EN

VA

LUE

OF

A G

RA

PH

27

Analogous results for the deletion and the relocation of an edge are obtained

by Row

linson in (79). Further, Maas (64) deals w

ith HuvK

as a perturbation ofH

U K

and Row

linson (79) deals with K

i \7 G as a perturbation of G

(an example

of a global rather than a local modification of G

). The results in these cases are

somew

hat technical and we om

it the details. A special case of the construction

H uv K

is the addition of a pendant edge. Here w

e can simply use equation (4.7)

to show that if pi(Pi - P2):/ 1 then Ài(Gu).( pi + tu where tu = aiu/(Pi - (pi-

P2)-i) (79, Rem

ark 5.3): since the eigenvalues of G interlace those of G

u, it sufficesto check that rpG

u (pi + tu) :/ O

.In our consideration of estim

ates for the index of a modified graph w

e have sofar discussed only upper bounds. L

ower bounds are readily obtained from

Rayleigh

quotients since for any real symmetric matrix A' we have Ài(A') = SUPtZT A'z/zTz :

z f OJ' In particular if x is the principal eigenvector of A

then Ài(A

+ B

) :: Ài(A

) +xTBx; for example, Ài(G + uv):: Ài(G) + 2aiuaiv' Accordingly the index of a mod-

ified graph can be restricted to a certain interval, and the effects on the index of two

different modifications can be com

pared if the lower lim

it of one interval exceedsthe upper lim

it of the other.W

e can also find upper bounds for the index of a modified graph G

', with adja-

cency matrix A

' as follows: if y is a positive vector and p a scalar such that A

'y :: pyand A

'y f py then Ài(A

').( P (67, Theorem

1.3.1). This is often useful w

hen G' is

obtained from G

by the introduction in some w

ay of an additional vertex, becausethe perturbation theories described above apply directly only w

hen G and G

' have

the same set of vertices. S

imic (87, T

heorem 2.4) deals in this w

ay with a graph

G' obtained from

G by splitting a vertex of G

: if the edges containing v are vw(w

E W

) then G' is obtained from

G - v by adding tw

o new vertices V

i, V2 and

edges ViW

i (wi E

Wi) V

2W2 (W

2 E W

2) where W

iUW

2 is a non-trivial bipartitionofW

.

TH

EO

RE

M 4.5 (87) If G

is a connected graph and G' is obtained from

G by splitting

a vertex then Ài(G

') ~ Ài(G

).

Finally we consider the case in w

hich G' is the graph G

u,v obtained from G

bysubdividing the edge uv: thus G

' is obtained from G

- uv by adding a new vertex

wand edges w

u, wv. N

ote first that the subdivision of an edge does not necessarilyresult in a change of index: if G is an n-cycle Cn then always Ài (Guv) = Ài (G) =

2, and if G is the (n + 5)-vertex graph Wn depicted in Figure 1 then Ài(Gu,v) =

Ài(G) = 2 for any non-pendant edge uv.

Hoffman and Smith (53) define an internal

path of G as a w

alk vovi,...,vk(k:: 1)such that the vertices vi,..., V

k are distinct, deg(vo) :/ 2, deg(vk) :/ 2 and deg(vi) = 2

whenever 0 .( i .( k. T

hus an internal path gives rise to either a subgraph Wk or a

k-cycle with one pendant vertex attached. B

y constructing a suitable positive vectory in the case that the edge uv lies on an internal path of G

, Hoffm

an and Sm

ithprove the follow

ing.

TH

EO

RE

M 4.6 (53) L

et uv be an edge of the connected graph G.

(i) If uv does not belong to an internal path of G and if G

f- Cn then À

i (Gu,v) :;

Ài(G

).(ii) If uv belongs to an internal path of G

and G f- W

n then Ài (G

u,v) ~ Ài (G

).

d h I (. - 1 ) . iii (ml h -i( IiI). h

an t e ang es aiu,aiv i - ,...,m w

e require IUV

"",luV w

ere cos IUV

is t eangle betw

een Pieu and Piey (defined when these vectors are non-zero). If pi - P2

is large enough (say, greater than 4) then Ài(G

+ uv) is the sum

of a convergentseries pi + L~i Ck where the Ck are recursively defined functions of the invariants

Pi,aiU,aiv,lt¿ (i = l,...,m). A convergent series may of course be used to compute

the index to any degree of accuracy, but if we require m

erely an estimate for the

index then for an upper bound we m

ay turn to an algebraic theory of perturbations(99, C

h. 6). Maas drew

attention to this theory in a paper (64) which treated both

G +

uv (where G

is connected) and the addition of a bridge between tw

o disjointconnected graphs. T

he idea is to consider à +

ÉP w

here P is an appropriate projec-tion, Ã = -A - (Ài(B) + b)I and É = (Ài(B) + b)I - B. Here b is chosen positive

to ensure that É is a positive m

atrix and hence that ÀIi(Ã

+ É

P) :: Àn(Ã

+ É

). Sinceà + É = -A - B, we have Ài(A + B):: -ÀIi(à + ÉP). The parameter b is chosen

to optimise this upper bound for the index of the perturbed graph. If A

and A +

Bare the adjacency m

atrices of G and G

+ uv (w

here G is connected), and if P is

an appropriately chosen projection onto the eigenspace of pi then we obtain the

following.

THEOREM 4.4 (64) Let u, v be non-adjacent vertices of

the connected graph G. T

hen

Ài(G + uv):: Ài(G) + 1 + b -, where b:/ 0 and

= b(l +

b)(2 + b) =

G _ G

I (aiu+aiv)2+ó(2+Ó+2aiUaiy) pi() P2( ).

5. TH

E LA

RG

ES

T E

IGE

NV

ALU

E O

F R

AN

DO

M G

RA

PH

S

'êl

This short section has been included for the sake of completeness. There has

been a remarkable developm

ent of the theory of random graphs in recent years:

several topics from the theory of (usual) graphs have prompted the study of

anal-

ogous questions in random graphs, and a review

of eigenvalues of random graphs

may be found in S

ection 3.7 of the monograph (27). W

e mention here just tw

o re-sults concerning the largest eigenvalue of undirected random

graphs.Let G

n,p denote a random graph on n vertices, each pair of vertices being con-

nected by an edge with probability p (0 ~ p .( I). Juhász proved the follow

ing result.

THEOREM 5.1 (55) Lim,l-oo(l/n)Ài(Gn,P) = p with

probability 1./

The second result is an application of the m

ethod of bounded differences de-scribed by M

cDiarm

id in (66). In order to apply Lem

ma 3.3 of that paper w

e need toobserve that if the graphs G

and G' differ in only one edge then IÀ

i(G) - À

i(G')1 ::

1. We m

ay assume that G

is connected and G' is obtained from

G by deleting the

edge ij. If G has adjacency matrix A and principal eigenvector x = (xi,X2,...,xn)T

then Ài(G'):: xT Ax - 2XiXj = Ài - 2XiXj :: Ài - xr - xJ:: Ài -1. The hypotheses

of (66, Lemm

a 3.3) are therefore satisfied and we deduce the follow

ing.

.:1

28D

. CV

ET

KO

viC A

ND

P. RO

WL

INSO

NLA

RG

ES

T E

IGE

NV

ALU

E O

F A

GR

AP

H29

6. APPLICATIONS

may be used to estim

ate the rate of growth of som

e combinatorial sequences. T

hisapplies in particular to the transfer m

atrix method used in several enum

erations inphysics and chem

istry (cf. for example, (28), pp. 245-251).

We m

ention in passing an application to tournaments (98) (see also (6) or (28, p.

226)): when ranking the participants of a tournam

ent (a complete directed graph)

one can use coordinates of the principal eigenvector.For any graph G, let E(G) = Ài(G) - d(G), where d denotes the mean degree.

Collatz and S

inogowitz (18) proposed E

as a measure of irregularity: note that

by Theorem

1.1, E;: 0 w

ith equality if and only if G is regular. B

ell (5) shows

that the largest possible value of E for an n-vertex connected graph lies betw

eenln - î +

21n and ln - 1 + Iln. A

natural measure of irregularity is the variance of

the vertex degrees, that is v(G) =

(lln)'£?=i(di - d)2. R

owlinson (80) gave exam

-ples of m

aximal outerplanar graphs G

i, G2 such that v( G

i) = v( G

2), E( G

i) :; E( G

2);and of associated trees Gi*,Gi* such that v(Gi*) = v(Gi*), E(Gi*) -( E(Gi*). Sub-

sequently, Bell (5) established that E

and v are actually inconsistent as measures of

irregularity in respect of the graphs Gll,k and H

Il,k of Theorem

3.12. (Note that for

prescribed numbers of vertices and edges, E

-ordering coincides with À

i-ordering.)Finally we note that Pötschke (73, 74) discusses the rôle of Ài in the graph iso-

morphism

problem.

TH

EO

RE

M 5.2 If E

(Ài) denotes the expected value of À

i = À

i(GIl,P) then for t :; 0,

Pr(IÀi - E(Ài)1 ;: t) :S 2exp t -2t2 / (~) ) .

In this section we give a brief com

mentary on som

e applications of the results de-scribed in previous sections. W

e divide the applications into two groups: applications

to other mathem

atical problems (m

ainly again in graph theory) and applications toother disciplines (physics, chem

istry, computer science, geography).

6.1. Applications within mathematics

Theorem

s about graph spectra can sometim

es be used to prove results in graphtheory and com

binatorics which them

selves make no m

ention of eigenvalues. Suchm

eans of proof are often referred to as 'spectral techniques'. Well-know

n instancesare structure theorem

s for strongly regular graphs and existence theorems for M

ooregraphs. Indeed spectral techniques often prove their w

orth in extremal graph theory:

see Chapter 7 of the m

onograph (28). Som

e further examples are given below

.U

pper and lower bounds on the index of a graph, as described in Section 1, m

aybe com

bined to provide inequalities relating non-spectral invariants. For example

Wilf (100) has com

bined Theorem

s 1.5 and 1.7 to obtain an upper bound for thechrom

atic number of a graph G

with n vertices and e edges:

,( G) :S 1 + V 2e (1- ~).

If G is connected and w

e use Theorem

1.6 instead of Theorem

1.5 then we obtain

il

Nk = ¿ciÀr

i=i

6.2. Applications in other disciplines

The theory of graph spectra has several applications in physics and chem

istry: see(28, C

hapter 8) and (27, Chapter 5). F

or example, a m

embrane (w

ith fixed bound-ary) m

ay be represented by a lattice graph whose eigenvalues determ

ine the har-m

onic oscilations of the mem

brane: the largest eigenvalue corresponds to the os-cillation w

ith least energy. In HückeI's theory of m

olecular orbitals, the eigenvaluesof the graph G

representing the carbon skeleton of a hydrocarbon molecule deter-

mine the quantum

energy levels for the molecule: again the largest eigenvalue of

G corresponds to the low

est energy state. Several physical and chemical properties

of saturated hydrocarbons (e.g. viscosity, surface tension, boilng point, density etc.)depend on the "extent of branching" of G

. A num

ber of topological indices (i.e.graph-invariants) have been proposed and studied as a m

eans of treating branchingin a quantitative m

anner. Am

ong them w

e find the largest eigenvalue of G, sug-

gested by Cvetkovic and G

utman in (30): there the asym

ptotic formula (6.1) is used

to justify empirical findings that À

i(G) is a suitable m

easure of branching. Calcu-

lation of the largest eigenvalue in a chemical context features in (72), and further

methods for calculating graph indices are described in (58).The idea of the index of a graph as a measure of btánching has also been ap-

plied in a quite different area, namely the theory of algorithm

s-in particular algo-rithm

s for so-called N P-problem

s. An N

P-problem is intractable in the sense that

all known algorithm

s for its solution have non-polynomial com

plexity. But the com

-plexity of an algorithm

is determined by the num

ber of elementary steps required

to deal with a worst case, and in practice exponential algorithms often behave as

polynomial algorithm

s in many cases. In (24) the authors suggest the use of an in-

dex, computable in polynom

ial time, w

hich estimates in advance the com

plexity of

,( G) :S 1 +

v2e - n + 1.

Theorem

1.10 has been used in (70) to prove and extend Turan's theorem

(cf.(28), pp. 221-222). S

ee also (40), where T

heorem 1.2 has been used to obtain a

lower bound on the num

ber of triangles in a graph.A

s a third application we m

ention the problem of determ

ining the number of

walks in a graph. L

et Nk be the num

ber of walks of length k in a graph G

with

adjacency matrix A

. It follows from

the spectral decomposition of A

k that

where the C

i are quantities which depend on the eigenvectors of A

but not on k (cf.(28), p. 44). W

e imm

ediately derive the following asym

ptotic formula

Nk rv DÀ1 (k -+ +00) (6.1)

where D

is a constant. Several combinatorial enum

eration problems can be reduced

to enumeration of w

alks in a graph (see (28), section 7.5). Thus the graph index

30D

. CV

ET

KO

viC A

ND

P. RaW

LIN

SON

'i~Ii

a particular case of an N P

-problem. (T

hen only an approximate solution w

ould besought for a case of high com

plexity.) They ilustrate their ideas w

ith a branch-and-bound algorithm

for the travelling salesman problem

, that is, the problem of finding

a Ham

iltonian path of minim

al weight in a w

eighted graph. As one index of com

-plexity they use the index of a spanning tree of m

inimal w

eight: the underlying ideais that this index is a m

easure of branching, the least value being attained when the

tree is a path. Experim

ental results show a m

oderate to good correlation between

index and running time of the algorithm

.Finally we note that the index À1 of a graph has featured in the geographical

liter-

ature (15, 16, 48, 94, 95) in the context of traffc networks. H

owever, claim

s that theentries of the principal eigenvector of a connected graph G

serve as a measure of

'connectedness of a vertex', with ordering of these entries consistent w

ith orderingby vertex degree, w

ere refuted by Maas (63). P

erhaps the only comm

ent relevanthere is that if j denotes the all- 1 vector and if G

has adjacency matrix A

, thenlimk__"x) Àik Akj is a non-zero scalar multiple of the principal eigenvector, while Aj

itself has entries the vertex degrees. (The result concerning À

ik Akj m

ay be provedby first expressing j as a linear com

bination of orthonormal eigenvectors.)

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mith, S

ome properties of the spectrum

of a graph, Com

binatorial Structures and T

heirA

pplications, R. G

uy, et a!. (Ed.), G

ordon and Breach, N

ew Y

ork-London-Paris, 1970,403-.

(92) R. P

. Stanley, A

bound on the specral radius of graphs with e edges, Linear A

lgebra and Appl. 87

(1987), 267-269.(93) M

. M. Syslo, A

djacency matrix equations and related problem

s: research notes, Com

ment. M

ath.U

niv. Caro/in. 24 (1983), N

o.2, 211-222.

(94) K. J. T

inkler, The physical interpretation of eigenvalues of dichotom

ous matrice, Inst. B

rit, Geog

Pub/. 55 (1972), 17-4.(95) K. J. Tinkler, On the choice of methods in the factor analysis of connecivity matrice, a reply,

Inst.

Brit. G

eogr Pub/. 66 (1975), 168171.

(96) A. T

hrgašev, On infinite graphs w

hose spectrum is uniform

ly bounded by ""2 + .¡, G

raph The-

ory, Proc. 4th Y

ugoslav Sem

inar on Graph T

heory, Novi S

ad, April 15-16, 1983, D

. Cvetkovic, I.

Gutm

an, T. Pisanski, and R

. Tošic (eds.) U

niv, Novi Sad Inst. M

ath" Novi Sad, 1984,239-30.

(97) N. S

. Wang, O

n the specific properties of the characteristic polynomial of a tree, J. LanzJou R

ail-w

ay Coli. 5 (1986), 89-94.

(98) T. H

. Wei, T

he algebraic foundations of ranking theory, Thesis, C

ambridge, 1952.

(99) A. W

einstein and W. Stenger, M

ethods of intermediate problem

s for eigenvalues, Academ

ic Press,N

ew Y

ork, 1972.

(100) H. S. W

ilf, The eigenvalues of a graph and its chrom

atic number, J. L

ondon Math. Soc. 42 (1967),

330-332.

(101) H. S. W

ilf, Graphs and their spectra-uld and new

results, Congressus N

umerantium

50 (1985),37-42.

(102) H. S. W

ilf, Specral bounds for the clique and independence nu;nbers of graphs, J. Com

binatoriiT

heory B 40 (1986), 113-117.

(103) Hong Y

uan, A bound on the spectral radius of graphs, Linear A

lgebra and Appl. 108 (1988), 135-

139.

(104) Fu Ji ZIJdng, Z

hi Nan, and Y

un Hu Z

hang, Some theorem

s about the largest eigenvalue of graphs,(C

hinese, English sum

mary) J. X

injiang Univ. N

at. Sci. (198), no. 3, 8490. (M

R88:05128)

(105) V. V

etchý, Estim

ation of the index of G2, A

rh. Math. (B

rno) 24 (1988), 123-136.

10 Contents

4.3. A generalization of the divisor concept . . . . , , . , , . , .

4.4. Symmetry properties and divisors óf graphs. , , , , , , , . .

4.5. The fundamental

lemm

a connecting the divisor and the spectrum4.6. The divisor - an effective tool for factoring the characteristic polynomial.

4.7. The divisor - a mediator between structure and spectrum

4.8. . Miscellaneous results and problems. . . . . . ,

5. The Spectrum and the Group of Automorphisms.

5.1. Symmetry and simple eigenvalues . . . . . . .

5.2. The spectrum and representations of the automorphism group

5.3. The front divisor induced by a subgroup of the automorphism group

5.4. Cospectral graphs with prescribed (distinct) automorphism groups

5.5. Miscellaneous results and problems. . . . . . .

6. Characterization of Graphs by Means of Spectra .

6.1. Some families of non-isomorphic cospectral graphs

6.2. The characterization of a graph by its spectrum .

6.3. The characterization and other spectral properties of line graphs

6.4. MetricaUy regular graphs . , , . , . . . . ' . , .

6.5. The (-1, 1, OJ-adjacency matrix and Seidel switching

6.6. :ßiiscellaneous results and problems. . , , , . , . .

7. Spectral Techniques in Graph Theory and Combinatorics

7.1. The existence and the non-existence of certain combinatorial objects

7.2. Strongly regular graphs and distance-transitive graphs . .

7.3, Equiangular lines and two-graphs . . . . . . . . , . .

7.4. Connectedness and bipartiteness of certain graph products.

7.5. Determination of the number of walks . . , .

7.6. Determination of the number of spanning trees

7.7. Extremal problems. . , . . . . .

7.8. Miscellaneous results and problems. ,

8. Applications in Chemistry and Physics

8.1. Hückels theory , . . , . . . . . .

8.2. Graphs related to benzenoid hydrocarbons,

8.3. The dimeI' problem ".....

8.4. Vibration of a membrane . . . , ,

8,5, Miscellaneous results" and problems.

9. Some Additional Results . .

9.1. Eigenvalues and imbeddings.

9,2. The distance polynomial . ,

9.3. The algebraic connectivity of a gmph ,

9.4. Integral graphs . . . . , , . . ,

9.5. Some problems . , . . , . . ' .

Appendix. T

ables of Graph Spectra.

Bibliograiihy. . .

Index of Symbols.

Index of Nam

es

Subject Index .

118118121125128131

134

134141149153153

156

156161168178183185

189

189193199203209217221223

228

228239245252258

260

261263265266266

268

324

360

361

364

o.Introduction

This introductory chapter is devoted m

ainly to the reader who is not fam

iliar with

graph theory to help him to enter the topic of the book. T

he basic definitions andfacts about the spectra of graphs are given together with a description of some

general graph theoretic notions and necessary facts from m

atrix theory. For a generalintroduction to graph theory the reader is referred to the books (B

eCh2), (B

el' 1),(Ber2), (Ber3), (BoMu), (Deo), (Har4), (Maye), (Nolt), (Sac9), (Wi,RJ2), (YHJIC),

(Xapa), a chem

ist may be especially interested in (B

ala)t, and for a survey of matrix

theory we recom

mend the books (G

ant), (MaM

i).

0.1. What the spectrum

of a graph is and how it is presented in this book

By a graph G

= (qc, 'W

) we m

ean a finite set qc (whose elem

ents are called vertices)together with a set 'W of two-element subsets of qc (the elements of 'Ware called

edges). Similarly, a digraph (diTected graph) (qc, 'W) is defined to be a finite set qc

and a set ql of ordered pairs of elements of qc (these pairs are called directed edges

or arcs). The sets of vertices and edges are som

etimes denoted by "f(G

) and &'(G

),respectively.

If multiple undirected or directed edges are allow

ed, we shall speak of m

ultigraphsor m

ulti-digraphs, respectively. These tw

o cases include the possible existence ofloops (a loop is an edge or arc w

ith both of its vertices identical). The term

inology isthat of F

. HA

AR

Y (H

ar4) except for the fact that in this book multi-(di-)graphs

are allowed to have loops. A

lthough the term graph denotes w

hat in many graph

theoretical papers is called "a finite, undirected graph without loops or m

ultipleedges" (or, briefly, a "schlicht graph"), for the sake of readability w

e shall sometim

es(when there is no danger

of confusion) use the term graph in the m

ost general meaning,

i.e., we shall m

ean undirected graphs, digraphs and even multigraphs and m

ulti-digraphs.

Tw

o vertices are called adjacent if they are connected by an edge (arc). The ad-

jacency matrix A

of a multi-(di-)graph G

whose vertE

jx set is jXi, X

2, ..., xnJ is a squarem

atrix of order n, whose entry aj.j at the place (i, j) is equal to the num

ber of edges

t Recently tw

o books on Hückel theory have appeared (see the end of p, 359).

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0.2. Some m

ore graph theoretic notions and conventions15

14O

. Introduction

is of interest, especially in some cases w

hen the spectra of compound graphs can be

expressed in terms of the spectra of sim

pler graphs.

The plan of the book is as follow

s:In Section 0.2 som

e general graph theoretic notions are given together with som

econventions used throughout this book. Section 0.3 contains the necessary theorem

sfrom

matrix theory and describes som

e basic facts about graph spectra.In C

hapters 1, 3, 4, 5, 6 relations between spectral and structural properties of

graphs are described.C

hapter 2 describes the relations between the spectrum

of a graph constructed byoperations on som

e given graphs and the spectra of these graphs themselves.

Chapters 7 and 8 describe the applications of the theory developed in C

hapters 1to 6. C

hapter 7 is related to the applications in graph theory and combinatorics and

Chapter 8 gives the applications beyond m

athematics, i.e. in chem

istry and physics.C

hapter 9 contains some additional m

aterial which did not fit into the clàssification

of the other chapters. The A

ppendix contains numerical data on graph spectra and

the corresponding characteristic polynomials.

The last section in each of Chapters 1 - 8 has the title Miscellaneous results and

problems. A

t these places some additional m

aterial is reviewed, partly in form

ofexercises and problem

s. Section 9.5 gives a list of unsolved problems.

The B

ibliography contains more than 650 references from

both the mathem

aticaland the chemical

literature. Although the authors believe that all im

portant papersfrom

the viewpoint of this book are included, they are aw

are that a complete biblio-

graphy on graph spectra is almost im

possible to compile because of the thousands of

chemical papers w

here graph spectra are only mentioned in passing and the hundreds

of papers on association schemes, block designs, and related combinatorial objects

where eigenvalues are also involved, although not alw

ays in an important w

ay.

If there is an arc from vertex x to vertex y, w

e shall sometim

es indicate this bywriting y' x; x and yare neighbours of each other, x is a rem' neighbour of y and y

is a front neighbour of x.A

cycle of length n, denoted by Õn, is a digraph w

ith the vertex set ¡xi, ..., :rnlhaving arcs (Xi, x¡+1), i = 1, ..., n - 1, and (xm Xl)' A linear directed graph is a di-

graph in which each indegree and each outdegree is equal to 1 , i.e., it consists of cycles.

A spanning linear subgraph of a m

ulti-(di-)graph G, i.e., a linear subgraph of G

which contains all vertices of G

is sometim

es called a linear factor of G. A

linearfactor of a multi

graph consists of disjoint copies of K2.

A regular spanning sub

graph of degree s of a multi

graph G is called a (regular)

factor of degree s or, briefly, an s-factm' of G.

I,n a multi-(di-)graph any sequence of consecutive edges (arcs) (having in m

indthe orientation in directed case) is called a w

(ilk. The length of the w

a.lk is the number

of edges (arcs) in it. A w

alk can pass through the same edge (arc) -m

ore than once.A path of length n - 1 (n ~ 2), denoted by Pn, a is graph with n vertices, say

Xl' ..., X

n, and with n - 1 edges in w

hich Xi and X

i+1 are connected by an edge for

i = 1, ..., n - i.

A multi-(di-)graph is (strongly) connected if any two of its vertices are joined by a

path (walk). A

multigraph is disconnected if it is not connected, and it then consists

of two or more parts called components,

two vertices being in different com

ponentsif they cannot be joined by a path. A

vertex ;¡ is called a mdpoint and an edge u is

called a bridge if the deletion of x or u, respectively, ca,uses an increase of the num-

ber of components.

The length of a shortest path betw

een two vertices is called the distance betw

eenthe vertices. T

he diameter of a connected m

ultigraph is the largest distance between

the vertices in it.A

circuit Cn of length n is a regular connected graph of degree 2 on n vertices. C

on-sidered as a subgraph, C

l is a loop, C2 is a pair of parallel edges, C

3 is atriangle, C4

is a quadrangle. The girth of a m

ulti-(di-)graph is the length of a shortest circuit(cycle) contained in it.

A m

ultigraph G is said to be properly coloured if each vertex is coloured so that

adjacent vertices have different colours. G is k-colourable if it can be properly

coloured by k colours. The chrom

atic number X

(G) is k if G

is k-colourable and not(k _ l)-colourable. G

is. called bipartite if its chromatic num

ber is 1 or 2. The vertex

set of a bipartite multigraph G

can be partitioned into two parts, say 2l and i!, in

such a way that every edge of G connects a vertex from 2l with a vertex from i!.

If 0l denotes the edge set of G, we have also the following notation: G = (2l, i!; qt).

If G is connected and has an edge, then 2l and i! are non-void and (up to an intei:-

change) uniquely determined. If the vertices are labelled so that

0.2. Some more graph theoretic notions and conventions

. We shall now

give some m

ore defintions of the graph theoretic notions frequentlyused throughout the book. W

e shall also point out some standard notations and

explain some conventions used in the subsequent text.

A graph H

= (i!, 1/) is said to be a subgraph of the graph G

= (2l, 0l) if i! c 2l

and 1/ cOl. T

he graph H is called a spanning subgraph or a partial graph of G

ifi! = 2l. If 1/ consists of all the edges from 0l which connect the vertices from i!,

then H is called an induced subgraph. A

n induced subgraph is said to be spanned byits vertices, and a partial graph is som

etimes said to be spanned by its edges.

The num

ber of edges incident with a vertex in an undirected graph is called the

degree or the valency of the vertex. Note that an undirected loop is counted tw

ice,thus its contribution to the valency of the vertex to w

hich it is attached is equalto 2. If all the vertices have the sam

e valency r, the graph. is called 'regular of degree r.In digraphs w

e shall distinguish between the indegree or rear valency and the

outdegree or front valency (of a vertex) by indicating how m

any arcs go into and goout from

the vertex, respectively:

2l = ¡xu X2, ..., xm), i! = ¡xm+1' Xm+2, ..., xin+n)'

then the adjacency matrix of G

takes the form

A = (0 BT)

B 0 '

where.B

is an n X m

matrix and B

T is the transpose of B

.

16O

. Introduction0.3. Som

e theorems from

matrix theory

17

A m

ultigraph is called semiregitlar of degrees ri, 1'2 (possibly ri =

1'2) if it is bi-partite, each vertex has valency ri or 1'2' and each edge connects a vertex ofvalency ri w

ith a vertex of valency 1'2'K

" denotes the complete graph on n vertices (any tw

o distinct vertices of K" are

connected by an edge). Km

,,, is a complete bipartite graph on n +

m vertices; K

i." iscalled a star. The complete k-partite graph on ni + n2 + ... + nk vertices is denoted

by K"lO

"...... "k'

A forest is a graph w

ithout circuits, a tree is a connected forest.T

he complem

ent G of a graph G

is the graph with the sam

e vertex set as G, w

hereany tw

o distinct vertices are adjacent if and only if they are non-adjacent in G.

Obviously, G

= G

. A graph w

ithout any edges is called totally disconnected, its com-

plement is a com

plete graph.T

he subdivi'ion graph S(G) of a graph G

is obtained from G

by replacing each ofits edges by a path of length 2, or, equivalently, by inserting an additional vertexinto each edge of G

. Clearly, S(G

) is a bipartite graph (~, Il; Ol) w

here ~ and Ilare the sets of the original and of the additional vertices, respectively.

The line graph L

(G) of a graph G

is the graph whose vertices correspond to the

edges of G with two vertices being adjacent if and only if the corresponding edges

in G have a vertex in com

mon.

The vertex-edge incidence m

atrix R of a loopless m

ultigraph G =

(~, 0//) is definedas follow

s: Let

matrix where

Vii =

1 if ui issues from X

i'

Vii = -1 if Uj terminates in Xi'

Vii = 0 otherwise.

In the majority of cases we shall use the following standard notation.

The num

ber of vertices of a graph is denoted by n, the number of edges or arcs

by m. T

he degree of a regular graph is denoted by l' as is the index of a graph (seethe next section). T

he symbol I m

eans a unit matrix in general and 111 is a unit m

atrixof order n. T

he symbol J denotes a square m

atrix all of whose entries are equal to i.

The transpose of a m

atrix X is denoted by X

T, and rkX

is the rank of X.

The K

ronecker symbol O

ii is defined by Oii =

1 and Oii =

0 if i =+

i.a I b m

eans a divides b.In the case of undirected m

ultigraphs the spectrum consists of real num

bers. Inthat case, the eigenvalues )'i, À

2, ..., À" are ordered so that alw

ays Ài =

l' ~ )'2

~ ... ~ À".

Other notations and graph theoretic concepts w

il be given at the place of theiruse.

~ = rxi, X2, ..., x,,l, Ol = rui, 1L2, ..., uml.

0.3. Som

e theorems from

matrix theory and their application to the spectrum

of a graphR

= (bij) is an n X

m m

atrix where bij =

1 if Xi is incident w

ith (i.e., is an end vertexof) U

j, and bii = 0 otherw

ise. The edge-vertex incidence m

atrix is the transpose RT

ofR.

The adjacency m

atrix of a multi-(di-)graph G

is denoted by A =

A(G

). The

v(tlency or degree matrix D

of a multigraph is a diagonal m

atrix ~ith the valency Vi

of vertex Xi in the position (i, i).

It is not difficult to see that, for a graph G, the vertex-edge incidence m

atrix R,

the degree matrix D, and the adjacency matrices of G, L(G), and S(G) are connected

by the following form

ulas:

Some fundam

ental properties of spectra of graphs (or, more generally, m

ulti-digraphs)can be established im

mediately by using several theorem

s of matrix theory. W

e shallform

ulate in this section only the most im

portant matrix theorem

s. Others, w

hichare also useful, w

il be given in the subsequent chapters as lemm

as at the placesw

here they are needed.T

he set of eigenvectors belonging to an eigenvalue À along w

ith the zero vectorform

s the eigenspace belonging to À. T

he geometric m

ultiplicity of (tn eigenvalue À is the

dimension of its eigenspace. T

he algebraic multiplicity of À

is the multiplicity of À

considered as a zero of the corresponding characteristic polynomiaL. T

he geometric

multiplicity is never greater than the algebraic m

ultiplicity.A

matrix X

is called symm

etric if XT

= X

.A

(G)

= RRT - D,

A(L(G

)) = R

TR

- 21,

A(S(G)) = (~ ~T).

Theorem

0.1 (see, for example, (M

aMi), p. 64): T

he geometric and algebraic m

ulti-plicities of an eigenvalue of a sym

metric m

atrix are equal.

In the subsequent text the multiplicity of an eigenvalue w

il always m

ean the alge-braic multiplicity.

A m

atrix is called non-negative if all its elements are non-negative num

bers.Since the adjacency m

atrix of a multi-(di-)graph G

is non-negative, the spectrumof G

has the properties of the spectrum of non-negative m

atrices. For non-negative

matrices the following theorem holds.

The above definitions and form

ulas can easily be generalized for arbitrary multi-

graphs.The (0,1, -1)-incidence matrix V of a loopless multi-digraph G with vertices

Xl' X

i, ..., Xli and arcs ui, ui, ..., U

m is defined as follow

s: V =

(Vii) is an n X

m

2 CvetkoviclDoob/Sachs

180.3. Som

e theorems from

matrix theory

19O

. Introduction

Theorem

0.2 (see, for example, (G

ant), voL. II, p. 66): A

non-negative matrix

always has a non-negative eigenvalue l' such that the m

oduli of all its eigenvalues do notexceed r. T

o this "maxim

àl" eigenvalue there corresponds an eigenvector with non-

neg(itive coordinates.

If the adjacency matrix is sym

metric, the converse of the last statem

ent also holds,as show

n by the following theorem

.

Theorem

0.4 (see, for example, (G

ant), voL. II, p. 79): If the "m

aximal" eigenvalue

l' of a non-negative matrix A

is simple and if positive eigenvectors belong to l' both in A

ltid AT

, then A is irreducible.

Theorem

0.5 (see, for example, (G

ant), voL. II, p. 78): T

o the "maxim

al" eigenvaluer of a non-negative m

atrix A there belongs a positive eigenvector both in A

and AT

ifand only if A

can be represented by ii permutation of row

s iind by the siime perm

utationof colum

ns in quasi-diagonal form A

= diag (A

¡, ..., As), w

here A¡, ..., A

s iire irredu-cible m

atrices each of which hiis l' a,s its "m

aximal" eigenvalue.

We shall now list some more theorems from the theory of matrices showing

new spectral properties of graph.

Theorem

0.6 (see, for example, (G

ant), voL. II, p. 69): T

he "maxim

iil" eigenvalue1" of every principal subm

atrix (of order less than n) of ii non-iiegiitive matrix A

(oforder n) does not exceed the "m

aximal" eigenvalue l' of A

. If A is irreducible, then

1" ~ l' iilwiiys holds. If A

is reducible, then 1" = l' holds for at least one prin¡;piil sub-

matrix.

In the subsequent text a vector with positive (non-negative) coordinates wil be

called a positive (non-negative) vector. A matrix A is called reducible if there is a

permutation m

atrix P such that the matrix P-IA

P is of the form (X

0), where

X and Z are square matrices. Otherwise, A is called irreducible. Y Z

Spectral properties of irreducible non-negative m

atrices are described by thefollowing theorem of FROBENIUS.

Theorem

0.3 (see, for example, (G

ant), voL. II, pp. 53-54): A

n irreducible non-negative m

iitrix A iilw

iiys has a positive eigenvalue l' that is a simple root of the cham

c-teristic polynom

ial. The m

odulus of any other eigenvalue does not exceed r. To the

"miixim

al" eigenviilue l' there corresponds a positive eigenvector. Moreover, if A

has heigenvalues of m

odulus 1', then these numbers are all distinct and are roots of the equation

Àh _ rh =

O. M

ore generally: the whole spectrum

(À¡ =

1', ..2, ..., Àn) of A

, regarded as iisystem

of points in the complex À

-plane, is mapped O

'ìto itself under a rotation of the

plane by the angle 2n. If h ? 1, then by a permutation of row

s and the same perm

utationh

of columns A

can be put into the following "cyclic" form

Theorem

0.7 (see, for example, (C

oSi 1)): The increase of any elem

ent of a non-negative m

atrix A does not decrease the "m

aximal" eigenvalue. T

he "rnaximal" eigen-

val'ue increases strictly if A is an irreducible m

atrix.

Theorem

s 0.6 and 0.7 state that in a (strongly) connected multi-(di-)graph G

every subgraph has the index smaller than the index of G

.

Theorem

0.8 (see, for example, (M

aMi), p. 64): A

ll the eigenvalues of a Herm

itiantm

atrix are 1'eal numbers.

0A

¡20

...0

)

00

An

.. .0

A -I:

i.(0.1)

00

0.. .

Ah-i.h

Ah¡

00

...0

)

where there are square blocks. along the m

ain diagonal.

Theorem

0.9 (see, for example, (H

of1)): Let A

be (i real symm

etric matrix w

hosegreatest and sm

allest eigenvalues a-re denoted by l' and q, respectively. Let æ

be the eigen-vector belonging to r. For a principal subm

atrix B of A

, letti' be the smallest eigenvalue

whose eigenvector is denoted by y. Then q' ~ q. If q/ = q, vector y is orthogonal to the

projection of vector æ on the subspiice corresponding to B

.

Theorem 0.10 (see, for example, (MaMi), p.119): Let A be

a Hennitian m

atrix with

eigenvalues ),¡, ..., Àn and B

be one of its principalsubmatrices; let B

have eigenvaluesfL¡, ..., fLm' Then the inequalities Àn-m+i ~ fLi ~ Ài (i = 1, ..., m) hold.

These inequalities are know

n as Oauchy's inequalitie; and the w

hole theorem is

also known as interlacing theorem

.

Theorem 0.11 (C. C. SIMS, see (HeHi))H: Let A be a real symmetric matrix with

eigenvalues J.1, ..., Àn- G

iven a partition p,..., n) = L

l¡u L12 U

... u Llm

with IL

lil =ni? 0,

If h? 1, the matrix A is called imprimitive and h is the index of imprimitivity.

Otherw

ise, A is prim

itive.

According to T

heorem 0.3, the spectrum

of a multi-(di-)graph G

lies in the circle1..1 ~

1', where l' is the greatest real eigenvalue. T

his eigenvalue is called the indexof G

. The algebraic m

ultiplicity of the index can be greater than 1 and there existsa corresponding eigenvector w

hich is non-negative.Irreducibility of the adjacency m

atrix of a- graph is related to the property ofconnectedness. A

strongly connected multi-digraph has an irreducible adjacency m

atrixand a m

ulti-digraph with irreducible adjacency m

atrix has the property of strong con-nectedness (DuMe), (Sed 1). In undirected multigraphs the strong connectedness

reduces to the property of connectedness.

According to T

heorem 0.3, the index of a st1'ngly connected m

ulti-digraph is asim

ple eigenvalue of the adjacency matrix and a positive eigenvector belongs to it.

t The com

plex matrix A

= (aij) is called H

ermitian if A

T =

A, i.e. aji =

aij.H

Recently W

. H. H

AE

ME

RS (H

aem) has show

n that the interlaoing properties also hold form

atrices A and B

of this theorem.

2*

0.3. Some theorem

s from m

atrix theory21

20O

. Introduction

consider the corresponding blocking A =

(Aij), so that A

ii is an ni X n¡ block. L

et eiibe the sum

of the entries in Aii and put B

= (eii/ni) (i.e., ei¡/ni is an average row

sumin Aii). Then the spectrum of B is contained in the segment (Àm Àd.

If we assum

e that in each block Ai¡ from

Theorem

0.11 all row sum

s are equal,then we can say more.

A square m

atrix with the property that its m

inimal and characteristic poly-

nomials are identical is called non-derogatory. T

hus Proposition (e) says that a square

matrix w

hich has all eigenvalues distinct is non-derogatory.

Theorem

0.12 (E. V

. HA

YN

SW

OR

TH

(Hayn); M

. PE

TE

RS

DO

RF

, H. S

AC

HS

(PeS

1)t):Let A be any matrix partitioned into blocks as in Theorem 0.11. Let the block Aii have

constant row sum

s bii and let B =

(bii). Then the spectrum

of B is contained in the

spectrum of A

. (having in view also the m

ultiplicities of the eigenvalues).

The square m

atrices A and B

are called similar if there is a (non-singular) square

matrix X

transforming A

into B, i.e., such that X

-lAX

= B

. Each sym

metric m

atrixand each m

atrix which has all distinct eigenvalues is sim

ilar to a diagonal matrix.

If A is the adjacency m

atrix of a multigraph, then A

is symm

etric and, consequently,sim

ilar to a diagonal matrix D

, namely, D

= (Ö

iiÀi)'

We m

ention the famous O

ayley-Ham

ilton Theorem

which says that each square

matrix A

,¡satisfies its own characteristic equation, i.e.:

We shall now

describe some m

ore basic properties of the spectrum of an undirected

multigraph. The facts wil be given almost without an.y proof for the convenience

of the reader. The proofs can be found at the corresponding places in the subsequent

chapters.T

he adjacency matrix of an undirected m

ultigraph G is sym

metric (and, therefore,

Hermitian) and the spectrum ofG, containing only

real numbers, according to T

heorem

0.8 lies in the segment ( -1', 1').

Let (À

1, ..., À,.) be the spectrum

of a multigraph. T

wice the num

ber of loops is equalto the tm

ce of the adjacency matrix. T

herefore, we have for m

ultigraphs without loops

tr A =

0, i.e., )'1 + ... +

Àn =

O. T

he number of vertices is, of course, equal to n,

and for undirected gmphs w

ithout loops or multiple edges the num

ber m of edges is given by

m =

~ i: ).7 (see Section 3.2).2 i=1

It is stated in (CoSi1) that for the index r of a connected graph the inequality

2 cos ~1 ~ r ~ n - 1 holds. The lower bound is attained by a path, and the upper

n+bound by a com

plete graph. If we om

it the assumption of connectedness, then for

a graph without edges w

e have l' = 0 and otherw

ise r ;S 1.For the smallest eigenvalue q of the spectrum of a graph G the inequality

-1' ~ q ~ 0 holds. For the graph without edges we have q = O. Otherwise q ~ -1.

This is a consequence of T

heorem 0.9, since the subgraph K

2 corresponds to a prin-cipal submatrix with least eigenvalue equal to -1. We have q = -1 if and only if

all components of G

are complete graphs (T

heorem 6.4). T

he lower bound q =

-1' isachieved if a com

ponent of G having the greatest index is a bipartite graph (T

heo-rem

3.4). According to the foregoing, the follow

ing theorem describes the fundam

en-tal spectral properties of (undirected) graphs.

If f(À) =

IÀl - A

I, then f(A) =

O.

The m

inimal polynom

ia,l m(À

) of A is the polynom

ial m(J,) =

ÀI' +

... suchthat(i) m

(A) =

0,(ii) under condition (i), the degree tt of m

(À) has its m

inimum

value.Then the following

propositions hold:

(a) m(À) is

uniquely determined by A

.

(b) If F(À) is any polynom

ial with F(A

) = 0, then m

(À) I F(À

); in particular,m

(À) I f(À

).

(c) Let rÀ

(ll, ),(2),..., À(k)) be the set of distinct eigenvalues of- A

, À(') having algebraic

multiplicity m

.. Then

Theorem

0.13: For the spectm

m(À

i, ..., Àn) of an (undirected) graph G

the following

statements hold:

1 ° The numbers )'i, ..., Àn are real and Ài + ... + ),11 = O.

20 If G contains no edges, we have Ài = ... = Àn = O.

30 If G contains at least one edge, we have

f(À) = (À - ),(l))mi (À - À(2))m, ... (À - À(k))'n~

andm

(J,) = (ì, - ì,(l))ai (À

- À(2))a, ... (À

- À(k))a.

where the q. satisfy

o ~ q. ~

m. (x =

1, 2, . ", k).

(d) If ~4 is similar to a diagonal m

atrrx, then all q. are equal to 1:

~(À

) = (J, - À

(l)) (À - À

(2)) ... (À - À

(k)).

(e) Let A

have order n. If A has all distinct eigenvalues, then

m(À

) = f(À

) = (À

- ì/1)) (ì, - ì,(2)) ... (J, - À(n)).

1~r~n-1,(0.2)

(0.3)-1' ~

q ~ -1.

In (0.2) the upper bound is attained if and only if G isa complete graph, while the lower

bound is reached if and only if the components of G

consists of gmphs K

2 and possibly Ki.

In (0.3) the uppe1' bound is reached if and only if the components of q are complete

graphs, and the lower bound if and only if a com

ponent of G having the greatest index is

t See Theorem

4.7.

22 O. Introduction

a bipartite graph. If G is connected, the lower bound in (0.2) is replaced with 2 cos ~ .

Then equality holds if and only if G is a path. n + i

We shall now

list some spectral properties of regular m

ultigraphs. The index is

equal to the degree (CoSi i). It can easily be seen that this holds for disconnected

multigraphs tob, but then the index is not a sim

ple eigenvalue. The m

ultiplicity ofthe index is equal to the num

ber of components. It can be seen im

mediately that the

vector having all coordinates equal to i is an eigenvector that corresponds to the index.

The eigenvectors of the other eigenvalues are orthogonal to this vector, i.e., the

sum of their coordinates is equal to O

.Further spectral properties of graphs can be obtained using the fact that the

coefficients of the characteristic polynomial are integers. It follow

s from this that the

elementary sym

metric functions and sum

s of k-th powers (k a natural num

ber) ofeigenvalues are integers, too. Since the coefficient of the, highest pow

er term of the

characteristic,polynomial is equal to i, rationnl eigenvalues (if they exist) are integers.

i¡1.

Basic Properties of the Specirum

of a Graph

The ordinàry spectrum

of a (multi-di- )graph G

is the spectrum of its adjacency

matrix, but there are various other m

ethods of connecting a spectrum or a cha-

racteristic polynomial w

ith G. A

general method of defining characteristic poly-

nomials (in one or m

ore variables) and graph spectra is outlined, the most im

portantspectra currently used and their interrelations are discussed, and it is show

n howthe coefficients of the corresponding characteristic polynom

ials can be obtaineddirectly from the "cyclic structure" or from the "tree structure" of G, respectively.

Eventually, the generating function for the num

bers of walks of length k

(k -' i, 2, ...) in G is expressed in term

s of the ordinary characteristic polynomial

and some conclusions are draw

n.

1.1. The adjacency matrix and the (ordiary) spectrum of a graph

In order to obtain an arithmetic method for describing and investigating the

structural properties of a finite (directed or undirected) (multi-)graph G

, it seems

quite reasonable to start with the adjacency m

atrix A of G

.O

bviously, G is uniquely determ

ined by A, but the converse statem

ent does not,in general, hold true since the ordering (num

bering) of the vertices of G is arbitrary:

To each graph G there corresponds uniquely a class d = d( G) of adjacency matrices,

two adjacency m

atrices A and A

* belonging to the same class (i.e., determ

ining thesam

e graph) if and only if there is a permutation m

atrix P such that A* =

P-IAP.

Thus the theory of graphs G

may be identified w

ith the theory of these matrix

classes d and their invariants. An im

portant invariant of a class d is the charac-teristic polynom

ial PG

V..) =

1J. - AI w

ith A E

d(G), or, w

hat amounts to the sam

ething, the spectrum

Sp(G

) = P

'l, Â2, ..., Â

n), where the )..;'s are the roots of the equa-

tion PG(Â

) = 0 (i.e., the eigenvalues of A

).tT

he main question arising is this: how

much inform

ation concerning the structureof G

is contained in its spectrum, and how

can this information be retrieved from

thespectrum ~ Of course, the amount of information

'contained in the spectrum m

ust\

t In order to avoid confusion, this "ordinary" spectrum wil later sometimes be called the

P-spectrum of G

, and it wil be denoted by Spp(G

).

241. B

asic properties of the spectrum of a graph

not be overestimated, since the spectrum

remains invariant not only under the group

of permutations, but also under the group of all orthogonal (and even of all non-

singular) transformations: T

hus the spectrum reflects com

mon properties of all

those graphs the adjacency matrices of w

hich may be transform

ed into one anotherby som

e non-singular matrix. A

ny such matrix transform

ing the adjacency matrix A

of a graph G into the adjacency m

atrix AI of som

e graph GI not isom

orphic with G

is subject to stringent diophantine conditions as all entries of A and A

I are requiredto be non-negative integers: T

herefore it may be expected that the classes of iso-

spectralt graphs are, in a sense, not too extensive. Jsomorphic graphs are, of course,

isospectral, and it has been conjectured, conversely, that any two isospectral graphs

are isomorphic; how

ever, this is not true. It is very easy indeed to find isospectralnon-isomorphic digraphs, e.g., all digraphs with n vertices, containing no cycle,

have the same spectrum

(0, 0, ..., 0) (see 1.4, Theorem

1.2).An essentially different situation arises if only undirected (multi-)graphs are

taken into consideration, and the construction of pairs of isospectral non-iso-

morphic (m

ulti- )graphs becomes m

ore and more difficult if one passes from

multi-

graphs to graphs and from graphs to regular graphs. Thus the spectral method

may be expected to be particularly efficient, w

hen applied to the class of regulargraphs.

Nevertheless, in the theory of block designs it has been show

n that, even among

strongly regular graphs (which form

a narrow subclass w

ithin the class of all regulargraphs) w

ith sufficiently many vertices, pairs of isospectralnon-isom

orphic graphsHare in fact not uncom

mon; see C

hapter 6.T

his phenomenon m

ay, on the one hand, be taken as an indication of the scopeand the bounds of this special spectral m

ethod; on the other hand, it probablyreflects a peculiarity of the theory of block designs, show

ing that there are indeedclose relations between this theory and the spectral

method.

1.2.A general method for defing diferent kids of graph spectra

In this section we shall consider another very natural approach to the spectral

method w

hich, by appropriate variation, yields arbitrarily many different "spectra",

i.e., systems of num

erical invariants.L

et us start with the ordinary spectrurn Spp(G

), as an example. W

e consider a setof n (unspecified) variables X

i; being in (1, l)-correspondence with the set of vertices k

(k = 1,2, ..., n) of a given (m

ulti-di-)graph G =

(gr, 0/). We try to find num

ericalvalues xi for all of the X

,1c not all equal to zero and such that for each vertex i thecorresponding num

ber x? is proportional to the sum s? of all those xi corresponding

to the (front) neighbours of i (i.e., such that the ratio s?;x? is the same for all i).

In other words, the xi are to satisfy, in a nD

n-trivial way, the system

of hDm

ogeneous

t Graphs having the sam

e spectrum are called isospectral or cospectral.

't Such a Pair of Isospectral Non-isom

orphic Graphs is som

etimes given the acronym

PING

;m

ore information about the construction of PIN

Gs w

il be found in Chapter 6.

1.2. A general m

ethod for defining different kinds25

linear equatiDns

ÀX

i = L xi;k-

(i E gr),t

(1.1)

the value of À being suitably chosen; if G is a multi-( di- )

graph, the multiplicity ail;

of the adjacency k . i is to be taken into accDunt by considering X

l; exactly ail; times

as a mem

ber of the right side sum of (1.1). O

bviously, (1.1) may be given the shorter

fDrm

Àæ

= A

æ,

( 1.2)

A =

(ail;) being the adjacency matrix D

f G and æ

denoting a column vectD

r with

compD

nents Xl; (k E

gr). As a necessary and sufficient condition for the existence of

a nDn-trivial sD

lution of (1.1) or (1.2), we have

IÀl- AI = PG(À) = 0,

III

i.e., the possible proportionality factors À are identical w

ith the eigenvalues of G.

This w

ay of reasoning has the advantage of being particularly intuitive, as thecom

ponents of an eigenvector may be directly interpreted as "w

eights" of the cor-responding vertices; at a later stage w

e shall find that the imm

ediate rationale of thespectrum

(via equations (1.1)) by inspection of the graph itself and, particularly,simultaneous consideration of its eigenvectors, wil be very useful for a series of

investigations and proofs.

Certain applications necessitate the determ

ination of the weights xt of the ver-

tices in such a way that xt is proportional not to the sum

(as above) but to the mean

value of all those xt corresponding to the (front) neighbours of i, i.e., the xZ are

required to satisfy the system of equations

1ÀXi = - LXi;

di l;,i(iE ¿().H

(1.3)

(1.3) may be replaced by

),Dæ

= A

æ,

(1.4)

yielding imm

ediately

I),D - A

I = 0

as a necessary and sufficient condition for the existence of a non-trivial solution of(1.3) or (1.4). T

hus we are led to introduce as a m

odified characteristic polynomial

1

QG

(À) =

iD IÀ

D - A

I = À

" + qiÀ

,,-i + ... +

qn(1.5)

t k . i means that k is a (front) neighbour of i (and i is a

(rear) neighbour of k).tt di here denotes the (out- )degree or (front) valency of vertex i, i.e. the num

ber of arcs issuingfrom

vertex i; it is assumed here that di ? 0; the diagonal m

atrix D =

(Oikd¡) is called the

(out-)degree or (fmnt) valency m

atrix of G.

261. B

asic properties of the spectrum of a graph

with corresponding spectrum

8pQ(G

) = P

'I, ,12, ..., )'n)Q'

Note that

(1.6)

QG(Â) = IU - D-IAI = ¡U - AD-II.

(1.5)'

1 1 1

Let D2 . (biT

. Vd;) and A

* = D

2 (D-IA

)D-2. T

hen

A* =

D-f A

D-f =

(V::~l,,

QG(Â) = IU - A*I.

(1.5)"

For an undirected multigraph G

, A* is sym

metric and, consequently, 8pQ

(G) is reaL

.In (1.5) D

appears in a multiplicative m

anner; D m

ay also be introduced in anadditive w

ay: starting from

~=~+

E~=

E~+

~k~

~i

(i E f!)

( 1.7)

we obtain another characteristic polynom

ial

RG

(Â) =

IU - D

- AI =

,1" + r1Â

"-1 + ... +

rn

with corresponding spectrum

(1.8) .

8PR

(G) =

(,11' )'2' ..., Ân)R

(1.9)

(cf. L. M. LIH

TE

NB

AU

M (JIiix2), E

. V. V

AH

OV

SK

IJ (Baxl)).

J. J. SE

IDE

L (LiSe) defines a m

odified adjacency matrix 8 =

(SiT

.) for (schlicht)graphs in the follow

ing way:

r -1S

iT. =

1 1

Sii = O

.

(i k), I

if i and k are adjacent

if i and k are non-adjacent(1.10)

Obviously,8=

J-I-2A,

(1.11)

J denoting a square matrix all of w

hose entries are equal to 1. tT

he system of linear equations, the characteristic polynom

ial, and the spectrum

t Obviously, if S is the Seidel m

atrix of the graph G and S is the Seidel m

atrix of the graph Gcom

plementary to G

, then simply S

= -S

o

1.2. A general m

ethod for defining different kinds27

corresponding to 8 are

ÂXi = E SiT.XT.

kEgl

(i E f!),

(1.2)

8G(Â) = IU ~ 81 = IU - J + 1 + 2AI

= )," + SI.1,,-1 + .,. + S",

( 1.3)

(1.4)8P

s(G) =

PI' ,12' ..., )'n)S

,

respectively.In this connection, two more spectra derived from the matrix of admittancet,

C = D - A, should be mentioned. Some authors (W. N. ANDERSON Jr., T. D.

MO

RLE

Y (A

nMo); M

. FIE

DLE

R (F

ie 1)) consider the polynomial

GG

(Â) =

IU - ci =

!U- D

+ A

I = )," +

CiÂ

"-1 + ... +

c" (1.15)w

ith corresponding spectrum

8po(G) =

(.11, ,12' ..., ,1,,)0(1.6)

(using, of course, different notation); A. K. KEL'MANS (KeJI 1) introduces a poly-

nomial

B1(G

) = ~ 1,11 +

ci,1

( 1. 7)

of order n - 1; clearly,

( -1)"B

1(G) =

- GG

(-)')',1

so that no special symbol is required for the K

el'mans spectrum

.A

ll the spectra considered so far - and only these tt - are to be found in the litera-ture; we shall return to this point in the next section.

We observe that all of the spectra dealt w

ith up to this point may be derived

from system

s of linear equations the coefficients of which are connected w

ith localstructural properties of the graph in question. B

ut the idea of obtaining systems of

numerical invariants by exploiting the solvability conditions for a system

of equationsconnected w

ith the graph and depending on certain parameters is not at all restricted

to the use of linear equations; for example, a m

ost natural way of extending the

method consists in the transition to a system

of quadratic equations of the form

ÂX; = E XiXT. (i E f!) (1.18)

j.ik.;

taking the multiplicities of the adjacencies into account by sum

ming over all pairs

of different edges (arcs) which have i as a starting vertex. In term

s of the adjacency

t The nam

e matrix of adm

ittance is taken from the theory of electrical netw

orks: any multi-

graph G m

ay be considered as corresponding to a specia,l electrical network all branches of

which have adm

ittance (= conductivity) 1.

tt In addition, of course, mention should be m

ade of the "distance polynomial" and corre-

sponding spectrum; see Section 9.2.

i,\

281. B

asic properties of the spectrum of a graph

1.3. Some rem

arks concerning current spectra29

matrix A

= (aik), (1.18) can be expressed in the follow

ing form:

Then

(l.21)PuCA) = IU - AI = FG(À, 0),

1 1

QG(À) = -IJ.D - AI = - FG(O, À),

IDI IDI

RG(À) = IU - D - AI = FG(J" -1),

0G(À) = IÀl- D + AI = (-1)" FG(-À, 1).

( 1.23)

(1.24)

2 " (aij) 2À

Xi =

L aijaikX

jXk +

L X

ji~

j~k~

" j=I 2

(1.9)(i =

1,2, ..., n).( 1.22)

(The right-hand side of (1.18) and (1.19) is nothing other than the elem

entary sym-

metric function of the second order of all X

k corresponding to the (front) neighboursof i, taking into account the m

ultiplicities of the adjacencies.)T

he set of all values of À for w

hich (1.18) and (1.19) have solutions consists of allzeros of the resultant RG(À) of the system (1.18): so the polynomial RG(J,) and the

system of the roots of the equation R

G(À

) = 0 (condition of com

patibility) can beconsidered as a characteristic polynom

ial "of quadratic origin" and the corresponding"quadratic" spectrum

, respectively.Instead of a system

of quadratic equations a system of cubic (biquadratic, ...)

equations could be taken into consideration, and if G is not regular w

e may connect

a system of homogeneous equations depending on more than one parameter (one

parameter for each degree, see. next section) w

ith the graph G thus obtaining a

characteristic polynomial depending on several variables. W

e may even leave the

field of algebra and connect with G

a system of suitably chosen functional equations

(boundary value problem, system

of integral equations, .. .),t thus obtaining alsospectra w

ith infinitely many eigenvalues: the possibilities of connecting "spectra"

w~

th graphs are many and varied.

It would be very desirable to learn something about the correlations between

these different kinds of spectra and especially about the particular role which the

"linear" spectra play among them

: Perhaps it may be possible to specify som

efinite system

of suitable spectra of a graph G, w

hich, taken as a whole, com

pletelycharacterize G.

Interesting as these problems are, they seem

to be difficult ones, tt and, since thereare at present scarcely any know

n results worth m

entioning, we shall confine our-

selves in this book to investigations concerning linear spectra, as described above.

As for the Seidel spectrum

, we can only state( À

+1 )

SG(À) = IU - 81 = (-1)". 2"FG* -~' 0

= (-1)" . 2" P G* ( _ À ~ 1);

(1.25)

here G* stands for a "generalized graph" with weighted adjacencies having the

1"adjacency matrix" A - - J.

2

Remark. FG(À, ll) may be considered as a characteristic polynomial depending

on two variables. B

ut (1.20) is, of course, not the only possible way of introducing a

characteristic polynomial depending on several variables: If, for exam

ple, G is non-

regular with s different (out-)degrees V

i' V2, ..., V

s' we m

ake a parameter )'a correspond

to every vertex i with (out-)degree di =

Va (0" =

1,2, ..., s). Let À

(i) denote theparameter belonging to the vertex i (i.e., ),(i) = Àa with 0" satisfying di = Va)

and put

(À(i) 0)

A =

À (2). '

o ).(,,)(1.26)

1.3.Som

e remarks concernig current spectra

then we may generalize PG(À) = IU - AI to

PW'I, À

2. ..., )'8) = IA

- AI.

(1.27)A

ll spectra comm

only used have been listed in the preceding section; it may be

worth m

entioning that all of them can be derived from

a comm

on source (the Seidelspectrum

playing a somew

hat exceptional role): PutB

y the specialization

Àa =

À +

llva(a =

1,2, ...,8),(1.28)

FG(À,ll) = IÀl + llD - AI.

( 1.20)i.e.,

t A first step in this direction can be found in (PeS 1) (note that formulas (3) and (6) of (PeS 1)

are incorrect, they should be replaced by the above formula (1.19)). See also (Sac

15).tt E

xperimenting techniques applied to resultants of system

s of non-linear algebraic equationswil hopelessly fail as the orders of the resulting polynomials are in general beyond any reason-

able size - even in simple cases.

J.(i) = À

+ lldi (i =

1,2, ..., n),

from (1.27) the polynom

ial FG(À

, ll) is retrieved:

F G(À

, ll) = P~(À

+ llV

I' À +

/tV2, ..., À

+ llvs)

( 1.29)

which is also valid in the regular case.

301. B

asic properties of the spectrum of a graph

1.4. The coeffcients of P G(Ã)

31

It would certainly be an interesting though possibly diffcult task to investigate

the significance of these generalized characteristic polynomials, but w

e shall notpursue such questions in this book. (See also Section 4.5.)

We return to form

ulas (1.21)-(1.25) and assume G

to be a (multi-)graph w

hichis regular of a certain degree 1": w

e shall show that in this case the four spectra

Spp, SPQ,-i SPR

, Spc are equivalent, i.e., contain the same am

ount of information

about the structure of G, and that "alm

ost the same" is also true for S

ps'T

his is quite obvious in the first four cases: Since D =

1", we have

components of æ

O are positive, it follow

s from T

heorem 0.3 that l' is the m

aximal

eigenvalue contained in the P-spectrum

of G.

It is worth mentioning that there is stil another important class of multigraphs

for which the spectra Spp(G

) and SPQ(G

) are equivalent, namely, the class of sem

i-regular multigraphs of positive degrees. (Recall: a multigraph G is called semi-

1"egula1' of degrees 1"i, 1"2' if it is bipartite having a representation G =

(g(i, g(2; OZ

t)w

ith ¡g(il = ni, 19(21 =

n2, ni + n2 =

n, where each vertex x E

eli has valency riand each vertex x E g( 2 has valency 1'2') In this case, a straightforward calculation

shows that the vector æ

O =

(Vdi, V

d2, ..., vtl,T (w

ith d¿ = 1"i or 1'2) is an eigenvector

of the adjacency matrix of G belonging to the eigenvalue V1"11"2, and since all com-

ponents of æO

are positive, it follows again from

Theorem

0.3 that V1"i1'2 is the m

aximal

eigenvalue. Recall that the m

aximal eigenvalue is called the index of G

denotedby /2. A

ccording to (1.5)" (Section 1.2),

U +

tlD =

(À +

1'tl) 1

and consequently

FG

(À, tl) -- F

G(À

+ 1'tl, 0) =

PG

(À +

1"tl).

So, according to (1.22)-(1.24),

1QG(À) = -PG(1"),

1"n

( 1.30)

I 1 \ 1 1Q

i;(J,) = IJ. -A

*! = J. - - A

= -IÀ

/21 - AI =

- PG(q:1).

ri on on,'" ~ ~

(1.31)So w

e have proved

RG

(À) =

PG

(À - 1"),

GG

(À) =

(-l)nPG(-À

+ 1"),

( 1.32)

(1.33)

Theorem

1.1 (F. R

UN

GE

(Rung)): Let G

be a multigraph either 1"egula1" of positive

degree 1" or semi1"egula1" of positive deg1"ees 1"i, 1"2' and let /2 be the index of G. Then /2 = l'

01" /2 = ~, 1"espectively, and in either case

and from1

QG(J,) = - PG(/2À).

enS

pp(G) =

(À1, À

2, ..., Àn)H

we deduceS

G (À1 À2 ÀnJ

PQ( )= -,-,...,-,

1" 1" 1"

SP

R(G

) = (:11 +

1",:12 + 1", ..., J'n +

1"),

SpdG

) = (1" - À

n, 1" - Àn-1, ...,1" - J,d.

Note that a connected m

ultig1'aph G is 1"egula1" 01" sem

i1"egular of positive deg1"ee(s)if and only if the line graph of G is 1"egula1".

(1.31')

(1.32')

(1.33')

1.4. The coefficients of P GO. )

In the next three sections we shall be concerned w

ith relating the coefficients ofPi;(À), GG(:1), and QG(:1), respectively, to structural properties of the graph G.

Let G

be an arbitrary iiulti-(di-)graph and

PG(J,) = J,n + a1:1n-1 + ... + an

In the case of the Seidel spectrum

, by due computation m

aking use of the eigen-vectors, w

e obtain

SG(J,) = (-l)n. 2n À + 1 + 21" - n P (_ À + 1).

À+1+21" G 2'

,

(1.34)

its characteristic polynomiaL. It has been observed by several authors-i that the

values of the coefficients ai can easily be computed if the set of all directed cycles

of G (considered as a digraph) is know

n. The converse problem

of deducing structuralproperties of G

(for example, concerning the cycles contained in G

) from the values

of the a¡ is much m

ore difficult; we shall return to this problem

in Section 3.1.

the eigenvalues with respect to S are -2:1n+

2-; - 1 (i = 2, 3, ..., n) and, in addition,

n - 21" - 1. (See also Section 6.5, Lemma 6.6.)

If G is regular of degree 1", then, as can easily be checked, æ

O =

(1, 1, ..., 1)T is an

eigenvector of its adjacency matrix A

belonging to the eigenvalue 1', and since all

-i Here l' ? 0 is assum

ed.H

Note that Ã

1 = r (see Section 0.3).

-i See the remark on the history of the "coefficients theorem

" (p.36).

321. B

asic properties of the spectrum of a graph

1.4. The coefficients of P a(),)

33

The follow

ing theorem is som

etimes called the "coefficients theorem

for digraphs". Then

( 1.37)T

heorem 1.2 (M

. M:IL

IC (M

ili), H. SA

CH

S (Sac2), (Sac 3), L. SPIA

LT

ER

(Spia )t): Let

PG(),) = I.U - AI = Â,n + alÂ,n-i + ... + an

be the characteristic polynomial of an a1'bitrary (directed) m

ultigraph G. T

hen

ai = ~ (_l)p(L)

LE

se,(i =

1,2, ..., n)(1.35)

where .!i is the set of all

linear dÍ1ected subgraphs L of G

with exactly i vertices; p(L

)denotes the num

ber of components of L

(i.e., the number of cycles of w

hich L is com

-posed).

This statem

ent may be given the follow

ing form:

The coefficient ai depends only on the set of all lÙ

iear directed 8ubgraphs L of G

having exactly i vertices, the contribution of L to ai being +

1 if L contains an even,

and -1 ,if L contains an odd, number of cycles.

If G is an undirected multigraph, we may stil consider G as a multi-digraph Gf

(see Section 0.1, p. 12); all that is necessary to observe is that to every edge of Gw

hich is not a loop there corresponds a cycle of length 2 in Gf, and to every circuit

of G there corresponds a pair of cycles in G

f, oriented in opposite directions. Theorem

1.2 may now

be easily reformulated for m

ultigraphs as follows:

Theorem

1.3 (H. SA

CH

S (Sac2), (Sac3), L. SPIA

TE

R (Spia)*): L

et

P G(Â,) = IU - AI = Â,n + alÂ,n-i + ... + an

be the characteristic polynomial of an arbitrary undirected m

ultigraph G.

Ca.l an "elementary figure"

a) the graph K2, or

b) every graph Cq (q ~ 1) (loops being included w

ith q = 1),

call a "basic figure" U every graph all of whose components are elementary figures;

let p(U), c(U

) be the number of com

ponents and the number of circuits contained in U

,respectively, (ind let óli denote the set of all basic figures contained in G

having exactlyi vertices.

Then

ai = ~ (-l)P(U

) .2c(U) (i =

1,2, ..., n).U

Eo¿l,

This theorem

may be given the follow

ing form:

Define the "contribution" b of an elem

entary figure E by

(1.36)

b(K2) = -1,

and of a basic figure U by

b(Cq) = (-l)q+l. 2

b(U) =

IT b(E

).E

cU

t See the rem

ark on the histOry of the "coefficients theorem

" (p. 36).

( -1)i ai = L, b( U) .

UE

o¿t,

Proof of T

heorern 1.2. Let us first consider the absolute term

an = PG

(O) =

(-I)n IAI =

(-I)n l(tikl'

According to the Leibniz definition of the determ

inant,

a = "'(-I)n+

I(P)al,a2' ...a.

n £. 11 12 nin

p(1.38)

with sum

mation taken over all perm

utations

(1 2 .. . n)p=

'ii i2 ... in '

I(P) denotes, as usual, the parity of P. For the sake of simplicity, let us first

assume that there are no multiple arcs so that aik = 0 or 1 for all i, k. A term

Sp = (_I)n+

I(P) (tlii a2i,'" anin

of the sum (1.38) is different from

zero if and only if all of the arcs (1, iil, (2, i2),...,(n, in) are contained in G

. P may be represented as a product

P =

(1i1 ...) (...) ... (...)

of disjoint cycles. t

Evidently, if S p =

! 0, then to each cycle of P there corresponds a cycle in G: thus to

P, there corresponds a direct sum

of (non-intersecting) cycles containing all verticesof G

, i.e., a linear directed subgraph L E .!n- C

onversely: To each lineal' directed

subgraph L E .!n there corresponds a permutation P and a term Sp = ::1, the

sign depending only on the number e(L

) of even cycles (i.e., cycles of even length)am

ong all cycles of L:

Sp =

(_I)n+e(£).

Obviously,n

+ e(L) _ p(L) (mod 2)

hence(1.39)

an = ~

Sp =

~ (_I)p(L).

P LEse n

Now

, (1.39) remains valid even if aik ? 1 is allow

ed:C

onsider the set of all distinct linear directed subgraphs L E

.! n connecting the nvertices of G

in exactly the way prescribed by the cycles of a fixed perm

utationP =

(lil ...) (...)... (...). It is clear that this set can be obtained by arbitrarilychoosing for each k an arc from

vertex k to vertex ik; and doing so in every possible

t Note that I(P) == e(P) (mod 2), where e(P) is the number of even cycles among all cycles

of the cycle representation of P given above.

3 Cvetkovic/Doob/Sachs

341. B

asic properties of the spectrum of a graph

manner; and since for fixed k there are exactly aki. possible choices, the total num

berof subgraphs so obtained equals ali, a2i, ... anin' T

hus the total contribution ofall of these subgraphs to the sum

~ (- l)p(L) equals (- 1 )n+

I(P) aii,a2i, ... anin'

LE,f n

Summ

ation with respect to all perm

utations P confirms the validity of (1.39) in

the general case.

In order to complete the proof of (1.35) suppose 1 ~ i ~ n (i fixed). It is w

ellknown that (-1)i ai equals the sum of all principal

minors (subdeterm

inants) oforder i 'of A. Note that there is a (l,l)-correspondence between the set of

these minors and the set of all induced subgraphs of G

having exactly i vertices. By

applying the result obtained above to each of the (~) minors, and summing, the

validity of (1.35) is established. iR

emark. If, instead of the determ

inant, the permanent of A

,

Per A =

'" ai' a2' ... a '.t ii i2 nin,

p

is considered, we obtain by m

eans of analogous deductions the simple form

ulas

per A =

number of directed linear factors-i of G

,

and in the case of an undirected multigraph:

per A =

~ 2c(U

) .U

Eq¡ n

(1.40)

(1.41 )

Call perm-polynomial of an arbitrary square matrix A of order n the polynomial

per (U + A) = J," + a~Ji"-l +... + a:.

The analogues of T

heorems 1.2 and 1.3 are then:

Theorem

1.2*: Let

P"((Ji) =

per (U +

A) =

Ji" + aiJ,"-l +

... + a~

be the perm-polynom

ial belonging to an arbitrary (directed) multigraph G

with ad-

jacency matrix A

. Then

at = num

ber of linear directed subgraphs of G

containing exactly i vertices (i = 1,2, ..., n).

( 1.35*)

Theorem

1.3*. Let

P"G(J,) = per (U + A) = Jin + aiJi"-l + ... + a:

t A directed linea'r factor of a m

ulti-(di-)graph G is a linear directed subgraph containing all

vertices of G.

1

1.4. The coeffcients of PGP.)

35

be the perm-polynom

ial belonging to an arbitrary iindÙ'ected m

ultigraph G w

ith ad-jacency m

atrix A. T

hen

(i = 1,2, ..., n).

(1.36*)a,¡ = ~ 2c(U)

UE

Úl(,

Theorem

s 1.2 and 1.2* may be extended. to digraphs w

ith weighted adjacencies

imm

ediately:Suppose that adjacency k . i has (arbitrary) w

eight aik't and let A =

(aik) be thecorresponding generalized adjacency matrix.

Then T

heorems 1.2 and 1.2* stil

hold withai =

~ (_l)p(i) II (L)

LE

,f,(1.35)'

(i = 1,2, ..., ii)

anda¡ =

~ II (L)

LE

,f ,(i =

1,2, ..., n)(1.35*)'

instead of (1.35), (1.35*), respectively, II (L) denoting the product of the w

eights of allarcs belonging to L

.

If G is an undirected graph w

ith weighted adjacencies and U

is a basic figurecontained in G

, let

II (U) = II (w(u))~(U;ul,

UE

E(U

)

where E(U) is the set of edges of U, w(u) is the weight,

of the edge u, and

U 1 1 if u is contained in som

e circuit of U,

C(u; ) = '.

2 otherwise. '

Since U

contains exactly 2c(U) linear directed subgraphs L all having the sam

e weight

II (L) = II (U), (1.35)' takes the simple form

Ui = ~ (-1)P(U) 2c(U) II (U). (1.35)"

UE

új(,

With i =

n, we obtain from

(1.35)' a simple form

ula for the calculation of thedeterminant of an arbitrary square matrix A considered as a generalized adjacency

matrix of a digraph G:

¡AI =

(-1)" ~ (_l)p(L) II (L)

LE,f n

(1.42)

(note that .! n is the set of all dirécted linear factors L of G).

If, in particular, A, is the adjacency m

atrix of a multi-digraph or a m

ultigraph,(1.42) reduces to

¡AI = (-1)" ~ (_1)p(L)

LE,f n

( 1.42)'

-i We m

ay assume that for every pair i, k there is exactly one arc from

i to k, and that aik isthe w

eight of this arc (possibly equal to zero).

3*

361. B

asic properties of the spectrum of a graph

or¡AI = (-1)" L (-1)P(U) 2c(uL,

UE

o¿tn(1.42)"

respeetively.(1.42) m

ay be taken as an intuitive form of the L

eibniz definition of the determinant.

A theory of determinants based on this .observation was outlined by D. M. CVETKO-

VIó (C

ve 15).

Remai'k (concerning the history of the coefficients theorem). In order to show

that this approach is not only of purely theoretical interest, it should be noted thatthere are tw

o other fields in which determ

inants have been connected with graphs: elec-

tronics-cybernetics (signal flow graph theory) and chem

istry (quantum chem

istry,sim

ple molecular orbital theory).

Apparently (1.42) was given for the first time by C. L. COATES (Coat) (1959) in

connection with flow

graph considerations t; (1.42) is therefore sometim

es calledC

oates' formula. A

simple proof is given by C

. A. D

ES

OE

R (D

eso) (1960). F. H

AR

AR

Y

(Har2) (1962) considers the case w

hen A is the adjacency m

atrix of a digraph orof a graph. But before COATES other authors came close to formula (1.42)

(see D. K

ÖN

IG (K

ön1) (1916), (Kön2) (1936); see also T

. MuIR

(Mui2), footnote on

p. 260 concerning Cauchy's rule for determ

ining the sign of a summ

and in theexpansion of a determ

inant).For som

e small values of i, the coefficients ai of the characteristic polynom

ial of anundirected graph 0 were already determined by C. A. COULSON (Cou2) (1949) and

1. SAMUEL (Sam1) (1949) (see also (Sam2)) in the context of molecular orbital

theory, and, independently, by L. COLLATZ and U. SINOGOWITZ in their fundamental

paper (CoSi1) (1957)tt on graph spectra. COULSON (Cou2), however, does not use

the concept of "basic figures" but expresses the coefficients by means of the num

bersof all possible subgraphs of 0 w

ith the given number of vertices. In this connection,

E. HEILBRONNER'S papers (Heil) (1953), (Hei2) (1954) should also be mentioned;

he showed how

, in the case of special graphs arising in the molecular orbital theory,

the characteristic polynomial can easily be obtained by som

e intuitive "graphical"recurrence procedures.

It seems that the coefficients theorem

in full generality was first published by

H. SACHS (Sac3) (1964) (see also (Sac2) (1963)) and almost at the same time by

L. SPIALTER (Spia) (1964) (in a terminology appropriate for chemical applications)

and M. M

rLIÓ (M

ili) (1964) (in terms of flow

graph theory). Later it has been re-discovered several tim

es: J. PO

NS

TE

IN (P

ons) (1966), J. TU

RN

ER

(Turn2) (1968),

A. B

EC

E (B

eiie) (1968), A. M

OW

SH

OW

ITZ

(Mow

5) (1972), H. H

OS

OY

A (H

os2) (1972),F

. H. C

LAR

KE

(Clar) (1972); for trees it has also been given by L. LovÂ

sz and J.P

ELIK

ÂN

(LoPe) (1973). T

UR

NE

R'S

paper contains a somew

hat more general theorem

t With regard to signal flow graph theory, see the fundamental papers of C. E. SHANNON

(Shan) (1942) (which remained unnoticed for several of years) and S. J. MASON (Mas1)

(1953), (Mas2) (1956); for applications see C

. S. LO

RE

NS (L

ore) (1964). For proofs see alsoR. B. ASH (Ash) (1959) and A. NATHAN (Nath) (1961). A detailed treatment may be found in

the book of W.-K. CHEN (Chen) (1971).

tt Note that this paper had already been prepared during W

orld War II, see (C

oSi2).

1

1.5. The coefficients of OG(Å.)

37

concerning the coefficients of a generalized characteristic polynomial

Pp.) = dl.(A

- Ål),

dl. being a matrix function generalizing determ

inant as well as perm

anent given by

dl.(A) =

L X

(P) aliia2i2 ... a"inp

(12...n)with summation over all permutations P = ;. .; here x(P) denotes some

i¡ i2 ... i"

character defined on the symm

etric group Y" of all perm

utations P considered.

Some sim

ple consequences of Theorem

s 1.2 and 1.3

Proposition 1.1: The num

her of linear subgraphs with exactly q edges contained in an

undirected forest H is equal to (-l)q a2q. A

n undirected linear factor exists if and onlyif aii =

j O. In this G

ase, n is even, and, as there evidently cannot be more than one linear

n

factor, an = (-1) 2 .

Proposition 1.2: The num

bei' of directed linear factors contained in a multi-digraph 0

is not srnallei' than lalil.

The general problem

"Let the characteristic polynom

ial PG(À

) of some m

ulti-(di-)graph 0 begiven, w

hat information about the cycles (or circuits) contained in 0 can

be retrieved from the coefficients ai ~

"

wil be treated in Chapter 3, Sections 3.1-3.3.

1.5. The coefficients of CG().)

Next w

e shall express the coefficients Ci of the polynom

ial

GG(J.) = IÅl- 01 = À" + CiÀ"-l + ... + c"

(1.43)

in terms of the "tree structure" of 0, w

here 0 is any multigraph (recall that

0= D

- A =

(oiidi - aii) is the matrix of adm

ittance of 0; see Section 1.2).Let M

be any square matrix w

ith rows 1'1, r2, ..., 1'". and colum

ns Ci, C

2, ..., Cn,

let.A = 11,2, ..., nl and" = 11'1' 1"2' ..., 1'ql c .A; let M f denote the square matrix

obtained from M

by simultaneously cancelling row

s rii' ri2' ..., riq and columns

Cii' ci2' ..., cii F

or the sake of convenience write ltli instead of M

¡i)' etc.; as usual,the determ

inant of the empty m

atrix (case" = .A

) is assumed to be 1.

If 0 is any multigraph w

ith n vertices 1,2,..., n, and if ß =

j ø, let 0" denote

the multigraph obtained from

0 by identifying (amalgam

ating) the vertices 1'1' 1'2' ...,l' q' thereby replacing the set l1'ii 1'2' ..., 1'ql by a single hew

vertex i (by this processmultiple edges and loops

may be created); evidently, 01 = O2 = ... = 0" = O.

381. B

asic properties of the spectrum of a graph

The following well-known important theorem connects the number of spanning

trees of a multigraph w

ith its matrix of adm

ittance.

:ilatrix-Tree-T

heoremi": Let G

be a multigraph w

ith vertices 1,2, ..., n and let t(G)

denote the number of spanning trees contained in G

. H T

hen

t(G) =

ICil,

where C

= D

-A is the m

atrix of admittance of G

and j E (1, 2, ..., nJ.

Corollary: Let .I c .A, .I =l ø .

Then

t(G .I) = IC .II.

(1.44 )

(1.45)

Proof of the Corollary. If C

' denotes the matrix of adm

ittance of G .I' then C

; = C

.I'ar¡d according to the M

atrix-Tree-T

heorem, t(G

f) = IC

;I = IC

ß:I'In the sequel the convention t(G

ø) =

0 wil be adopted so that (1.45) holds for

every .I c .A (note that ICøl = ~e¡ = 0).

Now

we are in a position to calculate the coefficients C

i of 0o(À) =

IÀl - e¡.

Since (-I)i Ci is equal to the sum of all principal minors of order i of C,

cn_k=(-l)n-k L IC.l1 (k=O,l,...,n), (1.46)

f c,A

I.li~ki

where, according to the corollary of the M

atrix-Tree-T

heorem, IC

.l1 equals t(G .I),

Thus

we have proved

Theorem

1.4 (A. K

. KE

L'lVIA

NS

(KeJI3)): Let

0o(À) =

IÀl - e¡ =

co).n + ciJ,n-i +

... + C

n(co = 1),

where G

is an arbitrary multigraph and C

= D

- A is its m

atrix of admittance. T

hen

(i = 0, 1, ..., n).

(1.47)Ci = (_l)i L t(G.I)

Le",y,

i;)'-i=n-i

Let the forest F have k com

ponents Ti w

ith ni vertices (i = 1,2, ..., k) and put

y(F) = nin2 ... nko A

ccording to (KeC

h) (see formula (2.14) on p. 203), c¡ can be given

the following form

:

Ci = (_1)i 1. y(F)

FE,%

n-'(i =

0, 1, ..., n - 1),C

n=O

,(1.47)'

where ?k is the set of all

spanning forests of G w

ith exactly k components.

i" This theorem was proved in a paper by R. L. BROOKS, C. A. B. SMITH, A. H. STONE, and

W. T

. TU

TT

E (B

rSS

T) (1940), and independently by H

. M. T

RE

NT

(Tren) (1954), and others;

an elementary proof was given by H. HUTSCHENREUTHER (Huts) (1967). Some authors hold

that it is already implicitly contained in G. KIRCHHOFF'S classic paper (Kirc) (1847). (For

more details consult (M

o02) (Chapter 5).)

H t(G

) is sometim

es called the complexity of G

. - A sim

ple determinant form

ula for thecom

plexity of a bipartite graph is due to F. RU

NG

E; see Section t.9, no. 12.

1

1.5. The coefficients of GG(À)

39

A theorem for multi-digraphs with weighted adjacencies generalizing Theorem

1.4 was proved by M

. FIE

DLE

R and J. S

ED

LÁC

EK

(FiS

e).n

Rem

ark. For i = n - 1, (1.47) yields C

n-I = (_l)n-l L

t(G¡) =

(_l)n-l nt(G).

j=1

Hence

(i) t( G) = .. ( -1 )n-l Cn-l'

n

Let l-1' l-2, ..., l-n (in som

e order) be the eigenvalues of C. Since C

n = IA

- DI

n-l=

0, it follows that 0 E

SPc( G); let l-n =

O. T

hen ( _l)n-l Cn-l =

n fli, and from (i)

i=1

(ii)1 n-l

t(G) = - n l-i

n i=1

is obtained.If G

is connected, t(G) ? 0, i.e., l-i =

l 0 for i = 1,2, ..., n - 1. T

hus we have

proved

Proposition 1.3: Let G

be a connected multigraph. T

hen

1t(G

) = - II l-,n

where l- runs through all non-zero eigenvalues of C

= D

- A.

In terms of the polynom

ial Oo(À

) or the Kel'm

ans polynomial B

~(G) (see (1.17),

Section 1.2), this result can also be expressed in the following form

:

, (_l)n-l 1 1 n

t(G) = OG(O) = - Bo(G).

n n

(iii)

If G is regular of degree 1', form

ulas (1.33) and (1.33') apply and we deduce from

(i),(ii), and (iii) (recall that À1 = 1')

Proposition 1.4 (H. HUTSCHENREUTHER (Huts)): For any regular multigraph G

of degree 1',

1 n 1t(G) = - II (1' - Ài) = - P~(r),

n i=2 n

where the Ài are the ordinary eig€nvalues of G.

By adding an appropriate num

ber of (simply counted) loops, any m

ultigraph G of

maxim

al valency r can be made a regular m

ultigraph G1 of degree 1'. S

ince thisprocess has no influence on the num

ber of spanning trees, Proposition 1.4 can beapplied to an arbitrary m

ultigraph G, provided the À

¡are taken to be the eigenvaluesnot of G

but of G'. T

his observation, due to D. A

. WA

LLER

((Wall), (W

a12), (Wa13);

see also (Ma12)), is equivalent w

ith Proposition 1.3.(S

ee also Section 1.9, nos. 10, 11.)

401. B

asic properties of the spectrum of a graph

1.6. The coefficients of QGo.)

By a procedure very sim

ilar to the method used in the proof of the preceding theo-

rem, the eoefficients of Q

G(À

) can be determined. (R

ecall: QG

(À) =

~ IÀD

- AI

= qoÀn + qiÀn-1 +... + qn (qo = 1); see Seetion 1.2.) IDI

Let G

be an arbitrary multigraph w

ithout isolated vertices. Consider Q

G(À

) as apolynomial in .1 - 1:

QG(J,) = !U - D-1AI = 1(.1 - 1) I + D-1(D - A)I

= 1(.1 - 1) I + D-1C¡ = qo(À - l)n + qi(À - 1)n-1 + ... + qn,

where qi equals the sum

of all principal minors of order i of D

-1C. A

ccordingly,

with

qn-k = ~

I(D-1C

) flf c,A

Ifl~k(k =

0,1, ..., n)

t(G f)

II di'

IE,A- f

the last equation following from

the Corollary to the M

atrix-Tree-T

heorem (Section

1.5) (if f = %, then II di = 1 is assumed). Thus

IE,A - f

_ t(G ,,)

q,,-k = ~ -

fc,A II di

Ifl=k IE,A- f

. IC"I_

1 (D-1C)fl =J(D-1)fC fl = ID fl -

(k = 0,1, ..., n),

and since g.-k = i: (j) (-l)i-q"_j' we obtain with k = n - i:

j=k k,. ( . ) t(G )

qi = (_l)n-i ~ J. (-l)j~. ~

j=,.-i n - i f;: II d

l"l~j lE

A/"- J

(i = 0,1, ..., n).

So we have proved

Theorem

1.5 (F. RU

NG

E (R

ung)): Let

1Q

G(À

) = - IÀ

D - A

I = qo),n +

qi),n-1 + ... +

q,.ID

I(qo =

1ì,

where G

is an aTbitraT

Y rnultigraph w

ithout isolated vertices. Then

, ,. ( j) , t( G )

,-- -1 n-i '7 -1 J '7 f .q, - ( ).: . ( ) _ (i = 0,1, ..., n),

j=,.-i n - i fcJV

II diIfi~J IE,A- f

where the conventions t(Gø) = 0 and II di = 1 are adopted.

IEØ

(1.48)

1

1.7. Cyclic structure and tree structure

41

Theorem

1.5 has also been extended to graphs and digraphs with w

eighted ad-jaeencies by F

. RU

NG

E (R

ung).

RernaTk. In order to obtain a coefficients theorem for QG(À) based on the cyclic

structure of G, recall that Q

G(À

) = IU

- A*I w

ith A* =

( a:ik ) (see (1.5)", Section1.2). Vdjdk

Now

formula (1.35)" (Section 1.4) w

hen applied to A* yields im

mediately

qi = ~ (- 1 )p(U) 2c(U) II (U)

UE

o/I,

with

( 1 )Ç((i.k);U) 1

II(U)=

II -= =

-,(j.k)E,g(U) V djdk II dh

hE"f'()

where g(U

), "f(U) denote the sets of edges and of vertices of U

, respectively. Thus

we have proved

Theorem

1.5a: Under the assum

ptions of Theorem

1.5,

2c(U)

qi = ~

(_l)p(U)_.

UEilll, II dh

hEf/(U

)

(1.48 a)

1.7. A form

ula connecting the cyclic structue and the tree structureof a reguar or sem

ieguar multigraph

There are tw

o strong connections between structural graph theory and linear algebra:

The first one consists of the fact that the m

ost important general invariant of linear

algebra, the determinant, m

ay be given a combinatorial form

(viz., the form it has

in its "Leibniz definition") that has an interpretation in term

s of the cyclic structureof a (di-)graph (w

ith weighted adjacencies), and the second one is the validity of the

Matrix-T

ree-Theorem

(see Section 1.5) w

hich, in a very simple w

ay, connects thetree structure of a' graph w

ith determinants form

ed from its m

atrix of admittance. B

othof these connections are taken advantage of by spectral theory: the coefficients theorem

sfor P

G(À

) (Theorem

s 1.2, 1.3) are based on the first one, and for GG

(À) (T

heorem 1.4)

and QG(J,) (Theorem 1.5) on the second one.

Of particular interest are those graphs G

which have the property that their

polynomials PG(),) and GG(J,) or QG(À) can be transformed one into another: in this

case, the coefficients can be expressed both in terms of the cyclic structure and in

terms of the tree structure of G, thus linking the basic structural elements, cycle

(or circuit) and tree, one to another.A

ccording to Theorem

1.1 (Section 1.3),

1QG(J,) = -; Pa(e),)

e( 1.49)

421. B

asic properties of the spectrum of a graph

for any multigraph G

which is regular or sem

iregular of positive degree(s) l' or ri, 1'2,respectively, and has index (=

maxim

al P-eigenvalue) (2, w

here (2 = l' or e =

Vrir2,

respeetively. From (1.49) w

e deduce

eiqi = ai

(i = 0,1, ..., n),

and applying Theorem

s 1.3 and 1.5, we obtain the follow

ing theorems.

Theorem

1.6 (F. R

UN

GE

(Rung)): Let G

be a regular multigraph of positive degree l'

with n vertices 1, 2, ..., n. T

hen

L (-l)P(U) 2c(U) = t ( f .) (_lyi+i-nri+i-n L t(G f) (i=O,l,...,n),

UEOJI, j~n-i n - i ,jc.f

ifl=j (1.50)

where for i =

0 the left-hand sum is taken to be 1.

Theorem 1.6

a (F. RU

NG

E (R

ung)): Let G

= (!!, qy; 0lt) be a sem

iregular multigraph,

where all vertices x E !! = iI, 2, ..., ni L have valency ri :; 0 and all vertices y E qy

= ini + 1, ni + 2, ..., n1 + 1i = n; have valency 1'2:; O. l'hen

for odd i E JV

,

, f,( f .)(-l)j ~ r~ir~'t(Gf)=O,

i~n-i n - i fc.f

ifl=j

(1.51)

(lnd for even i E JV

,

n ( 7' ) i.+ii-ni

~ (-l)p(U) 2C(U) = ~ (_l)i+j-n '" 1'2 .

~ .:. ,/ i

UEOJI, i=n-i n - i ,jc.f

ifl=j

i . -n

-;+12- 2 t(G

tE),

1'- d

2

(1.52)where in the last sum of (1.51) and (1.52) 1i = l!! nfl, f2 = iqy n fi (i1 + f2 = f).

Rem

a~k 1. For regular multigraphs of positive degree 1', w

e may use the relation

Go(J,) = (-l)n PG(-), + 1')

( 1.33)

(Section 1.3) instead of (1.49), equate corresponding coefficients and apply Theorem

s1.3 and 1.4 (instead of 1.5). T

he relation connecting the coefficients ai of PG

().) andC

j of GO

(Å) is

ai = (- l),i I: ,( f .) ,/i+

i-n Cn-j (i =

0, 1, ..., n) (1.53)i~n-i n - i

with ao =

Co =

1, and with (1.36) (T

heorem 1.3) and (1.47) (T

heorem 1.4) w

e arriveagain at T

heorem 1.6.

By inversion of (1.53) w

e obtain

_ in (f ).i+j-n '_

Ci - (-1) L

, . 1 an-j (i - 0,1, ..., n),i=n-i n - i

( 1.53')

1!

1.8. On the num

ber of walks

43

and (1.36) and (1.47) now yield the follow

ing system of equations equivalent w

ith(1.50) :

L t(G

f)~~,( f .)ri+i-n'L

. (_1)p(U)2c(U

) (i=O

,l,...,n), (1.50')"tc.f l=n-i n - i UEOJI,,-j

i;;î=n-i

where for f =

n the last sum is taken to be 1.

With i = n - 1 we obtain from (1.50') a new formula for the number of spanning

trees contained in a regular multigraph, nam

ely

1 n

t(G) = - ~ f' ri-1 L (_l)p(U) 2c(U)

n i=i UEÓll n-j

( 1.54)

(see also Proposition 1.4).

Rem

ark 2. A general form

ula connecting cyclic strudure and tree structure of anym

ultigraph is, of course, contained in Theorem

s 1.5 and 1.5a (Section 1.6): From(1.48) and (1.48a), after m

ultiplication by IT d¡ w

e obtain

Theorem

1.7: Let G

= (!!, 4't) be any m

ultigmph w

ithoid isolated vertices, where

!! = JV

= r1, 2, ..., n). T

hen

L (-l)P(U) 2c(U) IT d/¡ ~ £ ,( f .) (_l)n-i+j 2: t(G f) IT di

UEiíll, hE.f-i/'(U) i=n-i n - i fc.f IEf

ifl=i

(i = 1,2, ..., n), where "Y(U) is the set of vertices of the basic ligiire U and where the

conventions t(Gø) = 0 (in(Z IT di - 1 are adopted.

IEØ

By specialization T

heorems 1.6 and 1.6a are obtained from

Theorem

1.7, but inthe general case the significance of Theorem 1.7 is constrained by the fact that the

terms depending on the valencies d/¡ or ell cannot be elim

inated.

1.8.O

n the number of w

als

,I,

In this section, "spectrum" alw

ays means "P-spectrum

".L

et A be the adjacency m

atrix of a multi-digraph G

with vertices 1, 2, ..., n. If, in

addition to the spectrum of G

, the eigenvectors of it. are known, then, of course, m

orestatem

ents concerning the structure of G can be m

ade than without this know

ledge.M

oreover, a multi-digraph G

with a sym

metric adjacency m

atrix - in partiçular,a m

ultigraph - is completely determ

ined by its eigenvalues and eigenvectors. For,if V

i, Vi, . ", V

n is a complete system

of mutually orthogonal norm

alized eigenvectorsof A

belonging to the spectrum (I'i, )'2' ..., )'n), let 17 =

(vi, V2, . ", vn) =

(Vij) and

A =

(åijÅi): then, as is w

ell known, IT

is orHiogonal(i.e., y-i =

VT

) and

A =

17 AV

T. (1.55)

Since G is determ

ined by A, w

e have proved

441. B

asic properties of the spectrum of a graph

Theorem

1.8: A m

ultigraph is completely determ

ined by its eigenvalues and corre-sponding eigenvectors.

So, in principle, any multigraph problem

can be treated in terms of spectra and

eigenvectors. (For example, an algorithm

for determining w

hether two graphs are

isomorphic, w

hich is based on Theorem

1.8, has been developed in (Kuhn).) From

this point of view, w

e shall now investigate the problem

of the number of w

alks of. given length in a m

ulti-(di-)graph G~ (R

ecall: A w

alk of length k ~ 0 is a sequenceof arcs U1U2'" 'uk- where the starting vertex of Uj+l coincides with the end vertex of Uj

(j = 1,2,..., k - 1), repetitions and loops being allowed.) Some more problems

concerning eigenvectors wil be considered in Section 3.5.

The starting point of our considerations is the follow

ing well-know

n theorem.

Theorem

1.9: Let A

be the adjacency matrix of a m

ulti-digraph G w

ith vertices1,2, ..., n, let Ak = (air); further, let Nk(i, j) denote the number of walks of length k

starting at vertex i and terminating at vertex j. T

hen

Nlc(i, j) = aijk) (k = 0, 1,2, ...). (1.56)

Note that for k =

0, (1.56) agrees with the convention N

o(i, j) = Ö

ij'

Now

let G denote a m

ultigraph and let V =

(Vij) be an orthogonal m

atrix of eigen-vectors of A

, as described above. Then, according to (1.55),

n(k) ~ ,k

aij - ~ VivVjvJlv .

v=1

( 1.57)

The num

ber Nlc of all w

alks of length k in G equals

n (n )2

Nlc =

tr Nlc(i, j) =

ti air = P

~l iE

Vip 2:.

Thus w

e have proved

Theorem

I.IO:t T

he total number N

k of walks of length k in a m

ultigraph G is given by

"N

lc = ~ C

,ì.:v=

1(~

= 0, 1,2, ...),

( 1. 58)

(" )2

where Cp = ,~Vip .

i~lIn the next theorem, the generating function for the numbers Nlc is expressed in

terms of the characteristic polynom

ials of the graph G and its com

plement G

.

Theorem

1.11 (D. M

. CV

ET

Kovró (C

ve8)): Let G

be a graph with com

plement G

, and00

let HG(t)

= ~ Nktk be the generating function of the numbers Nk of walks of length k

k=O

t Part of this theorem was proved in another way by D. M. CVETKovrc (Cve9) who also

proved the theorem in the present form

when preparing the m

anuscript of this book; thetheorem

was also partly used in (C

vS 1). A

nother proof was given by F

. HA

RA

RY

and A. J.

SC

HW

EN

K (H

aS 1).

ii

1.8. On the num

ber of walks

45

in G (k =

0, 1,2, .. .). Then

H,l') ~ : ii-i/õ ~~;r) - iJ

( 1.59)

Proof. If M is a non-singular square m

atrix of order n, let Pl1l denote the matrix

formed

by the minors of order n - 1 so that (W

IlT =

IMI M

-1. Let sum M

denotethe sum

of all elements of M

, and let J be a square matrix all entries of w

hich areequal to 1; then, for an arbitrary num

ber x,

( 1.60)1M + xJI = IMI + x

sum (M)

which can be proved by straightforw

ard calculations. Now

, according to Theorem

1.9,N

lc = sum

A Ie and since

00

~ Aktk =

(I - tAt1 =

II - tAI-1 (I - tA

lk=

O(It i ~

. (max 2i)-1),

we obtain0

0 00

~ sum A

ktk = ~ N

lctk = II - tA

!-l sum (I - tA

l,k=O k=O

i.e.,

H (t = sum (I - tAl

G ) II - tA

lW

ith M =

I - tA, x =

t, (1.60) yields

(1.61)

1 ( -

sum (I - tA

l = - I(t +

1) I + tA

l - II - tAl),

t

where A = J - I - A is the adjacency matrix of the complement G of G, and by

inserting (1.62) into (1.61), the equation

( 1.62)

L i t +

1 -\ L-- i - A

HG(t) = ~ (_1)" t - 1

t \~I-AI

is obtained. Clearly, (1.63) im

plies (1.59), which proves the theorem

.

Theorem

1.11 has been proved in (Cve8) by another m

ethod. P. W

. KA

ST

ELE

YN

IKas2) gave the expression

for the generating function for numbers of w

alks between

two prescribed vertices of a graph.T

he generating function HG

(t) wil be used in Section 2.2. T

he numbers of w

alksfor graphs of som

e special types wil be determ

ined in Section 7.5.

( 1.63)

,.Ii

461.9. M

iscellaneous results and problems

471. B

asic properties of the spectrum of a graph

Let (,ui, ,u2, ..., ,urnI be the set of distinct eigenvalues of a multigraph G

. (1.58) canthen be rew

ritten in the form1.9.

Miscellaneous results and problem

s

Nlc = Di,uf + D2,u~ +... + Drn,u~ (k = 0, 1,2, ...), (1.64)

where D

i, D2, ..., D

rn are non-negative numbers uniquely determ

ined by G; som

e(but not all) of them may be zero.

In particular, with k =

0 the equation

Ci + C2 +... + Cn = Di + D2 +... + Dm = No = n (1.65)

is obtained from (1.58) and (1.64).

D. M. CVETKOVIC (Cve9) gave the following

Definition. T

he main part 01 the spectrum

of a multigraph G

is the set of all thoseeigenvalues ,uj for w

hich in (1.64) Dj =

1 0 holds.

For a regular m

ultigraph of degree r with n vertices, clearly, N

lc = nrk: hence, for

regular multigraphs (and, in fact, only

for these) the main part of the spectrum

con-

Y- k/-

sists of the index only. In this case, ~lc = T which motivates 1 :lc in the gene-

ral case to

be considered as a certain kind of mean value of the valencies, in general

depending on k. This gives rise to the follow

ing

Y- k

Definition. Let G be a multigraph and d = d(G) = lim Nlc = lim VNlc (it wil

. k-'oo n k-'oo

be shown that the limit exists). Then d is called the dynamic mean 01 the valencies of

the vertices of G.

1. Let G

be a multi-(di-)gm

ph with vertex-set 11,2, ..., nj and let N

lc(i, j) denote the number

of walks of length k in G

joining i to j. If ivii is the corresponding generating function (i.e.,00

wii = ~ Nlc(i, j) tk) and TV = (wii)' then TV = (1 - tA)-l.

k=O (P. W. KASTELEYN LKas2J)

, JI+1

2. Let IR be the set of the greatest eigenvalues of all graphs. Let T - - (the golden

mean). For n =

1, 2, ..., let ßn be the positive root of 2Pn(x) = Xn+l - (1 + x + x2 + ... + xn-l).

Let IXn = ß~2 + (3112. Then 2 = IXl ~ IX2 -c ... are all

limit points of IR smaller than 7:1/2 + .-i12

= lim IXn-

n-++

oo(A

. J. HO

FF

MA

N LH

of13))

3. If a digraph G has at least one cycle then the index of G

is not smaller than 1; otherw

ise alleigenvalues of G

are equal to zero.(J. S

ED

LÁC

EK

LSed 1))

4. Let G

be a digraph with vertices 1, ,.., n. For given vertices i and j (i =

F j), a spanningsub

graph of G in w

hich

1 ° exactly one arc starts and no arc ends in i,

20 exactly one arc ends and no arc starts in j,30 all the other vertices have in and out degrees equal to 1,is called a connect'ioii /TOm i to j and is denoted by O(i -)- j). For i = j the vertex i is an iso-

lated vertex of O(i -)- i) w

hile all the other vertices have property 3°.With a square matrix

A =

(aii)~ we associate a w

eighted digraph DA

, defined in the following

way. T

he n vertices of D A

are numbered by 1, 2, ..., n and for each ordered pair of vertices i, j

there exists an arc in DA

leading from j to i and having w

eight aii'The product TV = W(L) of the weights of the arcs of a spanning linear sub

graph L is called

the weight of L

. The num

ber of cycles contained in a linear subgraph L is denoted by c(L

);!I denotes the set of all spanning linear subgraphs L

of DA

.T

he weight T

V(O

(i -)- j) and the number of cycles c(O

(i -)- j) of a connection O(i -)- j) are

defined analogously.

Then the cofactor A

¡i of the element aij is given by

Aii =

(-1)n- ~ (_1)c(C

(i--j) W(O

(i -)- j)),C

(i-'j)

where the sum

mation runs through all connections O

(i -)- j) from i to j of the digraph D

A.

Consider further the follow

ing system of linear algebraic cquations:

Clearly, Nlc = O(dk) (k -- (0).

Theorem

1.12 (D. M

. CV

ET

KO

viC (C

ve9)): For a m

ultigraph G, the dynam

ic 'mean

d(G) is equal to the index 01 G.

Theorem

1.12, together with the existence of d, follow

s imm

ediately from T

heorem1. 10 and the fact that am

ong the eigenvectors corresponding to the index of G there

is a non-negative one. .

An application of this theorem

to chemistry is described in (C

vG4).

We quote w

ithout proof

Theorem 1.13 (F. HARARY, A. J. SCHWENK (HaS

1)) : For a m

ultigraph G, the

lollowing statements are equivalent:

10 JI is the main part 01 the spectrum;

20 ~I is the minim

um set 01 eigenvalues the span 01 w

hose eigenvectors includes thevector (1, 1, ...,I)T;

30 JI is the set 01 those e-genvalues which have an eigenvector not orthogonal to

(1, 1, ..., I)T.

The proof can be perform

ed by means of T

heorem 1.10.

ii~ aiixi = b¡ (i =

1, ..., n).j=

iW

ith this system w

e associate a digraph D having vertices 0, 1, ..., n in w

hich the vertices1, ..., n induce the digraph D

A, corresponding to the m

atrix A =

(a¡i)'i and in which there is an

additional arc from vertex 0 to vertex i having w

eight -bi for every i E 11,2, ..., nj. T

hen

~ (_1)(C(o--j)) W(O(O --j))

C(o--j)

xi=-

(j = l, ..., n),

~ (_l)c(L) W(L)

LE

!I

where in the upper sum

the summ

ation runs through all connections 0(0 -)- j) in D.

(C. L

. CO

AT

ES L

Coat))

481. B

asic properties of the spectrum of a graph

1.9. Miscellaneous results and problem

s49

5. Let G

be the digraph corresponding to a square matrix A

of order n (see il. 4). Let M

¡ibe the cofactor of the i,j-elem

ent of 1,1 - A. T

henC

omllary: If G

is either regular of positive degree i' or semiregular of positive degrees 1'1' 1'2'

and if 12 is the index of G (i.e., 12 =

l' or 12 = ¥i'1i-2, respectively), then

1 0 ~ ,0

m =

- 12" ¿" 1\;.

2 v=1

71

Mii =

L À

n-k L (_l)c(c,,(i-.j)) W

(Ck(i -+

j)),k=

2 c.(i-'j)w

here in the second sum the sum

mation runs through all connections C

k(i -)- j) from i to j

which have exactly k vertices.

(1.69)

Problem. Is condition (1.69) sufficient for a graph G

to be either regular or semiregular ?

(F. R

UN

GE

(Rung))

9. The determ

inant of the adjacency matrix A

of a multigraph G

is given in terms of the tree

structure of G by

(J_ PO

NS

TE

IN (P

ons))6. Some i-emai-ks concerning the Q-specti-m of a multigraph. If G is a multigraph without iso-

lated vertices having components G

1' G2, ..., G

k then, clearly,

k

QG(À) = n QG,().

i=1

71¡AI = (-1)71 L (-l)i L (n di) t(G f)'

j=1 fc.liE

fIfl~j

(1.66)

QG

may now

be defined for multigraphs G

having isolated vertices in the following w

ay:i) If P is the "point graph" having exactly one vertex and no edge, set

(F. R

UN

GE

(Rung))

10. Let G be a m

ultigraph having n vertices with positive valencies ai, ..., dl/ T

hen the com-

plexity of G is given by

Qp(À) = À - 1.

(ii) If G is any m

ultigraph having components G

1' G2, ..., G

h setk

QG(À).= n QG;(À)

i=1

(1.67)t(G

) = lQ

%(1) =

n d¡ IÎ (1 _ i,~'),

2m L di v=

2where in is the number of edges and where the À: are the Q-eigenvalues of G.

(F, RU

NG

E (R

ung), (RuSa); see also (Sac 12))

11. Let G =

(ít, qy; '1t) be a bipartite multigraph w

ithout isolated vertices where

ít = IX

1' ..., xml, qy =

(xm+

l' ..., xm+

nl; let V, W

be the valency matrices of the sets ít and qy,

respectively, so that the adjacency matrix A

and the valency matrix D

of G are of the form

which is consistent w

ith (1.66).

If G is a m

ultigraph without isolated vertices, then, according to (1.5)' and (1.5)" (Section 1.2)

where

QG(À) = IH - ÃI = ¡H -A*I,

(1.68)

A = (0 B),

BT

0Ð

= (V

0) ,O

W1 1

à = D-1A = (aik), A* = D -"2AD -"2 = ( a¡k ).

di Yd¡dk

If G is the point graph P, set A

* = Ã

= (1), consistent w

ith (1.67).T

he matrix A

* is symm

etric, Ã is stochastic, so all Q

-eigenvalues of G are real, the largest one,

being equal to 1.T

he Q-spectrum

has many properties analogous or very sim

ilar to properties of the P-spec-trum not to be itemized here; a few examples shall be quoted:

The num

ber of components of G

is equal to the multiplicity of the Q

-eigenvalue 1 of G.

- Let G

be 0, multigraph w

ithout isolated vei'tices. G is bipartite if and only if Q

G( -À

) = Q

G(À

).- Let G be a connected multigraph, not the point graph. G is bipartite if and only if QG( -1) = O.

Let G be a multigraph and let kG denote the multigraph derived from G by replacing every

edge by exactly k parallel edges. Let A(G

) denote the adjacency matrix of G

, etc. Clearly,

A(kG

) = kA

(G), butA

*(kG) =

A *(G

),Ã (kG

) = Ã

(G). T

hus G, though uniquely determ

ined by A,

is not determined by A* or Ã. Multigraphs G and

kG have the sam

e Q-spectrum

. This obser-

vation is, of course, not meaningful w

hen only graphs are considered.

7. Let G be a multigraph without isolated vertices and put Ãl = (ãW) (i = 1,2, ...). Then

ãh1 equals the probability of reaching vertex j as the last point in a random w

alk of length istarting at vertex i.

8. Let G

= (ít, '1t) be a graph w

ith in edges and n non-isolated vertices having Q-spectrum

Spo(G

) = (À

1, À2, ..., 1'71)0. T

hen

i: À; =

2 L --

v=1 (i,j)E

'1t d¡ dj

respectively (B is an n X

m m

atrix). Put

V-IB = lU, W-1BT = M,

ai~(À) =

IH - M

MI, ip~(À

) = IH

- MM

I,

where I denotes the identity matrix of order in or n, respectively. Then, by a well-known

theorem of the theory of m

atrices,

Ànai~(À) = Àmip~(À).

Put

iai~(À) if n ~ in,

aiG(À

) =ip~(À

) if n ~ m.

Thus the order of the polynom

ial aiG(À

) is equal to min (m

, n), Note that 'P

G(I') is invariant

under the interchange of the vertex sets ít and qy. The polynom

ial aiG(I') is connected w

ithQG(À) by the formula

Àmin(m,n)QG(À) = Àmax(m,n)aiG(À2)

so that essential information contained in Q

G(À

) is already contained in aiG(À

). Thus, for bi-

partite multigraphs, it may be more convenient to use aiG(À) (or the corresponding ai-spectrum)

than QG(À) (or the Q-spectrum). For example, in terms of the ai-spectrum the complexity of G

is given byt(G) = ¡VI. IWI IT (1 - Å.) = 2 n d¡ n (1 _ Å.),

i .~2 L d¡ .=2

50 1. Basic properties of the spectrum

of a graph

where I is the number of edges of G, k = min (m, n), and where the X~ are the rp-eigenvalues

of G. If the last form

ula is applied to the complete bipartite graph K

m,n' the w

ell-known

formula

t(Km,n) = mn-1 . nm-1

(see (FiSe)) is imm

ediately obtained. (See also Section 7.6, p. 219.)

(F. R

UN

GE

(Rung), (R

uSa), (S

ac12J)

12. Let G

= (fl, '!; ql) be a bipartite m

ultigraph without isolated vertices. .W

ith the notationof no. 11, the com

plexity of G is given by

t(G) =

¡Wi, IW

- BW

-1BT

)il = ¡V

I . I(W - B

TV

-1B)jl'

where .i, j are arbitrary num

bers taken from (1,2, ..., m

l or (1,2, .,' " nL, respectively. (Recall

that Mi denotes the matrix which is derived from the square matrix M by simultaneously

deleting the i-th row and the i-th colum

n.)

(F. R

UN

GE

(Rung), (R

uSa), (S

ac12))

13. The considerations of no. 11 m

ay be taken as a starting point for developing a spectraltheory for hypergraphs: A

ny hypergraph H can be represented by its incidence gm

ph (Levi

graph) G =

L(H

) which is a bipartite graph w

ithout isolated vertices; conversely, every con-nected bipartite graph G

with m

ore than one vertex uniquely determines a pair of connected

hypergraphs H, IÏ w

hich are duals of each other (so that G is the incidence graph of H

as well

as of IÏ): Thus the rp-spectrum

of a connected hypergraph H m

ay be defined as the rp-spectrumof L(H) - a definition which has the advantage of being invariant under dualization.

For som

e more results on various spectra connected w

ith hypergraphs and graphs derivedfrom

them see (R

ung).

14. A balanced incom

plete block design (BIB

D)t B

can be considered as a special hypergraph Hw

ith the varieties and blocks of B being the vertices and hyperedges of H

, respectively. So thecom

plexity t of B m

ay be defined as the number of spanning trees of the incidence graph

corresponding to H. It turns out that t = t(B) is completely determined by the parameters

v, b, '/, k, il of B:

t(B) = lcb-v+1ilv-1vv-2.

(F. R

UN

GE

(Rung); see also (R

uSa))

Hi. S

how that the relation betw

een the characteristic polynomial P

G(il) of a graph G

and thecharacteristic polynom

ial SG

(il) of the Seidel adjacency m

atrix S of G

can be written in the

form

p,I'i ~ I ;:io s,i~" ~-t~ )'

1 + - H

G -

2il ilw

here HG

(t) is the generating function for the numbers of w

alks in G.

(D. M

. CV

ET

Kovic (C

velS))

t For the definition of a BIBD see Section 6.2, pp. 165/166.

i ¡

r,;¡2.

Operations on G

raphs and the Resulting Spectra

In this chapter we shall describe som

e procedures for determining the spectra and/or

characteristic polynomials of (directed or undirected) (m

ulti-)graphs derived fromsome simpler graphs. In the majority of cases we have the following scheme. Let

graphs Gi, ..., G

n (n = 1,2, ...) be given and let their spectra be know

n. We define

an n-ary operation on these graphs, resulting in a graph G. T

he theorems of this

chapter describe the relations between the spectra of G

i, ..., Gn and G

. In particular,in some important cases, the spectrum of G is determined by the spectra of Gi,"" Gn-

At the end of this chapter, in Section 2.6, w

e shall use the theory we have developed

to derive the spectra and/or characteristic polynomials of several special classes of

graphs.

2.1.T

he polynomial of a graph

Let G

1 = (ge, ~il and G

2 = (ge, ~2) be graphst w

ith thB (sam

e) set of verticesge =

(Xi' ..., xn), w

here ~i and ~2 are the sets of edges of these graphs. The union

Gi u G2 of the graphs Gi and G2 is the graph G = (ge, ~), where ~ = ~i U ~2' It is

understood that every edge from ~i is different from

any edge from ~2' even w

henthe considered edges join the sam

e pair of vertices. Tf A

i, A2, and A

are the adjacencym

atrices of graphs Gi, G

2, and Gi u G

2, respectively, then A =

Ai +

A2.

However, Gi u G2 depends not only on Gi and G2 but also on the numeration of

the vertices of these graphs. Therefore, the spectrum

of the graph Gi u G

2 is, ingeneral, not determ

ined by the spectra of Gi and G

2. Som

e information about the

spectrum of the union of graphs is provided by the follow

ing theorem from

generalm

atrix theory.

Theorem

2.1 (the Courant-W

eyl inequalities; see, for example,' (H

ofl1)): Let

À1(X

), ..., Àn(X

) (;'I(X) ~ À

2(X) ~ ... ~ À

n(X)) be the eigenvalues of a real syrnrnetric

t The "graphs" considered in this section are, in fact, m

ulti-(di- )graphs (loops being allowed);

see the general remark in the Introduction (p. 11). .

4*

3. Relations Between Spectral and Structural Properties of Graphs

In this chapter we shall describe only a part of the know

n relations between the

spectra and the structure of (multi-)(di-)graphs. T

hese relations represent, in fact,the m

ain topic of this book, and they can be encountered in all other chapters.A

s is well, know

n, there are some structural properties that are not uniquely

determined by the spectrum

, but even in these cases we can, on the basis of the

spectrum, frequently specify a range of variation of these properties. Therefore,

many inequalities for various num

erical characteristics (chromatic num

ber, dia-m

eter, etc.) appear in this chapter.In all theorem

s of this chapter we assum

e that either the spectrum or the eigen-

vectors of the adjacency matrix of a graph, or both, are given and that a certain class

to which the graph belongs is specified. If the spectrum

of the graph is given, we

assume that its characteristic polynom

ial is also known, and conversely. T

he al-gebraic and num

erical problems w

hich appear here are assumed to be solved. N

otethat in som

e cases the class of graphs to which the graph w

ith the given spectrumbelongs can be determ

ined by means of the spectrum

.L

et, as usual, A denote the adjacency m

atrix, let

PG

(À) =

IU - A

I = À

n + (t¡À

n-1 +... +

an(3.1)

be the characteristic polynomial, and p,¡, ..., À

nl the spectrum of the graph G

.

3.1. Digraphs

First we shall assume that multiple oriented edges and loops are allowed in the

digraphs to be considered. Before form

ulating some theorem

s we shall note a few

simple facts.

The num

ber of vertices of G is equal to the degree n of its characteristic poly-

nomial, i.e. to the num

ber of eigenvalues of G.

The num

ber of directed loops is equal to the trace of the adjacency matrix, i.e.

to the sum ;.¡ + ... + Àn, i.e.

to the quantity -a¡.. If every vertex of G

has the same num

ber of loops, then the characteristic poly-nom

ial PH(À

) of the digraph H obtained from

G by deleting all of its loops is com

-

3.1. Digraphs 81

pletely determined by P G

P'): If every vertex of G has exactly h directed loops,

a¡then h = -" - and PH(À) = PG(J, + h).

nIf G

is a digraph without loops, then no pair of vertices of G

is joined by two

edges of opposite orientation if and only if a2 = O

. This fact can be easily realized by

considering all principal minors of the second order of the adjacency m

atrix.From

Theorem

1.2 we deduce im

mediately: A

digraph G contains no cycle if and

only if all the coeffcients ai (i = 1, ..., n) are equal to zero, i.e., if and only if the

spectrum of G contains no eigenvalue different from zero (J. SEDLÁCEK (Sed 1)).

According to T

heorem 1.9, the num

ber of closed walks of given length k contained

in a digraph G can be determ

ined by means of the spectrum

of G; this num

ber isn

equal to tr Ak =

L À

f.i=

lU

sing the Cayley-H

amilton T

heorem, w

e deduce from the characteristic poly-

nomial (3.1) the follow

ing relations:

An+k + a¡An+k-1 + ... + a.nAk = 0 (k = 0, 1, ...). (3.2)

By m

eans of Theorem

1.9, we can obtain from

(3.2) some inform

ation concerningthe digraph structure.

Now

we shall establish som

e theorems concerning the cycle structure of a digraph

G w

ithout multiple edges. Som

e statements given in the foregoing- are special cases

of these theorems.

The length g(G

) of a shortest cycle in a digraph G (if such a cycle exists) is called

the girth of G. If G

has no cycles, then g(G) =

+00. O

bviously, each linear directedsubgraph of G

with less than 2g vertices, w

here g = g(G

), is necessarily a cycle.From

Theorem

1.2 we deduce

(i .c 2g).

ai = L (-1 )C

(Ll = - L 1

LEg, ë,cG

Thus -a¡ is the num

ber of cycles of length i contained in G.

Theorem

3.1 (H. SA

CH

S (Sac 3)): Let G

be a digraph with the characteristic poly-

nomial (3.1) and let g(G) = g. Let further i ;? min (2g - 1, n). Then the number of

cycles of length i contained in G is equal to -ai' T

he girth g of Gis equal to the sm

allestindex i for which ai =l O.

This result can be extended so that the num

ber of cycles of length i for some

i ? 2g - 1 can also be ,determined. We shall introduce a new notion: the d-girth

of a digraph. For an

arbitrary integer d? 1, the d-girth gd(G) of a digraph G is

defined as the length of a shortest cycle among those cycles the lengths of w

hich arenot divisible by d. If there are no such cycles, then gd(G

) = +

00.

Theorem 3.2 (H. SACHS (Sac

3)) : Let G

be a digraph with the characteristic poly-

nomial (3.1) and let g(G

) = g and gd(G

) = gd' L

et /urtheri ;? min (g +

gd - 1, n),i =1 0 (mod d). Then the number of cycles of length i contained in G is equal to -ni'

The d-girth gd of G

is equal to the smallest index not divisible by d for w

hich ai =l O

.

6 Cvetkovic/Doob/Sachs

823. R

elations between spectral and structural properties of graphs

Rem

ark. If g is not divisible by d, then, trivially, ,gd = g and T

heorem 3.2 states

less than Theorem

3.1. But in the opposite case, w

hen d is a factor of g, we certainly

have gd ? g. If, further, gd ? g + 1, T

heorem 3.2 yields new

information that is

not obtainable from T

heorem 3.1.

Exam

ple. Let g =

9, gg = 15, ga =

20, Theorem

3.1 yields the numbers of cycles

of length c for c ~ 17. In addition, with d =

9 Theorem

3.2 provides these numbers

for c = 19, 20, 21, 22, 23, and w

ith d = 3 also for c =

25, 26, 28.

With d =

2 we have the follow

ing corollary.

Corollary: T

he length g2 of the shortest odd cycle in G is equal to the index of the first

non-v(inishing coefficient among ai, aa, a5, ...; the number of shortest odd cycles is

equal to -ago

Proof of Theorem

3.2. Let i ~ m

in (g + gd - 1, n), i =

$ 0 (mod d). T

hen each lineardirected subgraph in G

with i vertices is necessarily a cycle. A

s in the above argument,

ai = ~

(_l)C(L)=

- ~ 1

LE!I, ë,c G

which com

pletes the proof.

From the corollary of T

heorem 3.2 w

e can easily deduce the following theorem

.. í'

Theorem 3.3 (H. SACHS (Sac

3)) : A digraph G

has no odd cycles ifand only if itscharacteristic polynom

ial has the following form

:

PG().) =

iln + a2iln-2 +

a4),n-4 +... =

ilP. Q(il2),

where Q is a polynomial and p = 0 for n even, and p = 1 otherwise.

The follow

ing theorem can also easily be proved.

Theorem

3.4: A strongly connected digraph G

with greatest eigenvalue r has no odd

cycles if and only if -r is also an eigenvalue 0t G.

Proof. If G has no odd cycles then, by T

heorem 3.3, -r is also an eigenvalue of G

.

Conversely, if -r belongs to the spectrum

of G then the adjacency m

atrix of Gis im

primitive. A

ccording to Theorem

0.3 (Section 0.3), the index of imprim

itivity hcan in that case be only an even num

ber. By the sam

e theorem, there exists a per-

mutation matrix P such that PAP-I has the

form (0.1). S

ince h is even, G obviously

contains no odd cycles.

This com

pletes the proof.

A digraph G

is said to be cyclically k-partite if its vertex set fl can be partitionedinto non-empty

mutually

disjoint sets fli,""fl/c so that, if (x,y) (XE

fli,yEflj)

is an arc of G, then j - i _ 1 (mod k). Note that a cyclically k-partite digraph is

also cyclically l-partite if k is divisible by l. The adjacency m

atrix of a cyclicallyh-partite digraph has the form

(0.1). According to (D

uMe) w

e can formulate the

following theorem

.

3.1. Digraphs

83

Theorem

3.5: The characteristic polynom

ial of a cyclically k-partite digraph G has

the form

I

PG(Â) = ilP. Q(ilk), (3.3)

where Q

is a monic, Q

(O) =

! 0, and p is a non-negative integer.If G is a strongly connected digraph and if its characteristic polynomial is of the

form (3.3), then G

is cyclically k-partite.

The follow

ing theorem is taken directly from

the theory of matrices (see, for

example, (G

ant), voL. II, p. 63).

Theorem

3.6 : Let d-i, ..., d;; and dt, . . ., d-; be the indegrees and outdegrees, respectively,

of the vertices of a digraph G. T

hen, for the index r of G, the follow

ing inequalities hold:

min d¡ ~ r ~ max d¡ ,

i i

(3.4)

min dt ~ r ~ m

ax dt .i i

(3.5)

If G is strongly connected, then equality on the left-hand side or on the right-hand side

of (3.4) (or of (3.5)) holds if and only if all the quantities d¡, ..., d;; (or dt, ..., d-;) areequal.

Theorem

3.7 (A. J. H

OF

FM

AN

, M. H

. McA

ND

RE

W (H

oMe)): F

or a digraph G w

iththe adjacency m

atrix A:

1° There exists a polynom

ial P(x) such that

J = P(A),

(3.6)

if and only if G is strongly connected and regular.

2° The unique polynom

ial P(x) of least degree such that (3.6) is satisfied is nS(x)jS(d)where (x - d) S(x) is the minimal polynomial of A (ind d is the degree

of G.

3° If P(x) is the polynomial of least degree such that (3.6) is satisfied, then the degree

of G is the greatest real root of P(x) =

n.

Proof. Assum

e that (3.6) holds. Let i, j be distinct vertices of G

. By (3.6), there is

some integer k such that A

k has a positive entry in position (i, j), i.e., there is some

walk of length k from i to j. So G is strongly connected. Further, from (3.6)

follows

that J comm

utes with A

. Let ei, dj be the outdegree and the indegree of vertex i andvertex j, respectively. N

ow the (i, j) entry of A

J is e;, and the (i, j) entry of JAis di. T

hus ei = dj for all i and j, so G

is regular, i.e., all row and colum

n sums of A

are equal (A being not necessarily sym

metric).

To prove the converse assum

e G to be strongly connected and regular. D

ue tothe regularity, u = (1, 1, ..., l)T is an eigenvector of

' both A and AT, corresponding

to the eigenvalue d. Hence, if d has m

ultiplicity greater than 1, it must have at

least one more eigenvector associated w

ith it. But because of the strong connected-

ness, u is the only eigenvector corresponding to d. It follows that, if R

(x) is the

6*

843. R

elations between spectral and structural properties of graphs

minimal polynomial of A, and if Sex) = R(x)/(x - d), then Sed) =1 O. We then have

o = R(A) = (A -dI) S(A).

(3.7)

Let 0 be the zero-vector. Since R

(A) t' =

0 for all vectors v, it follows from

(3.7) that

(A - dI) SeA) v = 0,

so SeA

) v = IxU

for some ix.

Let (U

, v) be the scalar product of vectors u and v. If we take (V

, u) = 0, then

(AkV, u) = (v, (AT)k u) = dk(v, u) = 0 for every k and so (S(A) v, u) = O. Therefore,

o = (S

(A) v, u) =

(IXU

, u) = nix, i.e., ix =

o.T

hus SeA) v =

0 for all v such that (v, u) = 0; further, SeA

) u = Sed) u. H

encenS

(A)/S

(d) = J, i.e., (3.6) is satisfied w

ith

nP(x) = - S(x).

Sed)

This completes the proof of 1°; part 2° follows since the polynomial (3.8) has

smaller degree than the m

inimal polynom

ial of A. T

o prove 3° we note that A

isnon-negative and has royv and colum

n sums all equal to d. T

hus, the eigenvalues ofA are all of absolute value not greater than d. The roots of P(x) are eigenvalues

of A and hence, for real x? d, IP

(x)1 is an increasing function in x. From

(3.8),P(d) = n and so, since P(x) is a real polynomiaL, P(x) ? n for x ? d.

This com

pletes the proof of Theorem

3.7. We call (3.8) the polynom

ial belongingto G and also say that G belongs to the polynomial.

Note that som

e non-regular graphs can have a polynomial w

ith similar properties

(Bri1). I

(3.8)

3.2.G

raphs

If a multi-digraph H

has a symm

etric adjacency matrix A

with even entries on the

diagonal, then the matrix A

can be understood as the adjacency matrix of an (un-

directed) (multi-)graph G

. In such a way w

e can apply the result from Section 3.1

to graphs. But now

, due to the symm

etry of the adjacency matrix, w

e have some

further results.T

he eigenvalues of a graph are real numbers, and w

e can order them so that the

sequence Ji1, ..., Àn is non-increasing. T

his convention wil alw

ays be adopted.In the sequel we shall consider only undirected graphs without multiple edges

or loops.

The follow

ing theorem can be proved using argum

ents directly from m

atrixtheory.

Theorem

3.8 (L. CO

LLAT

Z, U

. SIN

OG

OW

ITZ

(CoS

i1)): Let ìl be the mean value of the

valencies and r the greatest eigenvalue of a graph G. T

hen

ìl ~ r,

where equality holds if and only if G

is regular.

(3.9)

3.2. Graphs

85

Proof. As is w

ell known, since the adjacency m

atrix A =

(aij)~i of G is H

ermitian,

the problem of finding the m

aximal value of R

ayleigh's quotient

I

II n

¿ ¿

aijXiX

jR

i=1 j~

1n'V 2

l. Xi

i=1

(3.10)

(the Xi being arbitrary real num

bers not all equal to zero) has the solution R =

1'.T

he maxim

um is attained if and only if the X

i (i = 1, ..., n) are the com

ponents ofan eigenvector of ~

4 belonging to 1'.If w

e put Xi =

1 (i = 1, ..., n) in (3.10), w

e have

_ 1 n

R =

d = - L di,n i=1

n

where di =

L aij is the valency of vertex i. So, ìl is a particular value of R

ayleigh'sj~1

quotient establishing (3.9).For regular graphs equality holds in (3.9), since in that case" the greatest eigen-

value of G is equal to the degree of G

. Let, conversely, equality hold in (3.9). T

heilthe values X

i = 1 (i =

1, ..., n) constitute an eigenvector for A belonging to r, and

n n

L aijxj = 1'Xi (i = 1, ..., n) implies di = L aij = r (i = 1, ..., n). Thus, G is regular.

j~1 j~ 1

This com

pletes the proof of the theorem.

Applying T

heorem 3.6 to graphs and using T

heorem 3.8, w

e get

dmin ~ d ~ l' ~ dmax'

where dm

in and dmax are the m

inimal and m

aximal values, respectively, of the valen-

cies in G.

We continue w

ith some m

ore propositions relating the coefficients ai of PG

(À) to

some structural properties of G

.Due to the absence of loops, we always have a1 = O.

The num

ber of closed walks of length 2 is obviously equal to tw

ice the number 11

1 ii 1 n

of edges, therefore m = - L Jil. In a similar way the formula t = - L ˴ for the

2 i~1 6 i=1

number t of triangles can be obtained. N

ow, T

heorem 1.3 gives m

= -ci2 and

t = - l- a3. According to the same theorem, the coefficient a4 is equal to the number

2 '

of pairs of non-adjacent edges minus tw

ice the number of circuits 04 of length 4

contained in G.

In a similar w

ay the coefficient as is equal to twice the num

ber of figures consistingof a triangle and an edge (triangle and edge being disjoint) m

inus twice the num

berof circuits 05 of length 5. These facts were noted in (CoSi 1).

863. R

elations between spectral and structural properties of graphs

An interesting conclusion can be draw

n from form

ula (1.36) for the coefficients ofthe characteristic polynomial (Sac

3). For i = n the 1-factors of G represent ony type

of basic figures. The contribution of a 1-factor to an is either 1 or -1, w

hile otherbasic- figures contribute an even num

ber to an' Therefore, the num

ber of 1-factorsof G

is congruent to an modulo 2. If an is odd, then there exists at least one 1-factor.

If G is a forest then, obviously, the num

ber of 1-factors is equal to janl with

nan =

(-1)2 if there is a 1-factor, and an = O

otherwise.t

For the proof of the following theorem

we need a sim

ple lemm

a which w

e statew

ithout proof. Both, L

emm

a 3.1 and Theorem

3.9, wil be used in Section 7.7.

Lem

ma 3.1: L

et lXI' ..., IX

k be real numbers and let 1', s (1' even, l' ~ s) be non-negative

integers. Then for a :: 0 the following implication holds:

IX~ + ... + ix!; ~ aT =? IlXf + ... + ixti ~ CtS.

Equality on the right-hand side of the im

plication holds if and only if the absolute valueof exactly one of the qiintities lX

I' ..., IXk is equal to a, the other quantities being all

eqiil to zero. Strict inequality on the left-hand side implies strict inequality on the right-

hand side of the implication.

Theorem 3.9 (E. NOSAL (Nos

1)): Let (J,i, ..., ,1n) be the spectrim

of a graph G. T

hen, he inequality

;,i :: ,1~ +

,1~ +

.., + ;,~

implies that G

contains at least one triangle.

Proof. A

ccording to Lemm

a 3.1, (3.11) implies

I it2,1T I ~

,1i

(3.11)

and we obtain for the num

ber t of triangles

1 3 1 ~ 3 1 3 1 I.~ 31 0

t = - ,11 + - ~ ;,¡ ~ - ;'1 - - kJ ,1i :: .

6 6 i~2 6 6 i=2

This com

pletes the proof.11

Since ¿

,1f = 2m

, where m

is the number of edges of G

, we get the follow

ing¡~l

corollary.

Corollary: If ;'1 :: V

m, then G

contains at least one triangle.

The corollary of T

heorem 3.2 can be reform

ulated for (undirected) graphs in thefollow

ing way. Let us consider, together w

ith a graph G, the digraph H

which has

the same adjacency m

atrix as G. T

o each shortest odd circuit of G there correspond

t From (1.35) foIlow

s: The num

ber of directed 1-factors (= linear directed subgraphs w

ith nvertices) of any digraph G

is not smaller than Ian!'

3.2. Graphs

87

I

exactly two shortest odd cycles (w

ith opposite orientations) of H and therefore the

number of shortest odd circuits in G

is half the number .of the shortest odd cycles

in H. T

hus, we have the follow

ing theorem.,

Theorem 3.10 (H. SACHS (Sac

3)) : Let G

be a grapht with the characteristic polynom

ial

(3.1). Then the length f of a shortest odd circuit in G

,is equal to the index of the firstnon-vanishing coefficient am

ong ai, a3, a5, ... The num

ber of shortest odd cÙ'cuits

1is equal to - - ai'

2

IiJII

An im

mediate consequence of this theorem

is the following:

Theorem

3.11: A gm

ph t containing at least one edge is bipartite if and only if itsspectrum

, considered as a set of points on the real axis, is symm

etric with respect to the

zero point.

Theorem 3.11 is one of the best-known theorems making

evident a close connection

between the structure and spectra of graphs. It seems that the necessity part of

this theorem was first recorded in chemical

literature by C. A. COULSON, G. S. RUSH-

BR

OO

KE

(CoR

u) (chemists usually call it the "pairing theorem

").T

he entire theorem w

as proved by H. S

AC

HS

(Sac 7) in the form

of Theorem

3.3.It is of interest that this theorem

has been rediscovered several times. V

arious versions ofthe theorem

can be found in (CoS

i1), (Hof3), (C

ve1), (CoLo), (R

ou1), (Mari), (S

ac3).T

he characterization of connected bipartite graphs by Theorem

3.4 is also possible.W

e shall now consider the problem

of determining the girth of a graph. A

s in di-graphs, the girth of a graph G

is the length of the shortest circuits of G.

If we try to form

ulate a theorem sim

ilar to Theorem

3.1 for graphs, we encounter

the following difficulties: T

ogether with the graph G

, consider the digraph H w

hichhas the sam

e adjacency matrix as G

. If G contains at least one edge, then g(H

) = 2,

while g(G

) can at the same tim

e be arbitrarily large. Thus the girths of G

and Hare

not related.But it is easy to see the following. For i ~ g( G) there exist basic figures only for

i = 2q even, and each basic figure U

2q consists of q non-adjacent edges, so thatp(U

2q) = q and c(U

2q) = O

. Therefore,

r 0 for odd i

ai = (i ~ g(G)),

. (- l)q bq for i = 2q

where bq is the num

ber of basic figures consisting of exactly q non-adjacent edges.For i = g(G) basic figures can be of the described type (consisting of non-adjacent

edges; only for even i), or they can be circuits of length g(G). In the second case the

contribution of each such basic figure to ai is -2. If

J a¡ for odd i,ai = 1 a¡ - (- l)q bq for i = 2q,

t Theorem

s 3.10, 3.11 hold for multigraphs, too

883.2. G

ra,phs89

3. Relations betw

een spectral and structural properties of graphs

then ai = 0 for i ~ g(G) and -ag(G) is equal to twice the number of circuits of

length g(G).

So w

e have the following

Theorem

3.12 (H. S

AC

HS

(Sac3)): Let G

be a (multi-)graph w

ith the characteristicpolynom

ial (3.1) and let bq be the number of basic figures consisting of exactly q non-

adjacent edges. Let further

where p_, P

o, p+ denote the num

ber of eigenvalues of G sm

aller than, equal to, 01' greaterthan zero, respectively.

There are gm

phs for which equality holds in (3.13).

Proof. Let s =

Po + m

in (p_, p+). Suppose that there is a graph G

for which

Ix(G) / s holds. Thcn there is an induced sub

graph of G w

ith Ix(G) vertices containing

no edges. Thus, a principal subm

atrix, of the order Ix(G), of the adjacency m

atrixof G

is equal to the zero-matrix. Since all eigenvalues of a zero-m

atrix are equal tozero, T

heorem 0,10 gives for the eigenvalues )'1' ..., ),,, of G

the inequalities

fa'

_ iai = ai _ (-l)q bq

for odd i,

for i = 2q.

).i ~ 0, À,,-o(G)+i ~ 0

(i = 1, ..., Ix(G)).

Then g(G

) is equal to the index of the first non-vanishing numbe1' am

ong lI, a2, ...,

and the number of circuits of length g(G

) is equal to - l- ag(G)'

(For regular graphs see also Theorem

s 3.26,3.27.) 2

How

ever, this contradicts the assumption Ix(G

) / s. Thus, (3.13) holds.

Equaliy holds in (3.13), for exam

ple, for complete graphs.

This com

pletes the proof of the Theorem

3.14.

A special case of this result is noted in (B

ax 1) This paper deals w

ith an adjacencym

atrix of a somew

hat different structure.Since the adjacency m

atrix A of a graph is sym

metric, w

e can determine its m

ini-m

al polynomial on the basis of the spectrum

. As is w

ell known, if rÀ

(l), ..., ),(m)) is the

set of all distinct eigenvalues of A, the corresponding m

inimal polynom

ial qi(íl) isgiven by

Theorem

3.15 (See (Cve9), (C

ve12), (Am

Ra), (R

of16)): Let P=

-i' P_i, andp~i denotethe num

ber of eigenvalues of the graph G w

hich are smaller than, equ(il to, or greater

than -1. Let ),* represent the smallest eigenvalue greater than -1. Let

further p = P=-l

+ P_i + 1 and s = min (p, P~i + P_i, l' + 1), where i' is the index (= maximum eigen-

value) of G, and let

qi(À) = (À -_ íi)) ... (À - íl(m)).

Let qi().) =

Àm

+ blÀ

m-i +

... + bm

. Then the follow

ing relations hold:

Am+k + bIAm+k-l + ... + bmAk = 0 (1: = 0, 1, ...). (3.12)

iU

sing these relations we can prove the follow

ing theorem ((N

os1), (Cve9); see also,

for example, (M

aMi), p. 123).

ix=f~

if ),* ~ p - 1,

if íl * / P - 1.

If K(G) denotes the 'maximum number of vertices

in a complete 8ubgraph of G, then

Theorem

3.13: If a, connected graph G has exactly m

distinct eigenvalues, then itsdiam

eter D satisfies the inequality D

~ m - 1.

Proof. Assum

e the theorem to be false. T

hen for some connected graph G

we have

D =

s ~ m. B

y the definition of the diameter, for som

e i and j the elements aW

)from

the i-th row and from

the j-th column of the m

atrices Ak (k =

1, 2, ...) areequal to zero for k ~ s, whereas aW =F O.

In (3.12), put k = s - m

. Making use of the relation so obtained, from

aW =

0(k = 1, ..., s - 1) we deduce aW = 0, which is a contradiction.

This com

pletes the proof of the theorem. '

The interior stabilty num

ber (\(G) of the graph G

is defined as the maxim

umnum

ber of vertices which can be chosen in G

so that no pair of them is joined by an

edge of G. .

Theorem

3.14 (D. M

. CV

ET

Kovic (C

ve9), (Cve 12)): T

he interior stability number

ix( G) of the graph G

satisfies the inequality

Ix(G) ~ Po + min (p_, p+), (3.13)

K(G

) ~ J sls-iX

(3.14)if s ~p,if s = p.

There are graphs for w

hich eq'uality holds in (3.14).

Proof. If G contains a complete subgraph with k vertices, then Theorem 0.10,

in a way sim

ilar to the proof of Theorem

3.14, yields the following inequalities:

Àn-k+

1 ~ 1: - 1 ~ )'1 = 1',

Àii-k+

i ~ -1 ~

)'i(i =

2, ..., k).

The greatest value of k satisfying these inequalities is given by the expression on

the right-hand side of inequality (3.14). Equality holds in (3.14), for exam

ple, for com-

plete multipartite graphs. T

his completes the proof of T

heorem 3.15.

In the paper (JIux 1) the author deals, among other things, with the connection

between the spectrum

of a graph and the maxim

um num

ber of vertices in a com-

plete subgraph. The proofs of the results announced in (JIux1) have, ,as far as w

eknow

, not yet been published.

903, R

elations between spectral and structural properties of graphs

Now

we shall discuss the relations betw

een the spectrum of a graph and its chro-

matic num

ber. It is surprising that on the basis of the spectrum, som

e information

about the chromatic num

ber (a quantity which in general cannot easily be deter-

mined) can be obtained. For som

e special classes of graphs the chromatic num

ber caneven be calculated exactly from

the spectrum (for exam

ple, for bichromatic

graphs: see Theorem

3.11; for regular graphs of degree n - 3, where n is the num

berof vertices: see Section 3.6). N

evertheless, in the majority of cases, w

e have some

inequalities for the chromatic num

ber. In general, these inequalities are not toosharp, but for each inequality there are graphs for w

hich this inequality yields agood (low

er or upper) estimate of the chrom

atic number. T

herefore, all known esti-

mates should be applied to the given graph, and then the best one should be

chosen.In general, how

ever, the chromatic num

ber is not determined by the spectrum

.M

oreover, A. J. H

OF

FM

AN

has proved that there is, in a certain sense, an essentialirrelevance betw

een the spectrum and the chrom

atic number of a graph (see Sec-

tion 6.1).

We shall now

present some theorem

s concerning the topic under consideration.W

e begin with a theorem

due to H. S

. WILF

.

Theorem

3.16 (H. S. W

ILF (W

il2J): Let X

(G) be the chrom

atic number and r ~he

index (= maximum eigenvalue) of a connected

graph G. T

hen

x(G) ;; r + 1.

(3.15)

Equ(ility holds if and only if G is (i complete graph or a circuit of odd length.

Proof. Let dmin(H) and dmax(H) denote the smallest and the greatest vertex degree

in a graph H and let ll(H) be the index of H. Since X(G) is the chromatic number

of G, there exists an induced subgraph H

of G w

ith dmin(H

) ~ X(G

) - 1. By T

heo-rem

s 0.6 and 3.8 we obtain

ll(G) ~

i.I(H) ~

dmin(H

) ~ X

(G) - 1,

(3.16)

and therefrom (3.15). L

et equality now hold in (3.15) and, consequently, also in

(3.16). Then ¡'i

(G) =

ll(H) im

plies G =

H, since G

is connected. Further ll(G)

= dmin(G), which implies, according to Theorem 3.8, that G is regular. Thus

X(G

) = 1 +

l' = 1 +

dmax(G

). The w

ell-known B

rooks Theorem

(see, e.g., (Sac9))now

implies that G is a complete graph or a circuit of odd length. This completes

the proof.

Before w

e quote a generalization of this theorem w

e shall give some definitions.

A graph G

is k-degenerate, for some non-negative integer k, if dm

in(H) ;; k for each

induced subgraph H of G

. The point partition num

ber ek(G) of the graph G

is thesm

allest number of sets into w

hich the vertex set of G can be partitioned so that

each set induces a k-degenerate subgraph of G. Since O-degenerate graphs are exactly

those which are totally disconnected, w

e see that eo(G) is the chrom

atic number

of G. Q

i(G) is called the point arboricity of G

, since 1-degenerate graphs are forests.

3.2. Graphs

91

l

It can be proved (see (LiW

h)) that every graph G contains an induced subgraph H

with dm

in(H) ~

(k + 1) (Q

k(G) - 1) O

n the basis of this fact the following theorem

(Lick) can be proved in a m

anner similar to the last one (for the special case k =

1,see also (M

i tc )).

Theorem

3.17 (D. R

. LIC

K (L

ick)): For any gralJh G w

ith inde;r r wid (m

y non-negative integer k,

Qk(G) S 1 + r~J.

- Lk + 1

For the proof of the following theorem

we quote a lem

ma w

ithout proof; both thelem

ma and theorem

have been proved in (Hof16).

L.em

ma 3.2: L

et A be a real sym

metricnw

trix of order n, and let Yi u .,. u Y

t(t ~ 2) be a partition of 11, ..., nl into non-empty subsets. Akk denotes the submatrix

of A w

ith row and colum

n indices from Y

k' If 0 ;; ik ;; IYkl, k =

1, ..., t, then

t-I tlii+i,+...+i+I(A) + ~ In-i+1(A) ;; ~ lik+1(Akk),

i=1 k=1

(3.17)

where Ài(X), i = 1, 2, ..., are the eigenva.lues of the matrix X in decreasing order.

Theorem

3.18 (A. J. H

OFFM

AN

(Hof16)): If r (r =

f 0) and q (ire the greatest and thesm

allest eigenvalues of the graph G, then its chT

omatic num

be'l' x(G) satisfies the in-

equality

l'

x( G) ~ - + 1.

-q

Proof. Let X

(G) =

t and let the vertices of G be labelled by 1, ..., n. T

hen there. existj, a partition Y

i u ... u Yt such that each of the subgraphs of G

induced by Yi

contains no edges. With ik =

0 (k = 1, ..., t), (3.17) yields for the eigenvalues

)'1 = 1', ,12' ..., 171 =

q of G

(3.18)

t-Ii' +

~,1n-i+l;; O

.i=

1(3.19)

.t-I

Since ~ In-i+1 ~ (t - 1) q, (3.18) follows from (3.19). This completes the proof of

i~1the theorem

.N

ote that from (3.19) there can be draw

n more inform

ation about the chromatic

number than from

(3.18). Actually, (3.19) yields the follow

ing bound:

X(G

) ~ 1 + m

in Ix I ,11 + ~ ,1n-i+

1 ;; ol.l( 1 i=

1 JT

he following theorem

provides another lower bound for the chrom

atic number.

923. R

elations between spectral and structural properties of graphs

Theorem

3.19 (D. M

. CV

ET

KO

vrC (C

vell)): If G is (¿

graph with n ve1.tices, w

ithindex r and chrom

atic number X

(G), then the follow

ing inequality holds

x(G)~~n - r'

Proof. Consider the characteristic polynomial of a k-complete. graph K"".."u"

which is given by (2.50) or. (2.51). T

he polynomial (2.50) has a single positive root

which is sim

ple. Indeed, as is seen in Theorem

6.7, complete m

ultipartite graphs areprecisely those connected graphs with a single positive eigenvalue. Thus for

x:: 0, PK (x) ~ 0 if and only if

x ~ 1.1'

ni'....nkN

ow consider the values of

Ie

L~

;=11. + n' ' 1.:: 0,

i

Ie

Ln; =n.

¡~1(3.20)

Assunie for the m

oment that the n;'s can assum

e positive real values. Then (3.20)

attains its maxim

um w

hen all the n;'s are equal. Indeed, if n¡ =1 ni' then by letting

ni = nj = l- (ni + nil and by leaving all other values unchanged, (3.20) is in-

2 k' t

creased. For the particular value 1. =

~ n, (3.20) is equal to 1 w

hen the n;'s arek

equal. Thus w

hen the n;'s are positive integers (2.50) is non-negative and hence

k -1íl~-n1 =

k(3.21)

with equalìty only w

hen the graph is regular.So w

e have proved

Lem

ma 3.3: T

he index r of K"".."u. satisfies

k - 1 kr ~ - n where n = L ni'

k ;=1

(In the Appendix the spectra of som

e k-complete graphs are given.)

H X

(G) =

k, the set of vertices of G can be partitioned into k non-em

pty subsetsso that the subgraph induced by anyone of these subsets contains no edges. If them

entioned subsets contain n1,..., nk (n1 + ... +

nk = n) vertices, respectivlIy,

then by adding new edges to G

we'can obtain K

n"....Uk It is know

n (see Theorem

0.7)tJiat the index of a graph does not decrease w

hen new edges are added to the graph.

Therefore, the index of G is not greater than the index of KU",..,uk According to

this and the foregoing. r ~ k - 1 n, which implies k ¿ ~ .

, - k -n-r

This com

pletes the proof of the theorem.

The follow

ing theorem of H

OFFM

AN

and HO

WE

S esta,blishes the existence of anupper bound of another type for the chrom

atic number.

3.2. Graphs

93

i1

Theorem

3.20 (A. J. H

OF

FM

AN

, L. HO

WE

S (H

oHo)): Let m

(G) be the num

ber ofeigenvalues of a graph G

not greater than - 1. Then there exists a function f such that

X(G) ~ f(m(G)). .

Proof. Let e =

e(G) be the largest num

ber such that G contains a set of 2e vertices

1, ..., e, 1', ..., e' with i and i' adjacent (i =

1, ..., e), other pairs of vertices beingnot adjacent. L

et K(G

) be the maxim

um num

ber of vertices in a complete sub-

graph of G. Using Theorem 0.10 we simply obtain K(G) ~ 1 + m(G), e(G) ~ m(G)

(see also Theorem

3.15). Hence, w

e have only to prove that X(G

) is bounded by some

function of K

(G) and e(G

). This w

il be done by induction on K(G

). If K(G

) = 1,

then X(G

) = 1. F

or a given graph G, let G

¡ (O¡,), i =

1, ..., e, be the subgraph of Ginduced by the set of vertices adjacent to i (i/). S

ince K(G

;), K(G

;,) -e K(G

) ande(G

;), e(Gi,) ~

e(G), the induction hypothesis can be applied to G

i (G;,). B

ut the setof vertices of G

contained in no G; or G

i, induces a subgraph without edges. T

hisfact is sufficient to prove the theorem

.

It was conjectured in (H

o!16) that f(m(G

)) = 1 +

m(G

). But as w

as observed in(H

oHo), this is false since C

7, the complem

ent of a circuit of length 7, provides acounter exam

ple. In (Am

Ha) it w

as mentioned that the inequality X

(G) ~ P +

1,w

here p is the number of non-positive eigenvalues of G

, might possibly be valid. It

can be shown that at

least one of the inequalities X(G) ~ rk (G), X(G) ~ rk (0)

(rk (G) being the rank of the adjacency m

atrix of G) holds (N

uf2). On the basis of

this and some other facts, it can be conjectured that, except for K

n, X(G

) ~ rk (G

),w

here equality holds if and only if the non-isolated vertices of G form

a complete

multipartite graph (Nuf2).

Some other bounds for the chrom

atic number w

il be given in Sections 3.3 and 3.6.

Now

we shall give som

e bounds for certain quantities connected with the par-

titioning of the edges of a graph G.

Let b( G) be the smallest integer k such that there exists a partition

Æ'i u... u Clk = qt,

Æingi=

0(i, j = 1,2, ..., k; i =1 j)

(3.22)

of the set úl of edges of G and such that the subgraph G

; of G, induced by g;, is a

complete graph for each i =

1, ..., k. Let £(G

) be the smallest integer k such that

(3.22) holds and each G; is a complete iiultipartite graph. Further, let #(G) be the

smallest integer k such that (3.22) holds and each G

; is a bicomplete graph.

Theorem 3.21 (A~ J. HOFFMAN (Hof 11)): Let 1.1, ..., J,u be the eigenvalues of a

graph G

.

Let p+, p_, P

be the number of eigenvalues w

hich are positive, negative, different fromboth -1 and 0, respectively. T

hen

£(G) ~ p+, #(G) ~ p_,

b(G) ~ -Jon, r + 2å(G)) ~ p.

This theorem

was proved by m

eans of the Courant-W

eyl inequalities (see Theo-

rem 2.1). In the proof several

lemm

ata appear. They are included in S

ection 3.6.

943. R

elations between spectral and structural properties of graphs

The essential irrelevance (in a sense) of the graph spectrum

with respect to å(G

)and 8(G

) has also been shown in (H

ofll); fJ(G), how

ever, is closely related to thespectrum

of G.

3.3. Reguar graphs

In the theory of regular graphs, numerous new

theorems are valid that do not hold

for non-regular graphs. Naturally, all theorem

s of section 3.2 hold also for regulargraphs.

We shall start with the question: How can it be decided by means of its spectrum

whether or not a given graph is regular?

Theorem

3.22: Let )'1 =

r, )'2' ..., )'n be the spectrum of a graph G

, T being the index

of G. G is Tegular if and only if

1 ~ ,2 =

T.

- ~/''.l

n ;=1

(3.23)

If (3.23) holds, then G is regular of degTee r.

PToof. Since the mean value d of the vertex degrees in G is given by

_ 2m in

d = - = - ~ Â~ (m is the number of edges), Theorem 3.22 is a corollary of

n n i=1

Theorem

3.8.

This theorem

is implicitely contained in (C

oSil). See also (Cve7). It can be easily

modified for the case w

hen the existence of loops in some vertices of G

is allowed.

If G contains m

ultiple edges or multiple loops, T

heorem 3.33 can be applied for the

establishment of regularity.

The follow

ing theorem is obvious.

Theorem

3.23: The num

beT of com

ponents of a Tegular graph G

is equal to the multi-

plicity of its index.

Theorem

s 3.22 and 3.23 wil be used several tim

es in this book. In many theorem

sa graph G is required to be 1° regular, or 2° regular and connected. These conditions

can be replaced by the following ones: 1° The spectrum of G satisfies (3.23), 2°

The

spectrum of G

satisfies (3.23), and r is a simple eigenvalue. T

hus, in such theorems

the assumptions concerning the general graph structure are only seem

ingly of a non-spectral nature.

The follow

ing theorem is taken from

(Fine).

Theorem

3.24 (H.-J. FIN

CK

(Fine)): Let z be il¡e num

ber of ciTcuits O

2 of length 2 ina T

egular multi-graph G

of degTee T

with n veT

tices and without loops. T

hen 4z = -2a2

- nT, w

heTe a2 is the coefficient of Â

n-2 in the characteristic polynomial of G

.

3.3 Regular G

raphs95

Proof. Let aij be the elem

ents of the adjacency matrix. T

hen a2 is given by

l'\ i 0 aij 1- - l- ~ ~ ?,"- -" .. g- (¿,¡ .

kj aji 0 2 i=1 j=1

If Zij is the num

ber of circuits O2 containing the vertices i and j, then Z

ij = (a~

j) andw

e get

1 n n n n

4z = 4. - )' "z,. = " ~ (a?, - a',) = -2a2 - nr

"- g- '1 g-.. ,¡ il .

2 i=1 j=1 i=1 j=1 .

a2

Corollary: G has no multiple edges if and only 1:j 2a2 = -nr.

Since for graphs the minim

al polynomial is obtainable from

the spectrum, T

heo-rem

3.7 takes now the follow

ing form. .

Theorem 3.25 (A. J. HOFFMAN (Hof3)): For (i graph G with adjacency matrix A

there exists a polynomial P(x), such that P(A

) = J, if and only if G

is regular and con-nected. In this case w

e have

n(x - ),(2)) ... (x - Â(m))

P(x) - ,

(1' - Â(2)) ... (r - Â(m))

where n is the num

ber of vertices, l' is the index, and Â(I) =

r, ),(2), ..., ),(m) are all distinct

eigenvalues of G.

This im

portant theorem provides great possibilities for the investigation of the

structure of graphs by means of spectra. It w

il be used many tim

es in the sequel.

We proceed now

to the investigation of the circuit structure of regular graphs. vVe

shall apply Theorem

3.12 and a result from (Sac4). C

onsider a regular graph G.

According to (S

ac4), in regular graphs the number bq occurring in T

heorem 3.12 can

for q -: g(G) be expressed in terms of q, the number of vertices n, and the degree 1"

of G. S

ince nand r are obtainable from the characteristic polynom

ial 'of G, the fol-

iowing result is im

mediately obtained.

Theorem 3.26 (H. SACHS (Sac

3)) : The girth g and the num

ber of circuits of length gof a regular graph G are determined by the corresponding characteristic polynomial

PG(Â

).

N ow

.we can go further and extend the w

hole of Theorem

3.12 to the case of regulargraphs. W

e shall again use a result from (Sac4).

Consider the basic figures U

i with i (g ~ i -: 2g) vertices contained in G

. Let U

~be those basic figures w

hich contain no circuits (i.e. which contain only graphs K

2as com

ponents) and let Ai be their num

ber. For odd i there are obviously no basicfigures U~. For i = 2q we have Ai = bq (numbers bq b~ing defined in Theorem 3.12),

and the contribution of U~ to the corresponding coefficient (- l)i ai of the charac-

teristic polynomial of G

is (- 1 )q\= (- 1) +

. Let us further consider those basic figures

U~ w

hich contain a circuit of length c. Clearly, g ~ c ~ i; U

¡ contains exactly one

963. R

elations between spectral and structural properties of graphs

circuit, since, by hypothesis, the number i of vertices of U

¡ is smaller than 2g. N

ow,

i - c vertices, not belonging to that circuit, are vertices belonging to i - c graphs2

K2; thus, c =

i (mod 2) m

ust be valid. The contribution of a basic figure' U

~ to

( -l)i a¡ is, according to Theorem

1.3, equal to

i-c i+c

- 1+-

(-1) 2 .(-1)C+

1.2=2.(-1) 2

The last form

ula holds also in the case c = i, w

hen Uj reduces to a circuit of length i.

Now

, for each c with g ;; c ;; .i, i _ c (m

od 2) the number B

¡ of different basicfigures U

¡ must be determ

ined.L

et a have exactly Dc circuits of length c and let these circuits be denoted by

C1 (j =

1, ..., Dc)' If c =

i, we have B

j = D

i, while in the case c ~ i w

e have thefollow

ing situation:Let a1 be the subgraph of a induced by those vertices not lying on C

1. Then the

number of basic figures U

¡ which contain a fixed,circuit C

~ is obviously equal to the

number E

~.c of forests, containing exactly i ~

c graphs K2 as com

ponents, in a1.

According to (S

ac4) the number E

L depends only on i, n, 1', and c, but neitheron j nor on the special structure of a. T

herefore we can om

it the upper index j and,since the num

bers nand l' are directly obtainable from Pe(J..), w

e can assume that

the numbers E

i.c are also given through Pe(J,).So we have for c ~ i

D,

Bi = L E~,c = Ei,cDc'

i=1

If b(Ui) is'

the contribution of the basic figure Ui to the corresponding coefficient

(-I)iai of Pe(J,), we obtain

(-l) ai = L b(U?) +

u?i

L L b(U~) + L b(Uj)

gS;c~i Uc

c ~ i (mod2) i

uii

and hence for even ii i+c

- 1+-

ai = (-1) 2 b ¡ + L (-1) 2 2Ei,cDc - 2Di,

"2 !JS;c~i

c~O

(mod2)

(3.24)

and for odd i

i+c

ai= L (-1)22Ei.cDc-2Di.

g~c.cic~1 (mod2)

(3.25)

These form

ulas hold if i is smaller than 2g.

By a recursive procedure, equations (3.24) and (3.25) can easily be solved w

ith

3.3. Regular graphs

97

respect to the desired numbers D

¡. If, for example, g is even, using T

heorem 3.12 w

eobtain in order from

(3.25):

1 _

D - -- ag+1'

g+1 2

from (3.24):

1Dg+2 - - 2 (ãg+2 - 2Eg+2.gDg),

where, according to T

heorem 3.26, D

g is the known num

ber of circuits of length g,from

(3.25):1

Dg+3 = - 2 (ãg+3 - 2Eg+3,g+1Dg+1),

etc. The num

bers ãj defined in Theorem

3.12 are, as has already been stated, deter-m

ined by 1', n, and aj, ~.e. by Pe(J,).T

hus, we have proved the follow

ing theorem.

Theorem 3.27 (H. SACHS (Sac3)): Let a be a regular graph with gÙth g and with

the characteristic polynomial (3.1). L

et h ;; n be a non-negative integer not greater than2g - 1. Then the number of circuits of

length h, which are contained in a, is determ

inedby the largest 1'00t r and the first h coefficients ai, a2, ..., a~ of the characteristic poly-

nomial of a.

The following theorem establishes a spectral property of self-complementary

graphs. A graph a is self-com

plementary if it is isom

orphic to its complem

ent G.

Self-com

plementary graphs w

ere primarily studied by G

. RIN

GE

L (Ring) and

H. SACHS (Sac1) (see also (Clap), (Rea 1)). If a is a regular self-complementary

gi'aph, then a is connected and has n = 4k +

1 vertices and degree r = 2k (R

ing),(S

ac1). We shall assum

e n? 1, i.e. k ~ 1. A

ccording to Theorem

2.6,

Å - 2k Pe(-Å

_ 1),PG(Å) = Pe(J,) = - À + 2k + 1

orPe(J,)

Å - 2k

Pe(-À - 1)

-Å - 1 - 2k

If Åi (i =

2, 3, ..., 4k + 1) are the eigenvalues of a different from

the index )'1 (i.e.,)'i =

f r = 2k), then4k+l 4k+l 41c+l

n (Å - J'i) = n (-Å - 1 - Ài) = I1 (Å + 1 + Àj).

i~2 i~

2 j-2To each eigenvalue Î'i =f 2k there corresponds another eigenvalue )'j = -Ài - 1,

where Å

j =f Å

i, since otherwise Å

i = - l- w

hich is impossible due to the fact that Å

i2

7 Cvetkovic/Doob/Sachs

983. R

elations between spectral and structural properties of graphs

1 1

is an algebraic integer. Thus, .1i+1 ? - - and .1j = .1n+1-i ~ - - fori = 1,2,..., 2k,

2 2'

and

)'i+1 +

.14k+2-i =

-1\ (3.26)

(i = 1,2, ..., 2k),

giving rise to

the following theorem

.

Theorem

3.28 (H. SA

CH

S (Sac1)): The characteristic polynom

ial of a regular self-com

plementary graph has the form

Ü+l Ü

PG

(.1) = (), - 2k) IT

(.1 - )'i) (.1 + .1i +

1) = (.1 - 2k) IT

(.12 + .1 - IX

i),i~

2 i~l

where IX

i = .1T

+ ¡ +

)'i+1'

~ote that this theorem

also follows from

Theorem

2.10 (p. 59); see also the foot-note to p. 57.

Formula (3.26) implies )'2 ~ 2k - 1, since in the opposite case we should obtain

the impossible relation )'4k+1 ~ -(2k - 1) - 1 = -1' (note that a self-complementary

graph with n ? 4 cannot be bipartite, thus .14k+1 ? -1'; see Theorem 3.11).

The converse of T

heorem 3.28 does not hold. N

amely, there are connected regular

graphs with 4k +

1 vertices which have the characteristic polynom

ial of Theorem

3.28 and which are not self-com

plementary. Such graphs w

il be mentioned in C

hapter 6(see exam

ples of cospectral pairs of graphs consisting of a graph G and of its com

-plem

ent a).A

statement ,sim

ilar to Theorem

3.28 can be made for non-regular self-com

ple-m

entary graphs. As a sim

ple consequence of Theorem

2.5, we obtain the follow

ingstatem

ent (Cve8), (C

ve9):

Let G

be a self-complem

enta1'Y graph. T

hen to each eigenvcilue )'i of G of m

ultiplicityp ? 1 (if the1'e'is such an eigenvalue) there corresponds another eigenvalue .1¡ whose

multiplicity q satisfies the inequality p - 1 ~

q ~ p +

1, where .1i +

.1j = - 1.

We shall now

discuss some theorem

s which are closely connected w

ith the conceptof the V

-product of graphs introduced in Section 2.2. A

graph is called V -prim

e ifit cannot be represented as a v-product of tw

o graphs.

Theorem

3.29 (H.-J. FIN

CK

, G. G

RO

HM

AN

N (FiG

r)): Let G

be a regula1' connectedgraph of degree l' with 1' vertices. G can be 1'ep1'esented as a v-product of p + 1 (p ~ 0)

V -prim

e graphs if and only if l' - n is a p-fold eigenvalue of G.

Proof. G can be represented as a v-product of p +

1 V-prim

e factors if and onlyif a has p +

1 components. A

ccording to Theorem

3.23, this situation arises if andonly if the graph a, whose index is ;¡ = n - l' - 1, has the number n - l' - 1 as a

(p + I)-fold eigenvalue. B

y virtue of Theorem

2.6, the last statement is equivalent

to the statement that l' - n is a p-fold eigenvalue of G. This completes the proof.

This theorem

enables us to calculate several lower and upper bounds for the

chromatic num

ber X(G

) of regular graphs G w

hich are not V-prim

e.

3.3. Regular graphs

99

Let us first consider lower bounds. Obviously, X(G¡ V G2) = X(G¡) + X(G2).

Therefore,

Ii

x(G) ~

k(3.27)

if G can be represented as a v-product of k v-prim

e factors.L

et G be a connected regular graph of degree l' w

ith n vertices, n ? l' + 1 (com

-plete graphs are thereby excluded, but this lim

itation is not essential). The m

ulti-plicity of an eigenvalue .1 of G

wil be denoted by Pi.'

Let Pr-n + 1 = k ? 1. According to Theorem 3.29, G can be represented as a

v-product of k v-prime graphs, say, G

. (v = 1,2, ..., k). E

ach of the G. is regular

and its degree r. and number n. of vertices satisfy the equation n. - r. = n - l' (see

Section 2.2, p. 57).S

uppose that among the k v-prim

e factors of G there àre exactly m

¡ monochro-

matic, i.e. totally disconnected graphs. F

or such factors G., X

(G.) =

1, and for theother k - m¡ factors G., x(G.) ~ 2. So,

x(G) ~ m¡ + 2(k - m¡) = 2k - mi'

(3.28)

Assum

e further that exactly m2 of the factors G

. are bichromatic, i.e. bipartite of

positive degrees 1'., and let m =

2m¡ +

m2' T

hen .

x(G) ~ m¡ + 2m2 + 3(k - m¡ - m2) = 3k - m.

(3.29)

So, any upper bound for m¡ or m

wil, by virtue of (3.28) or (3.29), autom

aticallyyield a (possibly trivial) low

er bound for X(G

). iB

efore we can outline a m

ethod of finding upper bounds for 1n¡ or m, w

e needtw

o more definitions.

').0 Let (.1~, .1~

, ..., ),;,,) be the spectrum of a graph G

'. Then the fam

ily (.1~,.1;, ..., .1~

,Jis called the reduced spectm

m of G

J.20 L

et ff¡, ff2, ..., ff. be subfamilies of a finite fam

ily ffo and let pu(e) be the multi-

plicity (possibly zero) with w

hich element e is contained inff u (0- =

0, 1, ..., s)., .

ff1, ff2, ..., ff. are called inclependent in ffo if ~ pu(e) ~ po(e) for each element e

of ffo' . u~¡

Now

, according to Theorem

2.9, the reduced spectra of the v-prime factors G

. ofG

constitute a family of independent subfam

ilies of the spectrum of G

. Evaluating

the conditions which the spectra of totally disconnected and regiilar bipartite graphs

must satisfy, w

e can in principle easily obtain upper bounds for m¡ and m

.Suppose G

* to be a totally disconnected V -prim

e factor of G. T

hen 1'* = 0 and

n* = n - r. T

he reduced spectrum of G

consists of n - l' - 1 ? 0 numbers all

equal to zero: so, zero is contained in the spectrum of G

with a m

ultiplicity notsmaller than m¡(n - l' - 1). Thus Po ~ m¡(n - l' - 1) and, consequently,

m¡ ~

( Po J .

n-r-1(3.30)

(3.30) is a fortiori true if m¡ =

O.

7*

1003. R

elations between spectral and structural properties of graphs

Recall that k =

P.r-n + 1. From

(3.28) we deduce

x( G) ~ 2p.r-n + 2 - ( Po J .

n-1' - 1

(3.31)

A better estim

ate may be obtained by taking possible bichrom

atic v-prime factors

into consideration. Denote by Pl the set of regular bipartite V

-prime factors of G

having positive degrees: then IPlI =

m;. F

or every G, E

Pl, 1', ~

~ n, (note that a

regular bipartite graph of positive degree has an even number of vertices); m

oreover,

1', ~ ~ n, - 1, because a regular bipartite graph G' with 1" = ~ n' is bicomplete

and therefore not v-prime. T

he above inequality, together with the relation n, - 1',

= n - 1', im

plies 1', ~ n - l' - 2.A

n arbitrary subfamily Y

= (,ui, ,u2, ...,,un -iJ of the spectrum

of G can be the

g'reduced spectrum of a regular bipartite V -prime graph G, E Pl of some degree

1', = i ? 0 only if the follow

ing conditions be satisfied:

a) n - l' + 1 ~ ng' =

n - l' + i ~ 2(n - l' - 1);

b) ng' is even;c) -i ~,ui ~ i (l =

1,2, ..., ng' - 1);d) the fàm

ily Y u iiJ =

(i, ,ui, ,u2, ..., ,un g'-i) is symm

etric with respect to the zero

point of the real axis (,u and -,u in Y u i iJ having the sam

e multiplicity);

n g'-i

e) ~,uf =

i(n - 1').i=

iT

hese conditions are either direct consequences of Theorem

s 3.11,2.9,3.22 or obvious.In the cases i = 1 and i = 2 stronger conditions that Y must satisfy can be

formulated.

Case i = 1: In this case, G, has only complete.

graphs K2 as com

ponents.

f') ng' = n - r + 1 ~ 4 (since K2 is not V-prime);

~ 1 n-1'-l

g') Y contains num

ber 1 with m

ultiplicity - n g' - 1 = and num

ber -12 2

. h 1 . 1" 1 n - r + 1 . d h b

wit m

u tip icity - n g' = . an no ot er num

ers.2 2

Case i =

2: In this case, G, has only circuits of even length ~ 4 as com

ponents.T

he characteristic polynomial is of the form

a ( 2:7l)

ff!(À) =

IT .I À

- 2 cos -- ,aE

f! i~i IX(3.32)

where fl may be any partition of n - l' + 2 into even numbers ~ 4, and IX runs

through all elements of fl (see end of Section 2.1).

f") ny = n - r + 2 ~ 6 (since a circuit of length 4 is not V-prime);

3.3. Regular graphs

101

I

g") Y u i2J is identical with the family

of roots of the equation 1 f!(À) =

0 (see (3.32))for som

e partition fl of n - l' + 2 into even num

bers ~ 4.N

ote that conditions c)"-e) are consequences of conditions I'), g') or f"), g"),respectively.

Now

let S be a family of U

i + U

2 ~ k independent subfamilies Y

of the spectrumof G, exactly Ui of them having zero as their only element with multiplicity n - l' - 1,

and each of the remaining U

2 families satisfying the conditions given above for som

ei; such a fam

ily S wil be called feasible. T

here is, in particular, a feasible S = S*

(with Ui = u~, U2 = u~) 'which is identical with the family of the reduced spectra of

all monochrom

atic andbichromatic v-prim

e factors of G, thus u~ =

mi, u~ =

m2,

2u~ + u~ =

m. C

onsequently, the maxim

um value M

of 2ui + U

2, taken over allfeasible S, is an upper bound for m

, and so

x(G) ~ 3k - M

.L. L. K

RA

US

and D. M

. CV

ET

KO

VIÓ

(KrC

l) noted that all constraints to be satis-fied (in particular the condition of independence) can be given the form

of linearinequalities so that M

may be obtained as the solution of an integer linear pro-

gramm

ing problem w

hich we shall now

formulate.

Let Yi be the family having zero as its only element, with multiplicity n - r - i.

Determ

ine the set iY2, Y

3, ..., Y/J of all distinct (not necessarily independent)

subfamilies Y

of the spectrum of G

satisfying, for some i, the conditions given above.

Let the spectrum

of G contain the distinct eigenvalues ),(i) (i =

1,2, ..., d) with

multiplicities P

i (Pi +

P2 +

... + P

d = n). Let P

ij be the multiplicity in Y

j of theeigenvalue ),(i). If Y

j appears e:xactly Xj tim

es as an element of the feasible fam

ilyJ

Xi =

Ui, ~ :"C

j = U

2, and so the following

j~2S then, with the notation used above:

inequalities hold:

(3.33)X

j ~ 0 (j = 1, 2, ..., f) ,

I~ X

j ~ k,

j=i.r

~PijXj ~ Pi (i =

1,2, ..., d);j=

i

the last inequality is equivalent to the independence of the familes Y of S as

Isubfam

ilies of the spectrum of G

. Since 2ui + U

2 = 2xi +

~ Xj' it is clear that the

I j=2

maxim

um value of 2xi +

~ Xj' w

here the Xj are integers subject to the contraints

j~2(3.33)-(3.35), is equal to the m

aximum

value M of 2ui +

U2'

SO we have proved

(3.34)

, (3.35)

Theorem 3.30 (L. L. KRAUS, D. M. CVETKOVIÓ (KrCl)): Under the assumptions

Im

ade above, let JJI be the maxim

um value 0/2xi +

~ Xj' w

here the Xj (j =

1,2, ..., I)j=

2

1023. R

elations between spectral and structural properties of graphs

are integers subject to the constraints (3.33)-(3.35). Then

x( G) ;S 3k - M

,where k = Pr-n + 1.

(3.36)A

long similar lines, som

e rougher (but more easily calculable) 100.;er bounds have

been obtained by H.-J. F

INC

K (F

inc) and again by L. L. KR

AU

S and D

. M. C

VE

TK

OV

IC

(KrC

1).N

ow w

e proceed to the determination of an upper bound for the chrom

atic number

of a regular graph which is not V

-prime.

Let, as earlier, G

be a connected regular graph of degree r with n vertices and let

the eigenvalue l' - n occur in the spectrum of G

with m

ultiplicity k - 1 (;S 0).T

hen the v-prime factors of G

are regular graphs G. of degree rv w

ith nv vertices,w

here r. = l' - n +

n. (v = 1, ..., k). A

ccording to the well-know

n theorem of

BR

OO

KS

(see, for example, (S

ac9)),

x(G.) ;: r. +

1(v =

1, ..., k),(3.37)

and so

k k

X(G) ;: ~ (1'. + 1) = ~ 1', + k = k(r + 1) - (k - 1) n.

(3.38)v=

lv=

!

This bound can be im

proved. In (3.37), equality holds only in the following four

cases

(a) rv = 0,

(b) 1'. = 1,

(c) r" = 2 and G

v contains a component w

ith an odd number of edges,

(d) rv;S 3 and Gv contains a complete graph with r" + 1 vertices as a component.

If s is the number of graphs G

, which satisfy one of these conditions, then

k

X( G) ;: ~ Tv + s.

v=l

We shall now

derive an upper bound for s. In order to simplify the analysis w

eshall assum

e that n - r is even. In this way graphs G

. satisfying condition (b)are excluded since such graphs, on the one hand, m

ust have n - l' + 1 vertices

(this is an odd number) and, on the other hand, m

ust have an even number of

vertices. ( J

The num

ber of graphs Gv satisfying condition (a) is not greater than P

o ,as mentioned earlier. n - r - 1

A graph G

v satisfying (c) cannot be connected since its number of vertices n.

= n - r + 2 is even. This means that the characteristic polynomial of Gv contains

a factor (íl - 2)2. Thus, according to T

heorem 2.9, the characteristic polynom

ialof G contains a factor íl - 2 which- stems from Gv' So the number of such Gv is not

greater than P2'

For each graph G

v which satisfies (d), a factor (íl +

l)'v appears in the characteristicpolynom

ial of G. S

ince rv ;S 3, the num

ber of such graphs Gv is not greater than

( p;il

3.4. Some rem

arks on strongly regular graphs103

In summarizing we obtain

I"

(Po J (p-iJ

s ;: + P2 + - ,

n-r-l 3

and, consequently,, (po J (p-i)

X( G

) ;: kr - (k - 1) n + +

P2 +

- .n ,- r - 1 3

(3.39)

Having in view

relations (3.38) and (3.39), we can form

ulate the following theorem

.

Theorem

3.31 (H.~J. FIN

CK

(Finc)): Let G

be a connected regular graph 01 degree rw

ith n vertices, where n - l' is even. L

et Pi. be the multiplicity 01 the eigenval1le íl in the

P-spectrum 01 G

. For the ch1'matic n1lm

ber X(G

) 01 the graph G the lollow

ing inequalityholds:

X(G

) ;: r + m

in (Pr-n +

1, (n _~~

_ 1) + P

2 + (p;iJ) - (n - 1') P

r-n'

If n - r is odd, a more com

plex analysis is necessary. It seems that a problem

of integer linear programm

ing, similar to the one treated above (see T

heorem 3.30),

wil have to be solved.

3.4. Some remai'ks on strongly regular graphs

Let x and y be any two distinct vertices of a graph and let LI(x, y) denote the num

berof vertices adjacent to both x and y. A

regular graph G of positive degree r, not the

complete graph, is called strongly reg1llar if there exist, non-negative integers e and 1

such that L1(x, y) = e for each pair of adjacent vertices x, y and L1(x, y) =

1 for eachpair of (distinct) non-adjacent vertices x, y of G

.The concept of a strongly regular graph was introduced by R. C. BOSE (Bas 1)

(1963), and at present there is already an extensive literature on this type of graph(see Sections 7.2 and 7.3).

From

Theorem

1.9 (Section 1.8) w

e deduce imm

ediately that a regular graph Gof degree r / 0 - not the com

plete graph - is strongly regular if and only if thereexist non-negative integers e and 1 such that the adjacency m

atrix A =

(aij) of Gsatisfies the follow

ing relation:

A2 =

(e - I) A +

IJ + (1' - /) I.

(3.40)

Theorem

3.32 (S. S

. SH

RIK

HA

ND

E, B

HA

GW

AN

DA

S (S

hBh)): A

regular connectedgraph G

01 degree l' is st'1ngly regular il and only il it has exactly three distinct eigen-val1les íl(i) = r, íl(2, ),(3). '

Ii G is strongly regular, then

e = r + ),(2)íl(3) + ),(2) + íl(3) and 1 = l' + íl(2)íl(3).

1043. R

elations between spectral and structural properties of graphs

Proof. Let G

be strongly regular. The eigenvalues of G

are not all equal, for if they.were, they would all be eqnal to zero - contradicting the hypothesis that G has an

edge. Nor can the spectrum

of G have exactly tw

o distinct eigenvalues since then Gw

ould have at least one edge and, according to Theorem

3.13, its diameter w

ouldbe equal to 1 - contradicting the hypothesis that G is not the complete graph.

Since the relation (3.40) holds for G, the m

inimal polynom

ial of the adjacencym

atrix A of G

has degree 3. Thus, G

has exactly three distinct eigenvalues.Let now

G have exactly three distinct eigenvalues .í(1) =

1', .í(2), .í(3). ¡Then, ob-

viously, l' ? 0 and G is not the com

plete graph and according to Theorem

3.25, therelation

aA2 +

bA +

c1 =J

(a =1 0)

(3.41 )

holds, where ),(2) and ),(3) are the roots of the equation a.í2 +

b.í + c =

O. A

com-

parison of the diagonal elements of the left and right side of (3.41) yields the equation

ar + c =

1, or c = 1 - ar.

If the vertices i, j are adjacent, i.e. if aii = 1, w

e deduce from (3.41) that the

number of w

alks of length two betw

een i and j èquals 1 - b . If i, j are distincta

and non-adjacent, the corresponding number of such w

alks is l- . Hence, G

is stronglya

regular. Com

paring (3.41) and (3.40), we obtain e =

l' + ),(2) +

),(3) + .í(2)),(3) and

f = l' + .í(2J.3).

This com

pletes the proof.

3.5. Eigenvectors

In Chapter 1 w

e have seen that the eigenvectors of the adjacency matrix of a (m

ulti-)graph G

, together with the eigenvalues, provide a useful tool in the investigation of

the structure of G. In this section we shall go into a bit more detaiL.

Sometimes valuable information about a (multi-)graph can be obtained from

its eigenvectors alone. Such a result is given by

Theorem

3.33: A m

ultigraph G is regular if and only if its adjacency m

atrix has aneigenvector all of w

hose components are equal to 1.

This theorem

is a consequence of a well-know

n theorem of the theory of m

atrices(see, e.g., (M

aMi), p. 133).

In (Bed), p. 131, the following result of r. H. WEI (Wei) is noted:

Let N

k(i) be the number of w

alks of length k starting at vertex i of a connected graph G

( n )-1

with vertices 1,2, ..., n. L

et sk(i) = N

k(i). ,~Nk(j) . T

hen, for k -)- 00, the vector)=

1(Sk(l), sk(2), ..., sk(n))T

tends towards the eigenvector of the index of G

.T

he question as to whether or not a given m

ultigraph is connected can be decidedby m

eans of Theorem

0.4: combining T

heorems 0.3 and 0.4, w

e obtain

3.5, Eigenvectors

105

I

Theorem

3.34: A m

ultigraph is connected 1:f a.nd only if its index is a simple eigenvalile

with a positive eigenvector.

Theorem

0.5 can also be translated into the language of graph theory:

Theorem

3.35: If the ÙuJ'ex of a m

ultigraph has multiplicity p, and if there Ù

a.positive eigenvector in the eigenspace corresponding to it, then G

has exactly p comi)onents.

Of particular interest are the eigenvectors of the line graph L

(G) of a connected

regular multigraph G

of degree 1'.'1 Let G

have 11 vertices and m edges. T

he relationconnecting the spectra of G

and L(G), nam

ely

PL(G)(/h) = (,u + 2)m-n PG(/h - l' + 2), (2.30)

has already been given in Theorem

2.15.Form

ula (2.30) establishes a (1, l)-correspondence between the sets of eigenvalues

.í =1 -1' of G and ,U =1 -2 of L(G): If ).=1 -r is a p-fold eigenvalue of Gtt, then

/h = .í +

l' - 2 =1 -2 is a p-fold eigenvalue of L(G

), and if /h =1 -2 is a p-fold

eigenvalue of L(G), then .í = /h - l' + 2 =1 -1' is a p-fold eigenvalue of G. Therefore,

we shall call (.í, p.) a pair of corresponding eigenvalues if ), =

1 -r is an eigenvalueof G, /h =1 -2 is an eigenvalue of L(G), and ), + l' = /h + 2.

Denote the eigenspace belonging to the eigenvalue .í of G

, or to the eigenvalue ltof L

(G), by X

V,) or Y

(/h), respectively. Then the follow

ing theorem holds.

Theorem

3.36 (H. SA

CH

S (Sac8)): Let G

be a connected regular multigraph of degree l'

with 11 vertices an(l rn edges, let (.í, /h) be a pair of corresponding eigenvalues of G

andL

(G), and let R

denote the 11 X m

vertex-edge incidence matrix of G

. ThenR

T m

aps theeigenspace X

(.í) onto the eigenspace Y(/h), and R

maps Y

(/h) onto X(.í).

Proof. As in Section 2.4, let A

and B denote the adjacency m

atrices of G and

L(G

), respectively.1. Let æ E X

(.í), æ =

1 0, and y = R

Tæ

. Then, by virtue of (2.27) (S

ection 2.4),R

y = R

RT

æ =

(A +

D) æ

= (A

+ 1'1) æ

= (J. +

r) æ =

1 0,

thus Y =

1 o. Using (2.28) and again (2.27), w

e obtain

By =

(RT

R - 21) Y

= R

TR

RT

æ - 2y =

RT

(.í + 1') æ

- 2y

= V, + r - 2) Y = /hY,

thus Y E

Y(/h), i.e.:

æ E

X(.í), æ

=1 0 im

plies RT

æ E

Y(lt), R

Tæ

=1 o.

I

t The line graph L

(G) of a m

ultigraph G w

ith edges 1,2, ..., m is a m

ultigraph with vertices

1, 2, ..., m arid adjacency m

atrix B =

(bik), where, for i =

1 k, bik = 0 if the edges i, k of G

have no vertex in common, bik = 1 if i, k are proper edges (i.e., not loops) having exactly

onevertex in com

mon, bik =

2 if i, k are proper edges having both their vertices in comm

on orif one of them

is a proper edge and the other one a loop haying a vertex in comm

on, bik = 4 if

i, k are both loops attached to the same vertex, and w

here bii = 0 if i is a proper edge, bii =

2if i is a loop."I R

ecall that Á =

~T

is an eigenvalue (of multiplicity 1) of G

if and only if G is bipartite

(see Theorem

3.4).

1063. R

elations between spectral and structural properties of graphs

2. In a similar w

ay it can be proved that

y E Y

(f-), y =1 0 entails R

y E X

(À), R

y =1 o.

3. If Y E

Y(f-), then there is a (unique) æ

E X

V,) such that R

Tæ

= y, nam

ely

1æ

=-R

y.f- +

24. If æ

E X

(À), then there is a (unique) y E

Y(f-) such that R

y = æ

, namely

1y=

-RT

æ.

À +

r

Theorem

3.36 is now proved.

Rem

ark. The m

appings Rand R

T considered in T

heorem 3.36 can be given a m

oreintuitive form

: the matrices R

, A, B

and the eigenvectors æ, y all refer to a fixed

labellng of the vertices and edges, respectively: for example, the com

ponent Xi of

an eigenvector æ of A

corresponds to vertex Vi, so w

e may w

rite x(v¡) instead of Xi,

or drop the subscript altogether and simply w

rite x(v) for the component of æ

thatcorresponds to vertex V

and, similarly, y(u) for the com

ponent of y that correspondsto edge u.

Now

let a, b denote both vertices, or edges, or one of them a vertex and the other,

one an edge, and let the symbol ~ m

ean that, for fixed b, the snmm

ation is to bea'b

taken over the set of all CL w

hich are adjacent to b or incident with b, respectively.

Then

y = R

Tæ

is equivalent to

y(u) = ~ x(v) for each edge u,

V'II

æ =

Ry is equivalent to

x(v) = ~ y(u) for each vertex v.

'U''V

Next the eigenvectors æ

of 0 and y of L(O

) belonging to -1', -2, respectively,shall be investigated.

Theorem

3.37: Let 0 be any connected m

ultigraph. Then 0 is bipartite il and only

il the system ol equations

~ x(v) = 0 lO1' each edge u ol 0,

(3.42)V

.'U

equivalent to RT

æ =

0,

has a (unique) non-trivial solution.Il, in particular, 0 is regular of degree l' and bipartite, the'l the solution ol (3.42)

equals the eigenvector belonging to )'11 = -1'.

The sim

ple proof may be left to the reader.

3.5. Eigenvectors

107

I

Theorem 3.38 (1\. DOOE (Do02), (Do08); for regular multigraphs see H. SACHS

(Sac8)): Let 0 be any connected m

ultigraph. Then y is an eigenvector ol L(O

) belongingto the eigenvalue -2 il and only il

~ y(u) = 0 lO

1' each vertex v ol 0, which is equivalent to R

y = o.

Ii'V

The proof is contained in the proof of T

heorem 6.11 (Section 6.3).

Corollary to Theorem 3.38: Il y is an ei'genvector ol L(O) belonging to

the eigenvalue

m

-2, then ~ Yi =

0, i.e.: the eigenspace Y( -2) is orthogonal to the vector (1, 1, ..., l)T

.i=

1Hence, in line graphs the eigenvalue -2 never belongs to the main part ol the spectrum.

Note that, in a regular multigraph, each eigenvector which does not belong to

the index is orthogonal to (1, 1, ..., 1)T.

Recall that a regular spanning subm

ultigraph (of degree s) of a regular multigraph

o is called a factor (s-factor) of O.

We shall now

establish, an interesting relation between the factors of 0 and the

eigenvectors of L(O).

A spanning subm

ultigraph Of of a m

ultigraph 0 can be represented by a vectorc =

(ci, C2, ..., C

m)T

, with c¡ =

1 if the i-th edge of 0 belongs to Of, and c¡ =

0otherwise; c is called the characte1'stic vector ol Of (with respect to 0).

Let 0 be a connected regular multigraph of degree r. According to Theorem 2.15,

the greatest eigenvalue of L(O) is 21' - 2 and the sm

allest eigenvalue is not smaller

than -2. Let ¡yI, y2, ..., yPl be a maxinal set of linearly independent eigenvectors

belonging to the eigenvalues of L(O) w

hich are greater than -2 and smaller than

21' - 2. It is easy to see that p = n - 2 if 0 is bipartite and p =

n - 1 in the oppo-site case; if n / 2, then p / O. Denote the m X p matrix with columns

yI, y2, . .., yP by JU.

The next theorem

provides a nieans for the investigation of the existence of ans-factor in 0, providing ltI is know

n.

Theorem

3.39 (H. S

AC

HS

(Sac8), (S

ac14)): Let 0 be a connected regular multigraph

ol degree r with n vertices and m

edges. A vector z w

ith m com

ponents is the characteristicvector ol an s-lactor ol 0 il and only il it satisfies the following conditions:

110 _ sn com

ponents of z are equal to 1, (ind the other components are equal to zero;

2

2° JUTz = O.

Proof. 1. Let z be the characteristic

vector of an s-factor Os of O

. Then 1° holds

trivially. In order to deduce condition 2°, consider the vector y = rz - syO, where

yO is a vector all of w

hose components are equal to 1 (note that yO

is the eigenvector

1083. R

elations between spectral and structural properties of graphs

of the index 21' - 2 of L(G

)). Then Y

i = l' - s if the j-th edge belongs to G

" andYi = -s otherwise. Further,

~ y(u) = s(r - s) + (1' - s) (-s) = 0

n"v

for each vertex v of G (or, briefly, R

y = 0). A

ccording to Theorem

3.38 this means

that y is an eigenvector belonging to the eigenvalue -2 of L(G), so y is orthogonal

to y1, y2, ..., yp. Since yO

has the same property, z =

~ (y +

syO) is also orthogonal

to each of y1, y2, ..., yp. Hence, MTZ = o. l'

2. Let now 1° and 2° hold for a vector z w

ith m com

ponents. Let y = 1'Z

- syo.T

hen

1 1 1

fyO, y) =

l' . -, sn - sm =

l' . - sn - s . - rn = 0

\, 2 2 2

and for each vector yi (i = 1,2, ..., p)

(yi, y) = r(yi, z) - S(yi, yO) = O.

Hence, y is orthogonal to yO

, yl, ..., yP and therefore y is an eigenvector belongingto the eigenvalue -2 of L(G). According to Theorem 3.38, ~ y(u) = 0 for each

u'vvertex v of G. Since the components of yare r - s or -s, it follows from the last

equation that for exactly s of the edges which are incident with v the equation

y(u) = r - s hplds and that y(u) = -s for the remaining r - s edges. Those edges

u for which y(u) =

l' - s (these are precisely the edges for which z(u) =

1) form an

s- factor of G.

This com

pletes the proof of the theorem.

Corollary to Theorem 3.39: If a vector z with m components, q of which a1'e equal

to 1 and the other m - q of w

hich are equal to 0, satisfies the conditionMT

z = 0, then

2q 0 (mod n) and z is the characteristic vector of ans-factor w

ith s = 2q.

nT

he proof of the Corollary is left to the reader.

Let"l be a subset of the set q¡ of edges of G. Then"l induces a regular factor if

and only if~ yi(U

) = 0

UE

"f(i=

1,2,...,p).

Let SP be the vector space generated by y1, y2, ..., yp. C

learly, each vector of SP is asolution of the follow

ing system of hom

ogeneous linear equations

L Y

( u) = 0,

UE

"f

where"l runs through all subsets of o¡ w

hich induce a regular factor. Therefore

the rank of the coefficient matrix of this system

is not greater than m - p. T

herows of this matrix are just the characteristic vectors of the regular factors of G.

3.5, Eigenvectors

109

Hence, the num

bèr of linearly independent characteristic vectors of regular factors isnot greater than m - p. So, if we call a set of regular factors independent (dependent)

if the corresponding characteristic vectors are linearly independent (dependent),

we have

Theorem

3.40 (H. SA

CH

S (Sac 8), (Sac14)): The num

ber of independent regula,rfactors of a connected regulaT

miÛ

tigraph G of degT

ee T w

ith n vertices and m edges is

not greater than

i ~m-n+

2

m-p=

1-m-n+

12

. if G is bipartite,

otheTw

ise.t

Moreover, H. SACHS (Sac 8) proved by direct construction that this bound is

attained for each n, T w

ith nT 0 (m

od 2) by some m

ultigraph, for n even in bothcases, and £01' n odd - naturally - only by non-bipartite multigraphs.

Several further results concerning eigenvectors of multigraphs can be found in

Sections 5.1, 5.2, 6.3.

Rem

ark (H. S.). M

any of the results stated above for regular multigraphs can be

generalized to arbitrary multigraphs if, instead of the ordinary spectrum

(= P-spec-

trum), the Q-spectrum and the corresponding eigenvectors (= Q-eigenvectors) are

utilized. (Recall that x is called a Q-eigenvector belonging to the Q-eigenvalue ).

if x =! 0 and x and ), satisfy A

x = À

Dx.)

The follow

ing propositions can easily be proved.

Let G

be a connected multigraph. T

hen

1 ° all Q-eigenvalues Ài are Teal and satisfy - 1 ~ Ài :: 1;

20 the maxim

al Q-eigenvalue À

1 (= "Q

-index") is a simple eigenvalue equal to 1;

30 the Q-eigenvector belonging to )'1 is (1, 1, ..., 1) T

;

40 any two Q

-eigenvectoTs x, X

l belonging to different Q-eigenvalues (/.re orthogonal in

the following sense: x T

Dxl =

0;

5° the following statem

ents are equivalent:(i) G is bipartite,

(ii) the Q-spectT

um of G

is symm

etric with T

espect to the zero point of the Teal axis,

(iii) -1 is a (necessaTily simple) Q-eigenvalue;

60 if G is bipartite, then the Q

-eigenvector x belonging to Àn =

- 1 satisfies Xi +

Xk =

0for each pair i, k of adjacent vertices.

In the sequel, suppose that G is an arbitrary connected m

ultigraph with n ~ 1

vertices and'm ~ 1 edges.

l' Note that-. rn - n + 1 = m - n + 1 is the cyclomatic number of G.

2

1103. R

elations between speG

tral and structural properties of graphs

Now

introduce two "m

odified incidence matrices"

R* = D-IR,

S* = -. RT.

2(3.43)

Both of them

are stochastic with respect to their row

s, and so are the "modified

adjacency matrices"

A* = R*S*,

B* = S*R*.t

(3.44)C

learly,

1 1 1

A* =

- D-IR

RT

= - D

-I(A +

D) =

- (D-IA

+ I),

2 2 2

(3.45)

B* = -. RTD-1R = B*T.

2(3.46)

Define the characteristic polynom

ials H

fG(t) = It I - A*I,

gG(t) = ItI - B*I

with corresponding spectra

P.i, 2~, ..., 2;,)¡,

(,ui, ,u~, ..., ,uii.)g'

By (3.44)tm

fG(t) =

t"gG(t),

(3.47)

and because of (3.46), all eigenvalues ).1', ,uj are real numbers.

By virtue of (3.45), D-IA = 2A* - I,

so

QG(t) = It

I - D-IAI-jtI - 2A* + II = 2" It ~ 1 I -A*I = 2"fGC ~ 1).

Consequently, if (21,22, ..., J,,,JQ

is the Q-spectrum

of a,

2:l _ 2i + 1

'i--2 '2i =

22: - 1(i =

1,2, ..., n).(3.48)

In connection with (3.47) and (3.48), Proposition 10 yields

o ~ 21 ~ 1

o ~,uj ~ 1

(i = 1,2, ..., n),

(j = 1,2, ..., m

),

(3.49)

t Note that the definitions ofR

*, S*,A*,B

* may be e:itended to hypergraphsw

ithR* =

D;lR

,S* =

D¡lR

T, w

here Dv is the "valency m

atrix of the set of vertices" and DB

is the "valencymatrix of the set of (hyper- )edges". These definitions lay a basis for a theory of a modified

"Q-spectrum of a hypergraph" and at the same time explain the apparent asymmetry in the

definitions of R*, S* given above (form

ula (3.43)). (See also Section 1.9, nos. 11, 13.)tt When dealing with hypergraphs, F. RUNGE (Rung) has used polynomials similar

to fG(t)

and gG(t); see S

ection 1.9, nos. 11, 13.

3.5. Eigenvectors

\111

and Proposition 5° (iii) m

ay now be restated as follow

s:

a is bipartite if and only if 2;' = O.

By (3.47) a sim

ple multiplicity preserving (1, I)-correspondence betw

een the sets ofall non-zero (i.e., positive) eigenvalues 2* of A

* and eigenvalues ,u* of B* is given,

namely 2* =

,u*.

If 2 is a Q-eigenvalue of a, let X

(2) be the corresponding eigenspace; then, clearly,

the eigenspace X*(J.*) of the eigenvalue ),* = 2 + 1 of the matrix A* is identical

with X(J,): 2

X*(J.*) =

X(2).

Denote the eigenspace of the eigenvalue ,u* of the m

atrix B* by Y

*(,u*). Then the

analogue of Theorem

3.36 holds:

Theorem

3.36': Let 2 * =

,u* ? 0 be corresponding eigenv(ilues of A *, B

*, respectively,and let J, =

22* - 1 be the corresponding Q-eigenvalue ? - 1. T

hen S* maps the

eigenspace X(J,) =

X*(J.*) onto the eigenspace Y

*(,u*), and R* m

(ips Y*(,u*) onto

X*(2*) =

X(J,).

Theorem

3.38 has the following analogue:

Theorem

3.38': y* is an eigenvector of B* belonging to the eigenvalue ,u* =

0 if andonly if R

*y* = o.

Now

define a generalized factor of an arbitrary multigraph a =

(!!, %') as follow

s:Let the valencies d(v) of the ve~

.tices v E !! have the greatest com

mon divisor 0, and

let 0 .c a ~ o. A spanning subm

ultigraph a' = (!!, %

") of a is called a a-factor iffor every vertex v E !! the valencies d(v) with respect to at and d(v) with respect

to a have the same ratio a: ö, i.e., if

ad(v) = - d(v) for every vertex v E !!.

o

A non-trivial a-factor can, of course, exist only if 0 ? 1.

Let ¡æ

1, æ2, ..., æ

Pj be a maxim

al set of linearly independent Q-eigenvectors

belonging to the Q-eigenvalues of a which are greater than - 1 and smaller than 1.

Clearly, p = n - 2 if a is. bipartite and p = n - 1 otherwise. Denote the n X p

matrix w

ith columns æ

i by E ançl put M

* = 2S*E

= R

TE

; then, according toT

heorem 3.36', the p colum

ns y*i of M* constitute a m

aximal set oflinearly independ-

ent eigenvectors belonging to the positive eigenvalues ,u* of B* w

hich are smaller

than 1. Now

the following theorem

which is a generalization of T

heorem 3.39 can

be proved in a way analogous to the proof of T

heorem 3.39.

Theorem

3.39': Let a be a connected multigraph w

ith m ~

1 edges. A vector z w

ithm

components is the characteristic vector of a a-factor of a if and only if it satisfies the

1123. R

elations between spectral and structural properties of graphs

following conditions:

a1° - In com

ponents of z aTe equal to 1, and the otheT

components of z aT

e equ,Û to 0;

o

2° J1tH Z = o.

An analogue to the C

orollary to Theorem

3.39 is also valid.T

he relations between the (generalized) factors of a (m

ulti-)graph and theeigenvectors of its line graph becom

e particularly evident when extended to hyper-

graphs (see also footnote on p. 110).

3.6.M

iscellaneous results and problemS

1. Let e and e be the indices of the graphs G and G

. Let G have n vertices. B

y use of Theorem

3.8 and relation (7.29), the following inequalities are easily obtained:

n - 1 ~ e + e ~ Y2(n - 1).

The left-hand inequality is actually an equality if and only if G

is regular.

(E. N

OSA

L (N

os1); A. T

. AM

IN, S. L

. HA

KIM

I (Am

Ha))

2. Prove that Ài ~ Ydm.x, where Ài = e is the index and dmax is the maximal valency of a graph.

(E. N

OS

AL (N

os1); L. LovÁsz, J. P

ELIK

ÁN

(LoPe))

3. Let G be a regular graph of degree r w

ith n vertices. Show

that for the number t(G

) of span-ning trees of G

the following form

ula holds:

(_1)nt(G

) =-P(j(-T

-1).n2

1 1

4. If G is a connected graph, neither a tree nor a circuit, then e :? T"2 + T -"2, where e is the

index of G and T

= .. (V

õ + 1).

2

5. Show

that the star has the largest index among all trees w

ith n vertices.

(A. J. H

OF

FM

AN

(Hof 13))

6. Each closed w

alk in a graph G can be represented as a sequence of vertices through w

hichit passes, for exam

ple, Xi' X

2, ..., Xn, X

i' The w

alks X¡, X

2' ..., Xk-i' X

h Xl and X

2' xa' ..., Xk, X

¡, x2are different because one starts from

. xi and the other from X

2 but are considered as cyclicallyequivalent: T

wo closed w

alks are called cyclically equivalent if one is obtained from the other

by rotating an initial segment to the end of the walk. Let Ck(G) be the number of cyclic equi-

valence classes of closed walks of length k in G

. Then

1 (k) n dCk(G) =-L ø - L À¡,

k dlk d i=i

where Ø

(k) denotes the Euler phi-function and À

i, ..., Àn are the eigenvalues of G

. À sim

ilarform

ula was obtained also for the num

ber of dihedral equivalence classes of closed walks in

a graph.

(F. HA

RA

RY

, A. J. SC

HW

EN

K (H

aS 1))

3.6. Miscellaneous results and problem

s113

7. Let A be'the adjacency m

atrix of a graph G w

ith n vertices, let f c: 11,2, ..., nj and letA

f be defined as in Section 1.5 (p.37). Then the num

ber of Ham

iltonian circuits of Gis,

given by

.. ~ (_1)8 L tr A

:ø.

2n 8~0 Ifi~

8(L. lV

1. LIHT

EN

BA

UlIi (JIl1x4j, (JIux5))

8. Let P_, P

o' p+ denote the num

ber of eigenvalues of a graph G, w

hich are smaller than, equal

to, or greater than zero, respectively. Then the chrom

atic number X

(G) of G

satisfies

X(G

)~ .n . ,P

o + m

m (p+

, p-)w

here n is the number of vertices of G

. This inequality is sharp; equality holds, for exam

plefor com

plete graphs.(D. M. CVETKOVIC (Cvell))

9. Let G be a regular graph of degree r = n - 3 with n (n ~ 3) vertices. Put,

(_1)n (), + n - 2)-i (À

- 2) PG(-Â

- 1) = À

"+ ai).n-i +

... + an

= (À

- 2)"" (À +

2)"'-' qi(Â),

where qi(2) =

1 0 and qi( -2) =1 O

. Then the chrom

atic number X

(G) of G

is given by

1X(G) = - (n + m2 - m_2 + aa)'

2(H

,-J. FIN

CK

(Fine))

10. Show

that the chromatic num

ber X of a graph G

is determined by the spectrum

of G if

the index e of G is sm

aller than 3. If G is connected, the sam

e statement holds also for e =

3.(D

. lV1. C

VE

TK

OV

IC (C

ve9J)

11. For a given k let a, b, c, d, e denote the numbers of eigenvalues Â

in the spectrum of a graph

G, which, respectively, satisfy the following relations: À ~ -k + 1, ), = -k + 1, -k + 1

~ À ~ 1, À = 1, Â:? 1. Further, let 8 be the smallest of the natural numbers k (k :? 1) for

which the inequality

min (b +

c + k(d +

e), k(a + b) +

(k - 1) (c + d)) ~ n

holds. Then X

(G) ~ 8.

(D. lV

1. CV

ET

KO

VIC

(Cve 12))

12. Let X

(G) be the chrom

atic number of the com

plement G

of a graph G. If G

is not a complete

graph, then_ (G) ¿ n + À2 - Âi

X - 1+À2 '

where n is the num

ber of vertices of G and À

i, )'2 are the first two greatest eigenvalues of G

.(A

. J. HO

FF

MA

N (H

of 16))

13. Let a(G

) be the smallest num

ber ,of subsets into which the vertex set of G

can be parti-tioned such that the subgraph induced by anyone of thè subsets is either a com

plete graphor a totally disconnected graph. L

et k be a positive integer and let G have eigenvalues À

i, ..., ).".T

hen there exist functions d k and éi k such that

a(G) ~ dk(À

2 - Ân-k+

i), a(G) ~ PJk(J'k - Â

n).(... J. H

OF

FM

AN

, L. HO

WE

S (H

oHo))

14. If 1/"(G) is the vertex set of a graph G

and if .Y c: "f(G

), then G!7 denotes the subgraph

of G induced by the set of ve,rtices of G

each of which is adjacent to all vertices in .Y

. Let

8 Cvetkovic/Doob/Sachs

1143. R

elations between spectral and structural properties of graphs

,1*(G) be the

smallest eigenvalue

of Gandlet k(H) denote

the cliquomatic num

berÎ of a graph H.

Then

10 there exists a function f such that if x E 1/'(G

),

k(G(x1) ~ Ip.*(G));

20 there exists a function g such that if x E "f(G

),

Ili i i is not adjacent to x, I"(G(x,iJ)I ? g(,1*(G

))lI ~ g(,1*(G

)).(A

. J. HO

FF

MA

N (H

of 15))

15. Prove that the Seidëi (-1, 1, OJ-spectrum (see Section 1.2) of a self-complementary graph

is symm

etric with respect to the zero point.

16. Let R be the vertex-edge incidence m

atrix of a connected multigraph G

having n vertices.T

hen

rk RJ n - 1

t n

if G is bipartite,

otherwise.

(H. S

AC

IrS (S

ac8); C. V

AN

NU

FF

ELE

N (N

uf1))

17. Let the edges U

i, U2,... U

2k form a closed w

alk (of even length) in a regular multigraph G

and let y be an eigenvector of L(G) not belonging to the eigenvalue -2. T

hen2k2: (_l)i Y(u¡) = O.

;=0

(H. SA

CH

S (Sac 8))

18. Let G

be a graph with eigenvalues ,1i, ..., ,1n- L

et c(G), #(G

), and o(G) be the quantities

defined as in Theorem

3.21. Then

c(G) =

1 if and only if ,12 ~ 0,

#(G) =

1 if and only if ,1,,-i ~ 0,

o(G) =

1 if and only if ,12 ~ 0 and ,1,, =

-1.(A

. J. HO

FF

MA

N (H

of11))

19. Let G

be a graph and let cm(G

) be the smallest integer k such that (3.22) holds and each

G; is a com

plete s-partite graph with s ~ m

. Then

cm(G) ~~,

m - 1

where p_ is the num

ber of negative eigenvalues of G.(D

. T. M

ALB

AŠ

KI, private com

munication)

20. A k-partition of a graph is a division of its vertices into k disjoint subsets containing

mi, m

2, ..., mk vertices, respectively, w

here mi ~ m

2 ~ ... ~ mk'

Let G

be a graph with adjacency m

atrix A, let U

be any diagonal matrix such that the

sum of all the elem

ents of A +

U is zero, and let fli, fl2' ..., flk (fli ~

fl2 ~ ... ~

flk) be thelargest k eigenvalues of A

+ U

. Then, if any k-partition a of G

is given, the number E

o ofedges of G

whose tw

o vertices belong to different subsets of a satisfies

1 k

Eo ~

- 2: (-mxflx)'

2 x~i

The right-hand sum

is a concave function of U.

(W. E

. DO

NA

TH

, A. J. H

OF

FM

AN

(DoH

o), cf. also (Fie3))

-r The cliquom

atic number of a graph G

is the chromatic num

ber of the complem

ent G of G

.

3.6. Miscellaneous results and problem

s115

21. The eigenvalues and eigenvectors of C = D - A (D is the valency matrix of a graph)

were used by K

. M. H

ALL (H

a, K) in a problem

of minim

izing the total length of the edgesof a graph w

hich is to be imbedded into a plane.

22. Define a relationship"" on the vertex set of a graph G

thus: x .. y if for every z =f x, y

the vertex z is adjacent to both or to none of vertices x, y, Let e(G) denote the number of

equivalence classes so defined. e(G) is not determ

ined by the spectrum of G

(see Fig. 6.1).B

ut A. J. H

OF

FM

AN

(Hof17) has proved that e(G

) is bounded from above and below

by some

functions of the number of eigenvalues of G not contained in the interval (~ (V'S - 1),1).

23. Let G be a regular graph of degree l' with n vertices. Let Gi be an induced sub

graph of Ghaving ni vertices and average vertex degree 1'1" Then

ni(i' - ,1,,) , , ~, .: ni(i' - ,12) + '

Î "'" = i i =

"'2'n n

This inequality w

as derived in (BuC

S) by the use of T

heorem 0.11. S

pecifying I'i = 0 and

Ti = ni - 1 we get the following inequalities for the-cardinalities Ix(G) and K(G) of an internal

stable set and of a complete subgraph of a regular graph G

, respectively:

Ix(G) ~

-nÀ"

'i" - ,1,,'K

(G) ~ (,12 +

1) nn-r+

,12'w

here ,12 and ,1,, are the second largest and the least eigenvalue of G. T

he first bound was found

by A. J. HOFF~IAN (unpublished). Together with Ix(G) X(G) ~ n it gives the bound from

Theorem

3.18 in the case of regular graphs. Similarly, the second inequality gives the bound

for the cliquomatic num

ber (see Section 3.6, no. 12). S

pecifying Ti in som

e other ways w

ecan get bounds for som

e more characteristics of a graph. T

he inequality for Ix(G) in the case

of strongly regular graphs was noted in (D

el1). The inequality can be extended to non-regular

graphs (W. HAEMERS (Haem)).

24. Further bounds for K(G

), defined in Theorem

3.15, can be obtained by the same technique

using the Seidel adjacency matrix instead of the (0, l)-adjacency m

atrix. For example, J. M

.G

OE

TH

ALS

and J. J. SE

IDE

L (GoS

4) found the inequality K(G

) ~ m

in (1 - 122' fli' fl2), where

G is a graph w

hose Seidel adjacency m

atrix has only two distinct eigenvalues 12i. 122 (l2i ? 122)

with the m

ultiplicities fli' fl2' respectively. Generalize this result for arbitrary graphs and,

in the case of regular graphs, express it in terms of the eigenvalues of the (0, l)-adjacency

matrix.

25. If D; (i =

3,4,5) are the numbers of circuits of length i in a regular graph of degree 1',

and if ai (i = 0, 1, ..., n) are the coefficients of the corresponding characteristic polynom

ial,then

1D3 = - - a3,

2

1 2

D4 = - (a2 + 2m2 - a2 - 2a4)

,41

D5 =

- (a3a2 + 3ra3 - 3a3 - a5).

2

26. If )'i' ..., ,1,, are the eigenvalues of a regular graph G,then the num

ber D4 of circuits of

length 4 in G is given by .

D4 =

.. (~ ,1t - nÂ

¡(2,1i - 1)) .8 ;=1

8*