M+(P) - UCSD Mathematics

SIAM J. MATRIX ANAL. APPL.Vol. 10, No. 3, pp. 407-423, July 1989

(C) 1989 Society for Industrial and Applied MathematicsOO8

THE RANKS OF EXTREMAL POSITIVE SEMIDEFINITE MATRICESWITH GIVEN SPARSITY PATTERN*

J. W. HELTON, S. PIERCE:, AND L. RODMAN

Abstract. Let P be a symmetric set of ordered pairs of integers from to n, and define M+(P) to be theclosed cone of all positive semidefinite Hermitian matrices whose (i, j) entry is zero whenever q: j and (i, j)is not in P. The extreme points ofM+(P) are considered. In some special cases, the maximum rank that suchan extreme point can have is calculated.

Key words, extremal matrix, order of a graph

AMS(MOS) subject classifications, primary 05C50; secondary 05B20, 15A57

1. Introduction. Let P be a symmetric set of ordered pairs of integers from to nand define M+(P) to be the closed cone of all positive semidefinite Hermitian matriceswhose (i, j) entry is zero whenever 4: j and i, j) is not in P. Two cases will be considered:(1) the matrices are over the field C of complex numbers; (2) the matrices are over thereal numbers R. Later, Pwill be naturally interpreted as an undirected graph with vertices

We say a matrix A in M+(P) is extremal ifA B + C for B, C M+(P) impliesthat B and C are scalar multiples ofA. Let J+(P) c M+(P) be the set of matrices thatare extremal points ofM+ (P). We say that P has order k if k is the maximum of ranksof matrices in the set J+(P). The problem of characterizing the orders of graphs P isimportant in several areas.

For one thing, it is related to the "positive completion problem" [DG], [GJSW]for matrices in the sense that we might think of the positive completion problem as astrictly easier one. Thus progress on the order problem is probably essential to makingprogress on the positive completion problem.

In this paper we put forward some general remarks and ideas that allow us to establishorders of new classes of graphs.

This paper is built on [AHMR] and heavily uses (in Part I) some ideas from theearlier paper M ].

The paper has three parts (in addition to the introduction and preliminaries sections).The first two address the theme of the relationship of the order problem to techniquesof Gaussian elimination for sparse matrices that are traditional in numerical analysis.This subject is devoted to doing the Cholesky decomposition LrDL of a sparse matrixwith the smallest number of algebraic operations. The fundamental problem is that fora given sparse matrixMwhen we perform a Gaussian elimination step to make a particularentry zero, usually several entries in Mthat are zero are made nonzero. This phenomenon

Received by the editors May 19, 1988; accepted for publication December 21, 1988.f Department ofMathematics, University ofCalifornia at San Diego, La Jolla, California 92093. The work

of this author was partially supported by grants from the National Science Foundation, the Office of NavalResearch, and the Air Force Office of Scientific Research.

Department of Mathematical Sciences, San Diego State University, San Diego, California 92182([email protected]). The work of this author was partially supported by a National ScienceFoundation grant.

Department ofMathematics, Arizona State University, Tempe, Arizona 85287, and School ofMathematicalSciences, Tel-Aviv University, Tel-Aviv, Israel. The work of this author was partially supported by a NationalScience Foundation grant. The work was done while this author, with support from the Office ofNaval Research,visited the University of California at San Diego, La Jolla, California.

407

408 J. W. HELTON, S. PIERCE, AND L. RODMAN

is known as fill in, generally, and the amount of it is generally very dependent on theorder in which one performs Gaussian elimination on the matrix. A major branch ofsparse matrix analysis is devoted to how we perform Gaussian elimination on matricesof a particular sparsity pattern so as to minimize fill-in (see P ], R], RT ], GL ).

In Part I we analyze a "divide-and-conquer" technique for the order problem andpositive completion problems. We show that if these problems can be solved for certainsubmatrices of a given matrix, then they can be solved for the full matrix (provided thatthe submatrices interact in a certain way). This is very similar to classical use of divide-and-conquer methods in the Cholesky decomposition for the case where there is noGaussian elimination fill in. While our results are simple, the matrix theoretic contentas opposed to the graph theoretic content ofall clear results on the order problem [PPS ],[M] are consequences of a few simple principles.

Use ofgraph theory techniques developed (by Rose) for the Cholesky decompositionwere introduced to the order problem by Grone et al. [GJSW]. Next Paulsen, Power,and Smith PPS used Rose’s methods more directly to give a very elegant proof of the[GJSW] theorem (that is close to the one we have here). They also found intriguingconnections with the completely positive maps that occur in operator theory.

While Part I analyzes the behavior oforder in situations that correspond to Gaussianelimination having no fill-in, Part II begins to treat cases with fill-in. There is a naturalmeasure a(G) ofthe minimum amount of fill-in producible by Gaussian elimination ona sparsity pattern G (see 5 ). We conjecture (for real matrices) that

order (G) _-< a(G) + 1.

In other words, that fill-in puts an upper bound on order. Indeed we suspect (on thebasis of numerous examples) that there is some relation between fill-in and order thatwe have not yet uncovered. Section 5 gives conjectures and examples.

Part III goes in a different direction and merely computes the orders of severalspecial classes of graphs.

All graphs G in this paper are finite, undirected, simple (i.e., without multiple edges),and without edges of the form (v, v), for a vertex v of G. The set of vertices of a graphG is denoted V(G), and the set of edges is denoted E(G).

For a given nonempty set S c V(G), denote by G(S) the graph obtained from Gby deleting all the vertices not in S together with all adjoining edges. So, V(G(S)) Sand i, j) E(G(S)) if and only if 4: j, i, j) E(S) and i, j) E(G).

As a graph is obviously preserved under natural graph isomorphisms, the statementsand proofs will be given modulo graph isomorphisms.

The k-dimensional vector spaces Re and C over the field of reals and the field ofcomplexes, respectively, will be represented as spaces of column vectors with k compo-nents, with the standard inner products in R and C.

2. Preliminaries. For the reader’s convenience we state here some results that willbe used frequently. All these results were obtained in [AHMR], and Theorem 2.1 in[PPS] as well.

THEOREM 2.1. A graph G has order ifand only ifG is a chordal, or triangulatedgraph, i.e.,for any cycle(v, 112), (112, 113), (11p- l, 11p), (11p, 111) E E(G), p >- 4 thereis a chord vi, v) E( G), where <= < j <= p and 2 <= j <-_ p 2.

The class of chordal graphs is an important class that appears in diverse problems(see, e.g., [G] for more information about chordal graphs and the problems in whichthey play a central role).

EXTREMAL MATRICES 409

THEOREM 2.2. Let G be a loop with n >= 3 vertices, i.e.,

E(G) ((v,, v2), (v2, v3), (1)n-1, l)n), (Un, I)1)>for some ordering v, Vn ofthe vertices ofG. Then the order ofG (over R) is n 2.

Introduce the partial order -< on the set of all (finite, undirected, simple, withoutedges of the form (v, v)) graphs G =< G2 if (and only if) G is isomorphic to G2(S) forsome set S V(G2).

THEOREM 2.3. IfG <-_ G2, then order (G) =< order (G2).We remark that another natural partial order =<e on the set of graphs G1 -<e G2 if

and only if V(GI) V(G2) and E(G) E(G2) generally does not imply any regularitybetween the orders. Indeed, let the graphs G1, G2, G3, be defined by

V(G) { 1,2, 3,4 }, i= 1,2,3,

E(G,) { (1,2), (2, 3), (3, 4) }, E( G_ E( G,) to (1, 4 ),

E( G3) E( G2)to(1, 3 ).

Then Gl =<e G2 <e G3 but

order (G1) order (G3) 1, order (G2) 2.

We shall often implicitly use the rather obvious fact that if Gl, Gp are theconnected components of G, then order (G) max order (Gi).

Part 1. Divide and Conquer Using Cut Sets and Cliques

3. Cut sets and cliques. A clique of a graph G is a subset S c V(G) such that everypair i, j) with i, j e S, # j belongs to E(G). A cut set of G is a subset S c V(G) withthe property that the graph G(V(G)\S) is not connected.

In this section we study the orders of graphs in terms of cut sets and cliques.THEOREM 3.1. Let S V( G) be a cut set, and assume that S is a clique. Then

(3.1) order G max (order G(SI tO S), order G(Sr tO S)),

where G(SI), G(Sr) are all the (nonempty) connected components ofG(V( G)\S).This theorem appeared (at least implicitly) in [M]; we shall provide an indepen-

dent proof.Proof. As the inequality >- in (3.1) follows from Theorem 2.3, we have only to

prove =<.Without loss of generality we may assume r 2 (otherwise, use an easy induc-

tion on r). Reorder the vertices in G so that Sl { 1, p ) $2 {p + 1, q);S {q + 1, ..., n}. Take MeM+(G)\ (0}. ThenMhas the form

M= 0 A2 Q2Q, Q, Q

As M is positive definite,

range Q1 c range A 1.

So for some matrix Wwe have Q1 A W. Form the matrix

I 0 W]E= 0 I 00 0 I


Then

A 0 0 ]M= E* 0 A2 Q2 E,0 Q’ Q

where Q Q W*AW. Since S is a clique, the matrix

M= 0 A2 Q20 Q’ Q

belongs to M+(G). Write

Q’ Q j

where M/(G(S2 tO S)), and rank . =< order (G(S_ tO S)). Now

(3.2)

M=E* 0 0 0 E+ ,E* E0 0 0

0 0Q{ 0 QA1Q

The matrix

[AI .Q ]N=Q{ QAQ

belongs to M+(G(S tO S)) and hence is the sum of matrices from M+(G(S tO S)) ofranks =< order G(S tO S). Substituting this sum into (3.2), we represent M as a sum ofmatrices from M/(G), the ranks of which do not exceed

max order G(S to S), order G(S 13 S)).

Some remarks concerning Theorem 3.1 are in order.Remark 3.2. Ifwe relax the assumptions and requirements that S becomes a clique

after addition ofjust one edge, then the disparity between the fight- and left-hand sidesof (3.1) can be arbitrarily large, as the following example shows. Let G be the loop withn vertices { 1, n } (so E(G) (1, 2), (2, 3), (n, 1) } ). Assuming, for instance,that n is even, let S { 1, n/2 + }. Obviously, S will become a clique after addition ofone edge. However, by Theorem 2.2 we have (in the real case)

order G n- 2, order G(SI US) order G(S2US) n/2- 1/2,

where G(S) and G(S2) are the connected components of G(V(G)\S).Remark 3.3. The following example shows that the fight-hand side in (3.1) cannot

be replaced by

(3.3) max (order G(Sl), order G(Sr)).

LetV(G)= {1,2,3,4,5};E(G)= {(1,2),(2,3),(3,4),(4,5),(5,2)};S= {2}.Inthis example the left-hand side of (3.1) is at least 2 (use Theorem 2.1 to verify this),while (3.3) is equal to by the same Theorem 2.1.


The following particular case of Theorem 3.1 deserves special attention.COROLLARY 3.4. Let u V( G) be such that the adjacent set

adj (u)= { ue V(G)I(u,)eE(G)is a clique. Then

order G order G(V(G)\ { u } ).

For the proofjust observe that adj (u) is a cut set, and apply Theorem 3.1.The condition of an adjacent set being a clique appears naturally in the analysis of

Gaussian elimination of sparse matrices (see R ], G ).As applications of Theorem 3.1 we can compute orders of many classes of graphs.

Consider one such class as an example.COROLLARY 3.5. Let G be a graph, and suppose V( G) S t.) Sp, where the

nonempty sets S have thefollowing properties:(i) For every 4 j either Si S or Si S consists oftwo vertices u and

with u, ,) E( G);(ii) Every edge in G belongs to some G(Si);(iii) The induced graph defined by V(O) { 1, ..., p }, (i, j) E(r) S

Sj. 4: is a forest, i.e., has no loops.Then

order(G)= max order(G(Si)).l_i_p

Proof. Induction on the number p. We can assume that 0 has at least one edge(otherwise, everything is trivial). Then 0 must have a pendant, i.e., a vertex with preciselyone adjacent edge. Say is a pendant, and 1, 2) E(0). The hypotheses ofthe corollaryeasily imply that the intersection S f) $2 is a cut set (consisting of two vertices), whichis a clique. We are now done in view ofTheorem 3.1 and the induction hypothesis.

The following result gives an upper bound for orders of graphs containing cliques.THEOREM 3.6. Let G be a graph with n vertices and let S V( G), S 4 V( G) be

such that G(S) is a clique. Then

(3.4) order (a) =< n SI.Here SI is the cardinality ofthefinite set S.Theorem 3.6 is contained in [M]. We shall include a proof anyway.Proof. Let X M/(G) be of rank greater than n sI, Enumerate all the vertices

so that S {1, k). Then

range Xspan {e, e } 4: { 0 },where e is the jth unit coordinate vector (with in the jth place and zeros elsewhere).Pick x range X span {e, e}, x 4:{0 }, and put R xx*. Then R M+(G)and range R range X. It is easy to see (for example, by writing the linear transformationsX and R in an orthonormal basis consisting of eigenvectors ofX) that X eR is positivesemidefinite for e > 0 sufficiently small. Now

x= 1/2(x-) + 1/2(x+ ),which shows that Xis not extremal in M+(G) unless rankX 1. However, the possibilitythat rank X is excluded because n > IS[. [2]

It is of interest to characterize those graphs for which equality holds in (3.4). Someinformation along these lines will be given later (Theorem 7.1).


X X

0

0

0 x, 0

FIG. 1.

We now indicate another sparsity pattern (see Fig. 1) whose graph contains a "lad-der," as in Fig. 2, and a large loop. We are able, however, to use the divide-and-conquermethod to split the computation of order into two more canonical looking graphs. Thusthe problem is worth considerable study, and even some compromises are justified.

For large n, consider the tridiagonal pattern whose graph is the line from to n.Fix a k, preferably close to n. Let G be the graph that includes the line from to n andin addition whose edges include (1, k), (2, k + 1), (n k + 1, n). Thus we obtaina pattern as indicated in Fig. 1. For n and k large, the matrix is still rather sparse and ina sense is not "too far" from the pattern that yields the n-loop. A careful drawing of thegraph yields the shape in Fig. 2. (Note that we must assume k > n k + 1.)

We now approach the problem with an obvious compromise. We lose only one zeroin the matrix in Fig. if we join vertex k and vertex n k + with an edge. Call thegraph G modified by this addition (, and note that these two vertices form a cut set thatis a clique for (. This cut set splits t into two graphs:

(1) A loop with 2k n vertices;(2) The ladder in Fig. 2 with k and n k + joined.

The loop is standard and has order 2k n 2. The graph in (2) is something like aloop ofdouble thickness and analyzing it suggests an interesting line of questions. Perhapsknown methods for studying loops will generalize.

Finally, in the special case k n 1, we observe that G contains an (n 1)-loopand hence in the real case has order at least n 3. Since G itself is not a loop, the orderof G is exactly n 3.

2 3 n-k n-k+1

FIG. 2.

n-k+2 k-1

EXTREMAL MATRICES 4 3

4. Completion problems. We study here implications ofthe "divide-and-conquer"technique for various completion problems.

Let G be a graph with n vertices, and call its vertices { 1, 2, n}. A partialHermitian matrix subordinate to G is, by definition, an n n array ofcomplex numbersand question marks such that the (i, j) entry is .9 if and only if q: j and i, j) is not inE(G), and that is Hermitian symmetric in the usual sense: if (i, j) E(G) (so that the(i, j) entry is a complex number ai), then the (j, i) entry is . We say that an n nHermitian matrix B is a completion of an n n partial Hermitian matrix A subordinateto G if the i, j) entry ofB coincides with the (i, j) entry ofA whenever the latter is nota question mark. In informal terms, B is obtained from A by replacing all ?’s by somecomplex numbers in the Hermitian way. Various completion problems have been studiedin DG ], GJSW ], JR], EGL], PPS ]; see also H, Chap. 8 ].

We consider three classes of completion problems:(1) For a given integer k, 0 -< k =< n, and a given partial Hermitian matrix A

subordinate to G, determine if A admits a completion with precisely k nonpositive ei-genvalues (counted with multiplicities).

(2) Same as (!) with replacement of "nonpositive" by "negative."(3) For a given partial Hermitian matrix A subordinate to G determine the com-

pletions B of A for which kmi (B) is maximal among all completions of A. Here kmi(B) is the minimal eigenvalue of B.

Given a partial Hermitian matrix A subordinate to G, and a subgraph F (soF -< G), the naturally defined restriction A IF is a partial Hermitian matrix subordinateto F.

In the rest of this section G is assumed to be a graph with a cut set S that is a clique.The connected components of G(F(G)\S) will be denoted G, G.

THEOREM 4.1. Let A be a partial Hermitian matrix subordinate to G. If each re-striction A IV(Gj) to S admits a completion Bj, whose size is equal to the cardinality ofV(G) tO S, with precisely kj nonpositive eigenvalues (j 1, p), then A admits acompletion with precisely max

_ _p kj nonpositive eigenvalues.

Proof. Order the n vertices of G so that

S={1,...,n}, G,={n+l,...,n2}..., Gp={np+l,’",n}.

Write

where Bj is n n. Since S is a clique, B is independent ofj. Consider the partialHermitian matrix

B B 2 Bz2 Bp2B’2 BI3 .9 ....9B’2 ? B23""" .9

The matrix/ is subordinate to the graph t obtained from G by adding all edges of thetype (q, q2), where (q, q2) is not in E(G) and

q,q2 - { 1, ,n,nj+ 1, ,n+


for some =< j -< p (by definition, np/l n). We check easily that 0 is chordal. Nowapplication of Theorem in [JR] finishes the proof. (For the reader’s convenience, wequote this theorem. Given a partial Hermitian matrix A subordinate to a chordal graphG, there is a completion B ofA such that the number of nonpositive eigenvalues of Bcoincides with the maximum number of nonpositive eigenvalues of any restriction A z,

where V is a clique of G.) [2]

Theorem 4.1 provides the best result in the following sense. IfA admits a completionwith precisely k0 nonpositive eigenvalues, and ko is maximal among all completions ofA, then no completion ofAI V(Gj) t_J S can have more than k0 nonpositive eigenvalues.This follows from the interlacing inequalities between eigenvalues ofa Hermitian matrixand eigenvalues of its principal submatrices.

The case when all kj are zero is of special interest.COROLLARY 4.2. Let A be as in Theorem 4.1. If each restriction A IV(Gj) t.J S

admits a positive definite completion, then A itselfadmits a positive definite completion.For the second class of completion problems we have the following result.THEOREM 4.3. Let A be a partial Hermitian matrix subordinate to G. Ifeach re-

striction A IV(Gj) t.J S admits a nonsingular completion with precisely k negative eigen-values, then A admits a nonsingular completion with precisely max1 _jzp kj negativeeigenvalues

Observe that nonsingularity is required in Theorem 4.3 (in contrast to Theorem 4.1).The proof is the same as that of Theorem 4.1.Note that, under the hypotheses of Theorem 4.3, the existence of a (not necessarily

nonsingular) completion of A with precisely max < <p k negative eigenvalues followsfrom Theorem 4.1 (applied to A + el for a small positive e).

We now pass to the third class of completion problems.THEOREM 4.4. Let A be as in Theorem 4.1. Let be the maximum ofkmin (nj)

taken over the set ofall completions B ofA V( G) t.J S. Then there is a completion B ofA for which

kmi (B) min { ,, . }.Theorem 4.4 follows immediately from Corollary 4.2 by subtracting a suitable mul-

tiple of I from A.Finally, we remark that all results of this section are true in the real case as well

(i.e., A is assumed to be real, and only real completions are allowed).

Part II. Gaussian Elimination Fill-In and the Order Problem

5. Gaussian elimination for sparse positive semidefinite matrices. Let G be an (un-directed) connected graph. We say that ( is obtained by one-step elimination from G iffor some vertex v in G the graph ( is obtained by removing v and all its adjacent edgesand by adding edges (x, y) for all pairs of vertices x, y different from v in G such that(x, v), (y, v) are edges in G but (x, y) is not. Thus, ( has one vertex less than G.

The one-step elimination procedure is the basic step in symmetric Gauss eliminationand has been studied extensively from the graph-theoretic point ofview (see R], RT ],[G], [GL]).

For a given graph G, let c(G) min { total number of edges added to E(G) inconsecutive one-step eliminations starting with G and ending in a one-vertex graph },the minimum being taken over all orderings of the vertices.

It follows from Theorem 2.1, combined with another characterization of chordalgraphs (see, e.g., [G]) that order (G) if and only if c(G) 0. Thus, it is of interestto find the relations between order (G) and c(G). We have the following conjecture.


CONJECTURE 5.1. There is a universal constant C such that

(5.1) order G -< a( G + Cin the real case, and

(5.2) order G -< 2a(G + Cin the complex case, for all graphs G.

The stronger conjecture is the following.CONJECTURE 5.2. The inequalities (5.1) and (5.2) are valid with C 1.We could not prove either conjecture. In this part we verify Conjecture 5.2 for some

graphs and prove that in some sense Conjecture 5.2 cannot be improved (Corollary 6.2).Two simple remarks are in order.Remark 5.3. Let G be a disjoint union of k copies of the graph Go. Then order G

order Go. However, a(G) ka(Go). This shows that there are no constantsC1, C2 with C > 0 such that

Ca(G) + C2 <- order G

for all graphs G.Remark 5.4. Let G be the ladder graph with 2m vertices:

Then a(G) 2m (see R ). On the other hand, a repeated application of Theorem3.1 (with the cut sets consisting oftwo vertices) shows that the order of G coincides withthe order of the four-vertex loop, which is two (over R as well as over C; see Theorem7.1 in [AHMR]). This shows that the difference a(G) order (G) can be arbitrarilylarge, even for connected graphs (the graph in Remark 5.1 was disconnected).

For a given graph G define (G) to be the minimal number of edges necessary toadd to G in order to obtain a chordal graph. It is not difficult to see that a(G) -/(G).Indeed, the inequality (G) _-< a(G) is obvious. To prove the opposite, let t be thechordal graph obtained from G by adding (G) new edges.

Let N" V(() -- { 1, n } be the enumeration of vertices given by a perfectelimination scheme for t (see [R], [G], [GL] for a definition and properties of thisnotion). Use the ordering Nto do consecutive one-step eliminations. Then it is necessaryto put 3(G) new edges in G in this procedure so a(G) =< 3(G).

PROPOSITION 5.5. IfG <-v G2, then a(G) <- a(G2).Proof. Clearly, (G) -</(G2). Now use the fact that a(G)= (G) for

j= 1,2.We can now prove the following.THEOREM 5.6. If order (G) _-< 3, then

(5.3) order (G) =< a(G) +in the real case.

Proof. If order (G)= 1, then, as we have observed already, a(G)= 0, and(5.3) holds.

Assume order (G) 2. Then G is not chordal (Theorem 2.1) and hence a(G) >= 1;so (5.3) holds again. Assume now order (G) 3.


Then G contains (in the sense of =<u-partial order) a minimal graph Go of orderthree. The list of all possible graphs Go (there are 16 of them) given by Theorem 8.2 in[AHMR] shows that a(G0) >-- 2. By Proposition 5.3, a(G) >_- 2. V1

The argument used in the proof of Theorem 5.4 also gives the following statement.As defined in [AHMR], a graph G is called a k-block if order (G) k and any graphstrictly -_<,-contained in G has order less than k. This notion depends on the choice ofthe field (R or C).

THEOREM 5.7. Fix positive integers k and C. Then in the real case the inequality

order (G) -< a(G) + C

holdsfor all graphs G oforder k ifand only if it holdsfor all k-blocks.An analogous statement is valid in the complex case concerning the inequality (5.2).In connection with Theorem 5.5, observe that for every fixed k the number of k-

blocks is finite [AHMR, Cor. 4.4]). Thus, in principle we could decide if (5.1) or (5.2)holds for all k-blocks using a finite procedure.

We conclude this section with a simple example.Example 5.1. Let G be a loop with n vertices. Then it is easy to see that a(G)

n 3. Combining with Theorem 2.2, we see that (in the real case)

order (G) a(G) + 1.

6. Fully bilmrtite graphs. In this section we compute orders of a large class of fullybipartite graphs, thereby giving another illustration of Conjecture 5.2 in the real, as wellas in the complex case.

The fully bipartite graph G(n, m) on n + m vertices is defined as follows (here m,n are positive integers)"

V(G)= {1, ,m+n};

i, j) e E(G(m, n)) if and only if 4: j and precisely one of the indices and j is inthe set { 1, n (so that the other index is in the set { n + 1, m + n } ). We shallassume n -< m, and (to avoid known cases) n >= 2. An easy inspection (see [R]) showsthat

n(n- 1)(6.1) a(G) .

2

THEOREM 6.1. For special n and m as indicated, we have

n2-n n2-n+2m if 4

<m<2

n2-n+2orderG(n,m)

n - n + 2ifm >

2 2

in the real case, and

morder G(n,m)=

n2-n+

if(n2-n)/3<m<=n2-n+ 1,

ifm>-n2-n+ 1.

Comparing with (6.1) we immediately get Corollary 6.2.COROLLARY 6.2. In the real casefor m > n2 n )/4 we have

order G(n,m) <=a(G(n,m))+ 1,


and equality holds for m (n 2 n + 2)/2. In the complex case for m > (n2 n)/3we have

order G(n,m)<=2a(G(n,m))+ 1,

and equality holdsfor m n2 n + 1.Thus, Corollary 6.2 confirms Conjecture 5.2 and shows that in a certain sense this

conjecture is best possible.The rest of this section will be devoted to the proof of Theorem 6.1.First, we need a simple lemma. For a real p p matrix A (a0), let

diag A (all, a22, app) TE Rp

be the diagonal part ofA.LEMMA 6.3. Let n >- 2 be an integer, and let n _-< m _-< (n 2 n + 2)/2. Then there

exists a linearly independent orthogonal set Y(1), Y n in Rm such that the vectors(the superscript "T" denotes transposition

diag (Y(i)Y(j)T+ Y(j)Y(i)r)ERm

span the linear space

F {(Xl, ,Xm)TRmlxl +"" +Xm-’0}.

Proof. The proof is by induction on n. The case n 2 is trivial. Pick m’ such that

n- <=m’<=((n 1)2-(n 1)+ 2)/2 and

m’>=m-(n-1), m’<=m-1.

Suppose vectors #(1), I7" (n 1) in Rre’with the desired properties are constructedalready. Then put

I7.(11) I7"(2)0Y(m-m’)=Y(1)= 0 Y(2)= 0

#(m-m’)0

0

Y(i)=[ #(i)]0 form-m’<i<n- 1;

where the numbers a l, -’-, am, satisfy the equation

-1

(.(n" lIT/ apl 0

(l

Y(n)= am,


The vectors Y(1), Y(n 1) are obviously linearly independent; Y(n) cannot bein the linear span of Y(1), Y(n 1) because Y(n) 4:0 and

Y(n)2_span {Y(1), ,Y(n- 1)).

Next, recall the notion of a representation of the graph G(n, m) introduced (forany undirected graph) and studied in [AHMR]. A function Y" { 1, n + m } Rkis called a k-dimensional representation of G(n, m) if the following properties hold:

(i) The set Y(1), (n) is orthogonal, and the set (n + 1), (n + m)is orthogonal (note that the vectors Y(j) need not be nonzero);

(ii) The vectors Y(1), Y(n + m) span Rk.If Y is a k-dimensional representation of G(n, m), then the n n matrix At[Y(1). Y(n)]r[Y(1) Y(n)] has rank k and belongs to M/(G(n, m)). The followingfact has been established in Corollary 3.2 of [AHMR].

PROPOSITION 6.4. The order ofG( n, m) in the real case coincides with the maximalk, for which there is a k-dimensional representation Y of G(n, m) with the additionalproperty that

(6.2) dim span {Y(i)Y(j)r+ Y(j)Y(i) r}_i4j_n

orn+ <=i4j_n+m

A representation Y that satisfies (6.2) will be called extremal.We now prove Theorem 6.1 in the real case. First, by putting

0Y(i)= ithplace, i=l,...,n,

0

0

-1-1

Y(n+ 1)= 0 Y(n+2)= 0

22

4(n+3) -1 ,Y(n+4)= -1

00

..., Y(2n-1)

Y(j)=0 for2n<-j<=n+m

k2+k-2

2

2n-3-1

we obtain an extremal n-dimensional representation of G(n, m). Thus, the order k ofG( n, m) is at least n. On the other hand, let Ybe a k-dimensional extremal representationof G(n, m).


The number of nonzero vectors in the set Y(1),..., Y(n) is at mostmin (k, n) n; the number of nonzero vectors in the set Y(n + 1), Y(n + m) isat most min (k, m). Comparing with (6.2), we obtain

k2+k-2 n(n-1)(6.3)

2 2min (k, m)(min (k,m)- 1)

This inequality implies easily that

(6.4) k<=(nZ-n-2)/2.

On the other hand, if m > (/7 2 n)/4, then k =< m. Indeed, assuming by contradictionthat k >- m + 1, we have with the help of (6.3)"

(m+ 1)2+(m+ 1)-2<=k2+k-2<=n2-n+m2-m,

which contradicts m > (n 2 n)/4.Now consider the case when m (n 2 n + 2)/2. The function

Y’{I,... ,n+m}-.-Rm,where Y(1), Y(n) is taken from Lemma 6.3 with m (n 2 n + 2)/2 and

0

(6.5) Y(n+ l)=0

Y(n+2)= 0 Y(n+m)=

0

is an irreducible representation of G(n, (n 2 n + 2)/2, (n 2 n + 2)/2). Thus, k >_-m. Together with (6.4) this shows

order G( n2-n+22 n2-n+2)2 n2-n+22As by Theorem 2.3

order G(n,m)>=order G(n, n2-n+2)2for all rn > (n 2 n + 2)/2, the inequality (6.4) shows also that

order G(n, m)n2-n+2

for all rn >- ((n 2 n + 2)/2).Now consider the case when m < ((n 2 n + 2) / 2).To prove the theorem in the real case it remains for us to exhibit an irreducible m-

dimensional representation of G(n, rn). To this end we define Y(n + i) as in (6.5) for1, rn, and Y(j) as in Lemma 6.3 for j 1, ..., n. This completes proof of

Theorem 6.1 in the real case.The proof of Theorem 6.1 in the complex case proceeds along similar lines. First,

we prove Lemma 6.5.


LEMMA 6.5. Let n >= 2 be an integer, and let n rn <- n 2 n + 1. Then there existsa linearly independent orthogonal set Y(1), Y(n) in Cm such that the vectors

diag (Y(i)Y(j)*)ECm

span (over C) the linear space

F= (zl, ,Zm)TECmlzl +’’" + Zm=O ).

Proof. The proof proceeds by induction on n. Suppose first that n 2; so 2 _-<

rn -< 3. For rn 2, put

For m 3, put

][1]rlY(2)=Y()= . _

-iY(2)

1-i

Suppose the lemma is already proved with n replaced by n 1. Let m’ be such that

n- <-m’<=(n 1)2-(n 1)+

and

<=m-m’<=2n-2.

By the induction hypothesis, there exist I7"(1), (n 1) e Cm’ with properties asin Lemma 6.5. Put

y(2) ;...;Y(p)= orY(p)=

depending on if rn m’ is even or odd; here p ((rn m’ + 1)/2). Put

I(oJ)Y(j)

Y(n)

for p<j<=n- 1,


(The vectors Y(1), Y(n) are m-dimensional.) The numbers al, am are deter-mined to satisfy the equations

()*?(2)* -1 +

-1+i

I?(p-1)* (a.l) -1+i

(p), x

(p+ ),, 0.

l?(n: 1)* 6

where x + ifm m’ is even and x if m m’ is odd. The vectors Y(1),Y(n) form a linearly independent and orthogonal set in Cm. Furthermore,

(6.6) diag (Y(n)Y(j)*)=(,, ,,, 1,-i,0, ,0)

forj 1, p 1, where and -i appear in the positions m’ + 2j and m’ + 2j,respectively (the stars in the fight-hand side of (6.6) denote entries of no immediateinterest to us). Also,

diag (Y(n)Y(p) * (,, ,, 1,-i)

or (,, ,, 1), depending on if m m’ is even or odd. These equalities together withthe properties of I? (1), I? (n 1) (assumed by induction) show that

diag (Y(k)Y(j)*), k4:j, <=k,j<=n

span the subspace F.The proof ofTheorem 6.1 is again based on the notion of a complex representation

of G(n, m) (this notion was introduced and studied in AHMR ). The definition of acomplex representation Y is the same as in the real case with the only modification thatY: { 1, n + m } - Ck. The analogue ofProposition 6.4 runs as follows (see AHMR,Cor. 3.2 ]).

PROPOSITION 6.6. The order of G(n, m) in the complex case coincides with themaximal k, for which there is a complex k-dimensional representational Y ofG( n, m)with

(6.7) dim span_i4j_n

orn+ _i4j_n+m

{Y(i)Y(j)*}=k2-1.

(The dimension here is the dimension ofa vector space over C.)A complex representation Ywith the property (6.7) will be called extremal. Return

to the proof of Theorem 6.1 in the complex case. Let k be the order of G(n, m) over C.As in the real case, there is an extremal n-dimensional complex representation ofG(n, m), so k >_- n. On the other hand, let Y be a k-dimensional (complex) extremalrepresentation of G(n, m). Arguing as in the proof of the real case and using Corollary3.2 of [AHMR] again, we obtain

(6.8) k2- <=n(n- 1)+min (k, m)(min (k,m)- 1).

This inequality easily implies

k<-n2-n+ 1.


Also, if m>(n2-n)/3, then k<=m (we prove this by contradiction assumingk >= m + and using (6.8)). Now we finish the proof as in the real case using Lemma6.5. V1

Part III. Graphs of Large Order

7. Graphs with largest order. Theorem 3.6 will be used to provide the followingdescription of graphs with the largest order (relative to the number of vertices).

THEOREM 7.1. The order of a graph G with exactly n vertices (where n >= 3) is

<=n 2. Moreover, in the real case when n >= 4, order G n 2 ifand only if G is aloop (see Theorem 2.2 for the definition ofa loop).

Proof. The first statement is just a particular case of Theorem 3.6 (because we canassume that G has at least one edge; otherwise, everything is trivial). Assuming thematrices are over R, the "if" part of the second statement is just Theorem 2.2.

Now let A be an extremal element in M+ (G), and let rank A n 2 (we continueto consider the real case). By Theorem 3.6, G has no 3-cliques, i.e., triangles. On theother hand, Corollary 3.2 in [AHMR] shows that the number of edges in G is atmost n.

Next, we show that G does not have pendants, i.e., vertices with degree 1. Indeed,assume that the first vertex has degree 1. Let A be an extremal matrix in M+(G) of rankn 2. We can suppose that the first column of A is nonzero (otherwise, the first rowand first column ofA are zeros, and by deleting the first row and column we reduce theproblem to the case of (n 1) (n 1) matrices). Let a be the first column ofA. PutR aa. Because the first vertex has degree 1, we have R M+(G). Then for smalle > 0 we have A eR M+(G) (cf. the proof of Theorem 3.6), and hence the equality

A=1/2(A-eR)+1/2(A+eR)

contradicts the extremality ofA (unless A is a scalar multiple ofR; however, this case isexcluded because rank A n 2 and n >= 4).

Furthermore, it is easy to see that G must be connected (because the order of adisconnected graph is the maximum of the orders of its connected components). Usingthe fact that the sum of the degrees of the vertices is twice the number of edges, and theabsence of pendants, we note that the degree of each vertex in G is precisely two, so Gmust be a loop. ff]

8. Graphs with six vertices. As an application of Theorem 7.1 we can describe theorders of all graphs with at most six vertices.

Only the real case will be considered in this section.We shall exclude from consideration chordal graphs (as their order is ), disconnected

graphs (as their order is the maximum among the orders oftheir connected componentsand loops (as their order is given by Theorem 2.2).

THEOREM 8.1. Let G be a connected nonchordal graph with at most six vertices andthat is not a loop. Then the order ofG is 2 except when G either <=,-contains a loop with

five vertices or G is one ofthefollowing six graphs:(1) Thefully bipartite graph G( 3, 3 (see 6 ).(2) The fully bipartite graph G( 3, 3 with precisely one edge added (the place of

the added edge is immaterial because ofgraph isomorphism).3 G( 3, 3) with precisely one edge removed.

(4) G( 3, 3 with precisely one edge added and one edge removed, where the addedand the removed edges are adjacent to the same vertex;


(5) V(G)={1,2,3,4,5,6};E(G)={(i,j)II<=i4=j<=6;1 < IJ-il <5}.(6) The graph as in (5) with the edge (1, 6) added.In all the exceptional cases the order ofG is 3.Proof. By Theorem 7.1 the order of G is either 2 or 3. If G _-<,- contains a 5-1oo19,

then order (G) 3 by Theorems 2.3 and 2.2. Graphs (1)-(6) have order 3 by Theorem8.2 of [AHMR (all ofthem are 3-blocks). Conversely, assume that G has order 3. ThenG must _,-eontain a 3-block. By Theorem 7.1 the only 3-block with five or fewer verticesis the 5-loop, while Theorem 8.2 of [AHMR] provides a list of all 3-blocks with sixvertices, which are precisely the six graphs listed in the theorem.

Acknowledgment. We thank P. DeWilde for pointing out that the sparsity patternin Fig. and similar double-banded patterns occur frequently in VLSI circuit analysis.

REFERENCES

[AHMR] J. AGLER, J. W. HELTON, S. MCCULLOUGH, AND L. RODMAN, Positive definite matrices with agiven sparsity pattern, Linear Algebra Appl., 107 (1988), pp. 101-149.

[DG] H. DYM AND I. GOHBERG, Extensions ofband matrices with band inverses, Linear Algebra Appl.,36 (1981), pp. 1-24.

[EGL] R. ELLIS, I. GOHBERG, AND D. C. LAY, Invertible selfadjoint extensions ofband matrices and theirentropy, SIAM J. Algebraic Discrete Methods, 8 (1987), pp. 483-500.

G M.C. GOLUMBlC, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.[GJSW] R. GRONE, C. R. JOHNSON, E. M. DE SA, AND H. WOLKOWITZ, Positive definite completions of

partial Hermitian matrices, Linear Algebra Appl., 58 (1984), pp. 102-124.[GL] A. GEORGE AND J. W. LIU, Computer Solution ofLarge Sparse Positive Definite Systems, Prentice

Hall, Englewood Cliffs, NJ, 198 I.[H] J.W. HELTON (WITH J. A. BALL, C. R. JOHNSON, AND J. N. PALMER), Operator theory, analytic

functions, matrices and electrical engineering, CBMS Vol. 68, American Mathematical Society,Providence, RI, 1987.

JR C.R. JOHNSON AND L. RODMAN, Inertia possibilitiesfor completions ofpartial Hermitian matrices,Linear and Multilinear Algebra, 16 (1984), pp. 179-195.

[M] S. MCCULLOUGH, 2-chordal graphs, Operator Theory: Adv. Appl., 35 (1988), pp. 133-192.[P] S. PARTER, The use oflinear graphs in Gauss elimination, SIAM Rev., 3 (1961), pp. 119-130.[PPS] V.I. PAULSEY, S. C. POWER, AND R. R. SMITH, Schur products and matrix completions, preprint.JR] D. RosE, A graph-theoretic study of the numerical solution of sparse positive definite systems of

linear equations, in Graph Theory and Computing, R. Read, ed., Academic Press, New York,1973, pp. 183-217.

[RT] D.J. Rosw AND R. E. TARJAN, Algorithmic aspects ofvertex elimination, in Proc. 7th Annual ACMSymposium on the Theory of Computing, Association for Computing Machinery, New York,1975, pp. 245-254.

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