ACADEMIE SERBE DES SCIENCES ET DES ARTSgvm/Teze/Bilten40n.pdf · G. Popivoda: On the origin of two degree–based topological indices ... 19 4. D. Stevanovi´c: Walk counts and the

ISSN 0561-7332

ACADEMIE SERBE DES SCIENCES ET DES ARTS

BULLETIN TOME CXLVIII

CLASSE DES SCIENCES MATHEMATIQUES ET NATURELLES

SCIENCES MATHEMATIQUES

No

R é d a c t e u r

GRADIMIR V. MILOVANOVIĆ

Membre de l’Académie

B E O G R A D 2 0 1 5

40




N 40o

ISSN 0561-7332

ACADEMIE SERBE DES SCIENCES ET DES ARTS




No 40

B E O G R A D 2 0 1 5

Publie et impime par

Academie serbe des sciences et des artsBeograd, Knez Mihailova 35

Tirage 400 exemplaires

c Academie serbe des sciences et des arts, 2015

TABLE DES MATI

`

ERES

1. M.M. Marjanovic: Hyperspaces of 0-dimensional spaces revisited . . . . . 1

2. M. Prvanovic: Bochner-flat Kahler manifolds and Rimanniancompability of the Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. B. Furtula, I. Gutman,

ˇ

Z. Kovijanic Vukicevic, G. Lekishvili,

G. Popivoda: On the origin of two degree–based topological indices . . . 19

4. D. Stevanovic: Walk counts and the spectral radius of graphs . . . . . . . . 33

5. M. Cvetkovic, V. Rakocevic: Fixed point of mappings of Perovtype for w-cone distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. V. Dragovic: Discriminantly separable polynomials: an overview . . . . . 75

6. B. Stankovic: Laplace transform of functions defined on a boundedinterval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bulletin T.CXLVIII de l’Academie serbe des sciences et des arts 2015Classe des Sciences mathematiques et naturelles

Sciences mathematiques, No 40

HYPERSPACES OF 0-DIMENSIONAL SPACES REVISITED

MILOSAV M. MARJANOVIC

(Presented at the 1st Meeting, held on February 27, 2015)

A b s t r a c t. An immediate reason for revisiting hyperspaces of 0-dimensional spacesis the paper of Sh. Oka [Topology Appl. 149 (2005), no. 1-3, 227–237], where the results ofM. M. Marjanovic [Publ. Inst. Math. (Beograd) (N.S.) 14 (28) (1972), 97–109] have beenreproved. Aside from that, we take this opportunity to highlight the concepts of accumulationorder and accumulation spectrum as a system of topological invariants which can be usedefficiently in some situations of determining topological types in this class of spaces.

In particular the Cartesian multiplication of the accumulation orders is an operationwith respect to which the set of natural numbers N becomes a semi-group that can be used toreduce some subtle topological problems (for example, the existence of non-homeomorphicspaces with homeomorphic squares) to simple arithmetic verifications. The paper summa-rizes central ideas and details of the main constructions and may serve as an overview andintroduction to this area of mathematics.

AMS Mathematics Subject Classification (2000): Primary 54B20; Secondary 54F65.Key Words: Hyperspaces, accumulation order, accumulation spectrum.

1. Introduction

When new results are considered to be particularly interesting, it is an academiccustom to present them a number of times (at some conferences, visiting some uni-versities, etc.). When it is done again after some forty years, then there must exist

2 M. M. Marjanovic

an exceptional reason as a justification. And in the case of this note a good reasonfor taking a look back to hyperspaces of 0-dimensional spaces is the coincidence ofresults in [Ma2] and [O1].

All spaces that we consider here are supposed to be compact Hausdorff and allmappings continuous. For a space X , let exp(X) be the set of all non-empty closedsubsets of X and for a sequence U

1

, U

2

, . . . , Un of open sets in X , let

hU1

, U

2

, . . . , Uni =F 2 exp(X) | (8i)F \ Ui 6= ;

and F U

1

[ U

2

[ . . . [ Un . (1.1)

When U

1

, U

2

, . . . , Un runs over all finite sequences of open sets in X , the setshU

1

, U

2

, . . . , Uni constitute the basis for a topology on exp(X) which is called theVietoris topology. For a mapping f : X ! Y , let exp(f) : exp(X) ! exp(Y ) bethe induced map defined by exp(f)(F ) = f(F ), and then the correspondence

X

f

exp(X)

exp(f)

!

Y exp(Y )

(1.2)

is a covariant functor from the category of all compact Hausdorff spaces and contin-uous mappings into itself.

Iterating the functor exp : exp(exp(X)) = exp

2

(X) and for n > 2, by puttingexp(exp

n1

(X)) = exp

n(X), hyperspaces of higher order are obtained. Let u :

exp

2

(X) ! exp(X) be the union mapping. By putting u

(1)

= u and for n >

1, u

(n)= exp(u

(n1)

), an inverse system is obtained

· · · ! exp

3

(X)

u(2)

! exp

2

(X)

u(1)

! exp(X)

(1.3)

whose limit exp(!)(X) contains X and exp(exp

(!)(X)) = exp

(!)(X) ([Ma1]). We

call the space exp

(!)(X) the hyperspace of X of the rank !.

We call a space X which is homeomorphic to its hyperspace exponentially com-plete. Evidently we were somewhat fascinated with this property of such spaces thatwe used the phrase “exponentially complete spaces”, entitling four of our papers.

2. Focus on hyperspaces of Peano continua

In the early 1920s, L. Vietoris ([V1] and [V2]) and T. Wazewski ([W]) proved thefollowing result:

Hyperspaces of 0-dimensional spaces revisited 3

• If X is a non-degenerate Peano continuum then exp(X) is a Peano continuum.

This apparently nice result was dramatically surpassed several times in the courseof a number of decades. Namely, in addition to be a Peano continuum, it has beenproved that:

• exp(X) has a subspace homeomorphic to the Hilbert cube, (K. Borsuk and S.Mazurkiewicz, in [B–M]).

• exp(X) is an absolute retract and a contractible space (M. Wojdyslawski, in[Wo]).

In this paper of Wojdyslawski, the famous hypothesis:

• If X is a non-degenerate Peano continuum, then exp(X) is homeomorphic tothe Hilbert cube,

was launched for the first time and according to the spoken reports of K. Kuratowski,the hypothesis was already known to Polish mathematicians in 1920s.

According to some results contributed to the hyperspace theory, it could be gues-sed that this hypothesis has been absorbing the interest of a number of outstandingmathematicians at least for some time. But finally, in 1972, R. M. Schori and J. E.West proved:

• exp(I), where I is the unit interval is homeomorphic to the Hilbert cube Q,(see [S–W1], proof with all details in [S–W2]).

And in 1974, D. W. Curtis and R. M. Schori proved:

• X is a non-degenerate Peano continuum if and only if exp(X) is homeomor-phic to the Hilbert cube Q ([C–S1], proof with all details in [C–S2]).

Without any doubt, confirmation of this hypothesis is the most significant resulton hyperspaces and it has motivated significant further research, e.g. characteriza-tions of the Hilbert cube manifolds, etc. ([T] et al.).

3. A long period without concrete examples

Leaving out the trivial case of finite spaces having 2n 1 points, no other con-crete example of hyperspaces was known until 1948, when G. Choquet proved thatthe Cantor set C is homeomorphic to its hyperspace exp(C), ([Ch]). In a situationwhich was lacking in concrete examples, it was quite natural to raise the question ofexistence of two non-homeomorphic spaces having their hyperspaces homeomorphic(V. Ponomarev, in [Po]). The answer came from A. Pelczynski ([Pe]), who provedthat:

4 M. M. Marjanovic

• When X is compact metric 0-dimensional space having the set of its isolatedpoints everywhere dense, then exp(X) is homeomorphic to the space T (C),being the Cantor set with a point interpolated in each of its removed intervals.

Pelczynski also quotes a result of A. Mostowski according to which there arecontinuum many different topological types of compact metric 0-dimensional spaceshaving the sets of their isolated points everywhere dense, but they still have theirhyperspaces homeomorphic to T (C).

4. A search for more concrete examples

The class Z of compact metric 0-dimensional spaces is a natural framework tolook for some other concrete examples of hyperspaces and to try to fix their topo-logical types. This led us to consider a classification of points of spaces in Z thatreveals some of their interesting topological properties. We give here a somewhatless formally tight description of this classification.

Let X 2 Z and let X0

be the set of isolated points of X . If X = X

0

, the set X0

is finite and we will also write s(X) = 0. Let X1

be the set of points of X , havinga neighborhood without isolated points. When X = X

1

, X1

is homeomorphic to theCantor set C and then we write s(X) = 1. When X = X

0

[ X

1

, X0

is a finite setand X

1

C and then we write s(X) = 0, 1. Let X(0)

be the set of accumulationpoints of X

0

, then X = X

0

[ X

1

[ X

(0)

. Let X2

be the subset of those points ofX

(0)

which are not accumulation points of X1

. When X

1

= ;, then X = X

0

[X

2

and we also write s(X) = 0, 2. When X

1

6= ;, then X = X

0

[X

1

[X

(0)(1)

.Let n > 1 and suppose that the sequence X

0

, . . . , Xn1

, X

(0)...(n2)

of disjointsubsets of X has been defined. Let Xn be the set of points of X

(0)...(n2)

whichare accumulation points of Xn2

and not of Xn1

and X

(0)...(n2)(n1)

those pointswhich are accumulation points of Xn1

. Then

X = X

0

[ . . . [Xn1

[Xn [X

(0)...(n2)(n1)

.

As soon as Xn1

= ;, then X

(0)...(n2)(n1)

= ; and X = X

0

[ · · · [Xn2

[Xn.In this case we write s(X) = 0, . . . , n 2, n.

When Xn1

6= ; and X

(0)...(n2)(n1)

= ;, then X = X

0

[ · · ·[Xn1

[Xn andwe write s(X) = 0, . . . , n 2, n 1, n. When for each n,X

(0)...(n2)(n1)

6= ;, theintersection X! of this descending sequence of compact sets is non-empty. In thiscase

X = X

0

[ · · · [Xn [ · · · [X!

and we write s(X) = 0, . . . , n, . . . ,!.


Definition 4.1. For x 2 Xn, (0 n !), the number n is called the accumu-lation order of x and denoted by n = ord(x). The sequence s(X) that is assigned toX is called the accumulation spectrum of X .

Examining the hyperspaces of spaces in Z, first we tried to see how the set(exp(X))n of elements of order n in exp(X) depends on the sets Xn, n = 0, 1, . . .,hoping to discover a regularity in this dependence. A great surprise arose when wefound that for each X in Z, (exp(X))

6

= ;, which meant that the spaces exp(X) hadno elements of order higher than 7. Namely, we established the following formulae:

(exp(X))

0

= hX0

i(exp(X))

1

= iX1

h := F 2 exp(X) | F \X

1

6= ;(exp(X))

2

= hX0

[X

2

, X

2

i(exp(X))

3

= hX0

[X

3

, X

3

i(exp(X))

4

= hX0

[X

2

[X

4

, X

4

i(exp(X))

5

= hX0

[X

2

[X

3

[X

5

, X

2

[X

5

, X

3

[X

5

i(exp(X))

6

= ;

This unexpected discovery turned an easily going consideration into a serioussearch for topological types of hyperspaces in Z.

Let us call a space X in Z full if for each n > 0, whenever the sets Xn arenon-empty they contain no isolated point. Then, we proved:

• For each X , exp(X) is a full space.

As a variation on Brouwer’s topological characterization of the Cantor set, we alsoproved:

• When X and Y are full spaces having finite spectra, then s(X) = s(Y ) impliesX Y .

At the end, a sequence of full spaces was constructed, starting with C1

= ;,C

0

= 1 (one point space), and C

1

= C (the Cantor set). Assuming that thesequence C1

, C

0

, C

1

, . . . , Cn, (n > 0) has been constructed, then Cn+1

is the spaceobtained from the Cantor set C when a (small enough) copy of Cn2

Cn1

isinterpolated in each of its removed intervals. For n > 1

s(Cn) = 0, . . . , n 2, n; s(Cn1

Cn) = 0, 1, . . . , n 1, n

(and for the Cantor set C1

, s(C

1

) = 1). Excluding the cases of finite spaces havingmore than one point, for some n, each other full space with finite spectrum is home-omorphic to either Cn or Cn1

Cn. Let us note that somewhat more complicated

6 M. M. Marjanovic

construction of full spaces was given in [Ma2] and the one that is presented here wasannounced in [Ma3].

Thus, our search for hyperspaces in Z culminated in the following statements:

• The only exponentially complete spaces in Z are

C

0

, C

1

, C

0

C

1

, C

2

, C

1

C

2

, C

3

, C

4

, C

5

, C

7

.

• Excluding the trivial case of spaces with a finite number of isolated points, theonly hyperspaces in Z are

C

1

, C

2

, C

1

C

2

, C

3

, C

4

, C

5

, C

7

.

The fact that, excluding the trivial cases, there exists only a finite number of topo-logical types of hyperspaces in Z, added some spectacularity to these results. Pub-lished in 1972, under the title “Exponentially complete spaces III” ([Ma2]) and com-municated several times in some lectures delivered by this author (Seminar Kuratow-ski–Engelking at Polish Academy of Sciences, Chair of General Topology at theMoscow State University, several international topological conferences, etc.) theseresults were widely known to the specialists in this area of mathematics.

A. N. Vybornov (and his mentor V. Ponomarev) followed the way of researchingfrom [Ma2] to investigate the hyperspaces of 0-dimensional Polish spaces ([Vy]).With some extra elegance, S. Todorcevic also presents these results in his Springer’smonograph [To].

Somewhat surprisingly the results from [Ma2] were reproved by Sh. Oka in hispaper [O1], without any referring to our paper. Thanks to J. van Mill and J. Vaughan,Oka wrote a corrigendum ([O2]) giving credit for my results.

5. The other cases of the use of accumulation orders

In a number of other cases the accumulation orders were used as an efficientsystem of invariants. In [Ma4], we announced that for X 2 Z, exp!(X) is only oneof the spaces

C

0

, C

1

, C

0

C

1

, C

2

, C

1

C

2

, C

4

and a complete proof of this fact was given in [Ma5].For the Cartesian product of spaces, there exists a corresponding Cartesian prod-

uct of accumulation orders. Namely, let x 2 X , y 2 Y and ord(x) = m, ord(y) = n.Then m n is defined as ord((x, y)), (x, y) 2 X Y . The product m n doesnot depend on the choice of the spaces X and Y and is commutative and associative


and for each n, n 1 = 1. Up to 7, this multiplication is given by the followingmultiplication table,

0 2 3 4 5 70 0 2 3 4 5 72 2 2 5 4 5 73 3 5 3 7 5 74 4 4 7 4 7 75 5 5 5 7 5 77 7 7 7 7 7 7

As each n 2 N can be written in the form n = 6k + r, where k = 0, 1, 2, . . . andr = 0, 2, 3, 4, 5, 7, the Cartesian product of m = 6k

1

+ r

1

and n = 6k

2

+ r

2

is givenby the following formula,

m n = 6(k

1

+ k

2

) + r

1

r

2

.

For example 8 13 = (6 · 1 + 2) (6 · 1 + 7) = 6(1 + 1) + 2 7 = 12 + 7 = 19.The semi-group (N,) was defined and its properties established in [Ma3]. Based

on the fact that in the case of two full spaces, equality of their accumulation spectraimplies their homeomorphism, it was a matter of simple verification to see that:

• The pairs of non-homeomorphic spaces,

C

6k+2

C

6k+3

, C

6k+5

, k = 0, 1, 2, . . .

have homeomorphic squares C12k+5

.

The existence of such pairs of spaces in Z solves a problem of P. R. Halmos,posed in the Boolean algebra terms in his book [H].

In full analogy with the case of spaces in Z, the class Z0

of countable metric0-dimensional spaces was considered in [Ma-Vu]. In Z

0

, the role of the Cantor set inZ, is reserved for the space Q

1

of rational numbers. In this class of spaces, a variationon Sierpinski’s topological characterization of the space Q

1

of rationals is somewhatmore demanding statement than its analogue in Z.

Let us note that Q1

can be taken, in a topologically equivalent way, to be the setof end points of the removed intervals of the Cantor set C. Starting with Q1

= ;,Q

0

= 1 (one point space), Q1

the set of rationals and proceeding inductively, Qn

is taken to be Q

1

together with a (small enough) copy of Qn3

Qn2

interpolatedin each removed interval of C. Then, it is easy to verify that, for example,

• Q

2

Q

3

, Q

5

is a pair of non-homeomorphic spaces having homeomorphicsquares.

The problem of existence of such pairs of spaces in Z0

was posed in [Ko–Tr] and[Tr].

8 M. M. Marjanovic

6. Some other properties related to hyperspaces

Though not specific for 0-dimensional spaces, we include here some propertiesrelated to the construction of the hyperspaces of rank !. In [M-V-Z] the followingresults have been proved:

• The union mapping u : exp

2

(X) ! exp(X) is open.

• The union mapping u : exp

2

(I) ! exp(I), where I = [0, 1] is universal in thesense that each continuous mapping of a compact metric space is a restrictionof the union mapping u.

• For X non-degenerate Peano continuum, exp!(X) is not locally connected.

In light of the fact that exp(exp!(X)) exp

!(X), an alluring idea that exp!(X)

is unique for all compact connected metric spaces X easily crosses the mind but, dueto the complexity of this construction has never been investigated.

At the end let us add that when throughout this whole construction closed sets arereplaced with closed connected sets, then:

• For a non-degenerate Peano continuum X , the hyperspace of rank ! of closeconnected subsets is homeomorphic to the Hilbert cube ([Ma–Vr]).

REFERENCES

[B-M] K. Borsuk, S. Mazurkiewicz, Sur l’hyperespace d’un continu, C. R. Soc. Sc. Varso-vie 24 (1931), 149–152.

[Ch] G. Choquet, Convergences, Ann. Univ. Grenoble, 23 (1948), 55–112.

[C-S1] D. W. Curtis, R. M. Schori, 2X and C(X) are homeomorphic to the Hilbert cube,Bull. Amer. Math. Soc. 80 (1974), 927–931.

[C-S2] D. W. Curtis, R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes,Fund. Math. 101 (1978), no. 1, 19–38.

[Ko-Tr] V. Koubek, V. Trnkova, Isomorphisms of sums of Boolean Algebras, Proc. Amer.Math. Soc. 66 (1977), 231–236.

[Ma1] M. M. Marjanovic, Exponentially complete spaces. I, Glasnik Mat. Ser. III 6(26)

(1971), 143–147.

[Ma2] M. M. Marjanovic, Exponentially complete spaces. III, Publ. Inst. Math. (Beograd)(N. S.) 14 (28) (1972), 97–109.


[Ma3] M. M. Marjanovic, Numerical invariants of 0-dimensional spaces and their Carte-sian multiplication, Publ. Inst. Math. (Beograd) (N.S.) 17(31) (1974), 113–120.

[Ma4] M. M. Marjanovic, Higher-order hyperspaces, Dokl. Akad. Nauk SSSR 278

(1984), no. 1, 34–37 (Russian).

[Ma5] M. M. Marjanovic, Spaces X

(!) for zero-dimensional X , Bull. Acad. Serbe Sci.Arts Cl. Sci. Math. Natur. No. 16 (1988), 23–36.

[M-V-Z] M. M. Marjanovic, S.T. Vrecica, R.T. Zivaljevic, Some properties of hyperspacesof higher rank, Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur. No. 13 (1984),103–117.

[Ma-Vr] M. M. Marjanovic, S. T. Vrecica, Another hyperspace representation of the Hilbertcube, Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur. No. 14 (1985), 11–19.

[Ma-Vu] M. M. Marjanovic, A. R. Vucemilovic, Two nonhomeomorphic countable spaceshaving homeomorphic squares, Comment. Math. Univ. Carolin. 26 (1985), no. 3,579–588.

[O1] Sh. Oka, The topological types of hyperspaces of 0-dimensional compacta, Topol-ogy Appl. 149 (2005), no. 1-3, 227–237.

[O2] Sh. Oka, Corrigendum to “The topological types of hyperspaces of 0-dimensionalcompacta” [Topol. Appl. 149 (1-3) (2005), 227–237], Topology Appl. 164 (2014),259.

[Pe] A. Pelczynski, A remark on spaces 2X for zero-dimensional X , Bull. Polon. Sci.,Ser. Math. 13, No 2 (1965), 85–89.

[Po] V. I. Ponomarev, A new space of closed sets and many-valued continuous mappingsof bicompacts, Mat. Sb. (N.S.) 48 (90) 1959, 191–212 (Russian).

[S-W1] R. M. Schori, J. E. West, 2I is homeomorphic to the Hilbert cube, Bull. Amer.Math. Soc. 78 (1972), 402–406.

[S-W2] R. M. Schori, J. E. West, The hyperspace of the closed unit interval is a Hilbertcube, Trans. Amer. Math. Soc. 213 (1975), 217–235.

[To] S. Todorcevic, Topics in Topology, Springer, 1997.

[T] H. Torunczyk, On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), 31–40.

[Tr] V. Trnkova, Representation of commutative semigroups by products of topologicalspaces, Proc. Fifth Prague Top. Symp., 1981, Berlin, 1982, pp. 631–641.

[V1] L. Vietoris, Bereiche zweiter Ordung, Monatshefte fur Mathematik und Physik, 32

(1922), 250–280.

10 M. M. Marjanovic

[V2] L. Vietoris, Bereiche zweiter Ordung, Monatshefte fur Mathematik und Physik, 33

(1923), 49–62.

[Vy] A. N. Vybornov, Exponents of zero-dimensional Polish spaces, Dokl. Akad. NaukSSSR 284 (1985), no. 5, 1053–1057 (Russian).

[W] T. Wazewski, Sur un continu singulier, Fund. Math. 4 (1923), 214–235.

[Wo] M. Vojdyslawski, Sur le contractilite des hyperespaces de continus localement con-nexes, Fund. Math. 30 (1938), 247–252.

Serbian Academy of Science and ArtsKnez Mihajlova 3511000 BelgradeSerbiae-mail: [email protected]



BOCHNER-FLAT KAHLER MANIFOLDS AND RIEMANNIANCOMPATIBILITY OF THE RICCI TENSOR

MILEVA PRVANOVIC

(Presented at the 3rd Meeting, held on April 24, 2015)

A b s t r a c t. In this paper we investigate Riemannian compatibility of Ricci tensor ofa Bochner-flat Kahler manifold, and specially of a such manifolds which is of quasi-constantholomorphic sectional curvature. Also, we extend our consideration to manifolds withoutBochner-flat condition. In all cases we found necessary and sufficient conditions on the Riccitensor of considered manifolds to be Riemannian compatible.

AMS Mathematics Subject Classification (2000): 53C20, 53C21.Key Words: Kahler manifold, Bochner-flat Kahler maniford Riemannian compatiblity.

1. Introduction

Let (M, g) be a Riemannian manifold. We denote by R, and the Riemanniancurvature tensor, the Ricci tensor and the scalar curvature respectively.

In [2], [3] and [4] it was introduced the algebraic notion of Riemannian compati-ble tensors as follows.

Definition 1.1. A symmetric tensor bij is compatible with Riemannian curvaturetensor if

biaRahjk + bjaR

ahki + bkaR

ahij = 0. (1.1)

12 M. Prvanovic

The metric tensor is trivially Riemannian compatible. If (M, g) is an Einsteinmanifold, then

iaRahjk + jaR

ahki + kaR

ahij = 0, (1.2)

that is, the Ricci tensor is Riemannian compatible. The relation (1.2) is also satisfiedif (M, g) is conformally flat, i.e. if

Rihjk =1

n 2

gikhj + gjhik gijhk ghkij

(n 1)(n 2)

gikghj gijghk

, n = dimM.

It is not the same for the Kahler manifolds. In general, the Ricci tensor of aBochner-flat Kahler manifold does not satisfy the condition (1.2). The aim of thepresent paper is to find the necessary and the sufficient conditions for such a com-patibility. This is done in Section 2. In Section 3, to find an example, we discussthis problem in the case of Kahler manifolds of quasi-constant holomorphic sectionalcurvature.

2. Compatibility of Ricci tensor for a Bochner-flat Kahler manifold

Kahler manifold (M, g, J) is a differentiable manifold M , dimM = 2n, en-dowed with Hermitian metric g and the parallel complex structure J . This meansthat, with respect to the local coordinates (x1, x2, . . . , x2n) we have

J

ia J

aj = ij , J

ai J

bj gab = gij , rkJ

ij = 0,

where r is the operator of the covariant derivative with respect to the Levi-Civitaconnection. It follows that the (0, 2) tensor

Fij = J

ai gaj

satisfies the conditionsFij = Fji, rkFij = 0.

The condition rkJij = 0 implies,

J

ah J

bk Rijab = Rijhk,

from which it follows

J

ai aj = J

aj ai, J

ai

2aj = J

aj

2ai, (2.1)

Bochner-flat Kahler manifolds and Riemannian compatibility of the Ricci tensor 13

where2ij= ia

ai .

The Bochner curvature tensor is

Bijhk = Rijhk 1

2(n+ 2)ijhk +

4(n+ 1)(n+ 2)Gijhk,

where

ijhk = gik jh + gjh ik gih jk gjk ih

+Fik Jaj ah + Fjh J

ai ak FihJ

aj ak Fjk J

ai ah (2.2)

2Fij Jah ak 2Fhk J

ai aj ,

and

Gijhk = gikgjh gihgjk + FikFjh FihFjk 2FijFhk. (2.3)

Thus, if the Kahler manifold is Bochner-flat, then

Rijhk =1

2(n+ 2)ijhk

4(n+ 1)(n+ 2)Gijhk, (2.4)

and therefore

iaRahjk + jaR

ahki + kaR

ahij =

1

2(n+ 2)

iaahjk + ja

ahki + ka

ahij

4(n+ 1)(n+ 2)

iaGahjk + jaG

ahki + kaG

ahij

.

But, according (2.2) and (2.3), we have

iaahjk + ja

ahki + ka

ahij = 2

h

FhiJaj

2ak FhjJ

ak

2ai FhkJ

ai

2aj

+J

ah

Fij2ak +Fjk

2ai Fki

2aj

i

and

iaGahjk + jaG

ahki + kaG

ahij = 2

h

FhiJaj ak FhjJ

akai FhkJ

ai aj

+J

ah

Fijak + Fjkai + Fkiaj

i

,

14 M. Prvanovic

such that the Ricci tensor is Riemannian compatible if and only if

FhiJaj

2ak

2(n+ 1)ak

FhjJak

2ai

2(n+ 1)ai

FhkJai

2aj

2(n+ 1)aj

+J

ah

"

Fij

2ak

2(n+ 1)ak

+ Fjk

2ai

2(n+ 1)ai

+Fki

2aj

2(n+ 1)aj

#

= 0. (2.5)

Transvecting (2.5) with J

ht and then contracting with g

tj , we find

2ij

2(n+ 1)ij = fgij , (2.6)

where f is a scalar function.Conversely, if the relation (2.6) holds, then

iaRahjk + jaR

ahki + kaR

ahij =

f

2(n+ 2)

FhiFjk FhjFki FhkFij

+FijFhk + FjkFhi + FkiFhj

= 0,

and the Ricci tensor is Riemannian compatible.Thus, we can state the following theorem:

Theorem 2.1. The Ricci tensor of a Bochner-flat Kahler manifold is Riemanniancompatible if and only if it satisfies the condition (2.6).

3. Kahler manifold of quasi-constant holomorphic sectional curvature

The Kahler manifold is said to be of quasi-constant holomorphic sectional cur-vature if

R = L0G+ L1 + L2 , (3.1)


where L0, L1 and L2 are some scalar functions and the tensors and are definedas follows

ijhk = gikVjh + gjhVik gihVjk gjkVih + FikJai Vah + FjhJ

ai Vak

FihJaj Vak FjkJ

ai Vah 2FijJ

ah Vak 2FhkJ

ai Vaj ,

ijhk = J

ai Vhj J

bh Vbk,

Vij = vivj + J

ai J

bj vavb,

where v is a vector field. Without loss of generality, in behalf of the functions L1 andL2 we can suppose thet v is unit vector field. Then

ViaVjb gab = Vij , Vab g

ab = 2, (3.2)

and because J

ai va? vi, we also have

J

ai J

bj Vab = Vij . (3.3)

The class of Kahler manifolds of quasi-constant holomorphic sectional curva-ture is analogous to the class of Riemannian manifolds of quasi-constant sectionalcurvature. It was appeared first in the papers [5], [6] and [7] dedicated to the holo-morphically subprojective Kahler manifolds. In [1] the authors considered the Kahlermanifolds having the following property: for any p 2 M , and any angle ↵ 2 [0,/2],all holomorphic planes in the tangent vector space Tp(M) making the angle ↵ withgiven vector v 2 Tp(M), have the same holomorphic sectional curvature. Theyproved, among others, that the Riemannian curvature tensor of such manifolds hasthe form (3.1). If L2 = 0, that is if

R = L0G+ L1 (3.4)

the manifold is Bochner-flat. The Ricci tensor, corresponding to the tensor (3.4), is

ji = a gij + b Vij , (3.5)

where

a = 2

(n+ 1)L0 + L1

, b = 2(n+ 2)L1, (3.6)

such that, in view of (3.2) and (3.3), we have

2ij= a

2gij + b(2a+ b)Vij .

16 M. Prvanovic

Also, = 2(na+ b).

Thus,

2ij

2(n+ 1)ij =

a(a b)

n+ 1gij +

b

n+ 1

h

(n+ 1)a+ nb

i

Vij .

The right hand side is of the form fgij if b = 0, or (n+ 2)a+ nb = 0.According (3.6), the condition b = 0 means that L1 = 0. But then (3.4) reduces

to R = L0G, and the manifold is of constant holomorphic sectional curvature. Asfor the condition (n + 2)a + nb = 0, it is L0 + L1 = 0 in view of (3.6). Thus, andin view of Theorem 2.1, we can state the following result:

Theorem 3.1. The Ricci tensor of Bochner-flat Kahler manifold of quasi-constantholomorphic sectional curvature is Riemannian compatible if and only if L1 = L0.

Theorem 3.1 gives an example of Bochner-flat Kahler manifold satisfying thecondition (2.5), i.e. satisfying Riemannian compatibility of the Ricci tensor. But,ifL1 = L0, the Ricci tensor is Riemannian compatible even if L2 6= 0, thet is eventhe manifold is not Bochner-flat. Namely, in the case (3.1), the Ricci tensor is

ij = a gij + b1 Vij , (3.7)

whereb1 = 2(n+ 2)L1 L2 = b L2.

Thus, taking into account that the tensors G, and satisfy the first Bianchiidentity, we have

iaRahjk + jaR

ahki + kaR

ahij = (b L2)

n

L0

ViaGahjk + VjaG

ahki + VkaG

ahij

+ L1

Viaahjk + Vja

ahki + Vka

ahij

(3.8)

+L2

Via ahjk + Vja

ahki + Vka

ahij

o

.

In view of (3.2 and (3.3), we have

ViaGahjk + VjaG

ahki + VkaG

ahij = Via

ahjk + Vja

ahki + Vka

ahij

= 2

FhjJak Vai FhkJ

ai Vaj FhiJ

aj Vak

+FjkJah Vai + FkiJ

ah Vaj + FijJ

ah Vak

,

whileVia

ahjk + Vja

ahki + Vka

ahij = 0,


such that the relation (3.8) becomes

iaRahjk + jaR

ahki + kaR

ahij

= 2(b L2)(L0 + L1)h

FhjJak Vai FhkJ

ai Vaj FhiJ

aj Vak

+FjkJah Vai + FkiJ

ah Vaj + FijJ

ah Vak

i

.

Thus, the Ricci tensor is Riemannian compatible if L1 = L0, or b1 b L2 = 0,or

FhjJakVai FhkJ

ai Vaj FhiJ

aj Vak + J

ah

FjkVai +FkiVaj +FijVak

= 0. (3.9)

If (3.9) holds, proceeding like as in the case of the condition (2.4), we get

Vij =1

n

gij , i.e., vivj + J

ai J

bj vavb =

1

n

gij ,

from which, contracting with v

j , we get vi = 1n vi. Thus, if dimM > 2, the relation

(3.9) can not hold.If b1 = 0, the relation (3.7) reduces to ij = a gij , (M, g, J) is the Einstein

manifold and the Ricci tensor is trivially Riemannian compatible.Thus, we can state the following result:

Theorem 3.2. The Ricci tensor of the Kahler manifold of quasi-constant holo-morphic sectional curvature is Riemannian compatible if L1 = L0 or if the mani-fold is Einstein one.

We get from (3.7)2ij= a2 gij + b2 Vij ,

wherea2 = (a1)

2 = a

2, b2 = b1(2a1 + b1).

In general,pij=

p1 ia

aj = ap gij + bp Vij ,

where ap and bp are some functions of L0, L1 and L2. Proceeding in the same wayas in the case of the condition (3.7), we can state the following theorem:

Theorem 3.3. The tensorpij of the Kahler manifold of quasi-constant holomor-

phic sectional curvature, is Riemannian compatible if L1 = L0 or if bp = 0. In thelast case it is trivially Riemannan compatible.

18 M. Prvanovic

REFERENCES

[1] C. L. Bejan, M. Benjounes, Kahler manifolds of quasi-constant holomorphic sectionalcurvature, J. Geom. 88 (2008), 1–14.

[2] R. Deszcz, M. Glogowska, J. Jelowicki, M. Petrovic-Torgasev, G. Zafindratafa, OnRiemann and Weyl compatible tensors, Publ. Inst. Math.(Beograd), (N.S.) 94 (108)

(2013), 111–124.

[3] C. A. Montica, L. C. Molinari, Riemannian compatible tensors, Colloqu. Math. 128,No. 2 (2012), 197–210.

[4] C. A. Montica, L. C. Molinari, Weyl compatible tensors, ar. XIV: 1212–1273 V3[Math.-ph] 21 Jan 2013.

[5] S. Yamaguchi, T. Adati, On holomorphically subprojective Kahler manifolds. I, Ann.Mat. Pura Appl. (4) 112 (1977), 217–229.

[6] S. Yamaguchi, T. Adati, On holomorphically subprojective Kahlerian manifolds. II,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 60 (1976), no. 4, 405–413.

[7] S. Yamaguchi, T. Adati, On holomorphically subprojective Kahlerian manifolds. III,Ann. Mat. Pura Appl. (4) 113 (1977), 111–125.

Serbian Academy of Sciences and ArtsKnez Mihailova 3511000 BeogradSerbia



ON AN OLD/NEW DEGREE–BASED TOPOLOGICAL INDEX

BORIS FURTULA, IVAN GUTMAN, ZANA KOVIJANIC VUKICEVIC,GIORGI LEKISHVILI, GORAN POPIVODA

(Presented at the 3rd Meeting, held on April 24, 2015)

A b s t r a c t. Let G be a graph with vertex sex V (G) and let d(x) be the degreeof the vertex x 2 V (G). The graph invariant F =

Px2V (G)

d(x)3 played some role in a

paper published in 1972, but has not attracted any attention until quite recently. In 2014an unexpected chemical application of the F -index was discovered, which motivated us toestablish its basic mathematical properties. Results obtained along these lines are presented.

AMS Mathematics Subject Classification (2000): 05C07, 92E10, 05C90.Key Words: F -index, forgotten topological index, degree (of vertex), Zagreb indices,

degree–based topological indices.

1. Introduction

Let G be a simple graph with n vertices and m edges, with vertex set V (G) andedge set E(G). The edge connecting the vertices x and y will be denoted by xy.

The degree of the vertex x , denoted by d(x) , is the number of first neighbors ofx in the underlying graph. Since the 1970s, two degree–based graph invariants havebeen extensively studied. These are the first Zagreb index M1 and the second Zagreb

20 B. Furtula, I. Gutman, Z. Kovijanic Vukicevic, G. Lekishvili, G. Popivoda

index M2 , defined as

M1 = M1(G) =X

x2V (G)

d(x)2, (1.1)

M2 = M2(G) =X

xy2E(G)

d(x) d(y) . (1.2)

Details on the two Zagreb topological indices, including their history, can befound in the reviews [26, 15, 5] published on the occasion of their 30th anniversary,and in the recent articles [13, 16]. The Zagreb index M1 was first time encounteredin a paper published in 1972 [19] where a series of approximate formulas for total-electron energy E were deduced. By means of these formulas, several structuraldetails have been identified, on which E depends. Among these was the sum ofsquares of the vertex degrees of the underlying molecular graph (in [19] denotedby 2

1). Eventually, it attracted much attention, has been subject of hundreds ofresearches, and became traditionally called the first Zagreb index and denoted by M1

(for details see [13]).In the same paper [19], in the same approximate formulas for E, there was also

a term equal to the sum of cubes of the vertex degrees (in [19] denoted by 31).

For reasons not easy to comprehend, this latter term did not attract any attention,and in the next more than 40 years was completely ignored by scholars doing re-search on degree–based topological indices. Recall that nowadays several dozens ofdegree–based topological indices are in the focus of interest of mathematicians andmathematical chemists, with a legion of published papers; for details see the books[30, 31, 28] and the surveys [18, 11, 12, 32].

In connection with the preparation of the article [13], we became interested in the“forgotten” topological index

F = F (G) =X

x2V (G)

d(x)3 . (1.3)

What first had to be decided was if this degree–based graph invariant deserves to bestudied at all. Under “deserves” is meant that it has some outstanding applicationor unexpected mathematical property. After a number of failures, we discovereda remarkable fact that the linear combination M1 + F yields a highly accuratemathematical model of certain physico–chemical properties of alkanes [10]. Thissuccess encouraged us to search for mathematical properties of the F -index. Thepresent article outlines the main results obtained so far.

On an old/new degree–based topological index 21

2. Encountering the F -index in previous works

The claim that between 1972 and 2014, the degree–based topological index F ,Eq. (1.3), was completely ignored, needs to be somewhat corrected.

2.1. Measures of irregularity

A graph whose all vertex degrees are mutually equal is said to be regular. If somevertex degrees differ, then the graph is irregular. Several approaches were proposedto measure the irregularity of a graph [17, 1]. Of those based on vertex degrees, themost thoroughly investigated are the Albertson index [2, 9]

X

xy2E(G)

|d(x) d(y)|

and the Bell index [4]X

x2V (G)

d(x) 2m

n

2

.

Interestingly, one of the most obvious such measures, namely

IRM(G) :=X

xy2E(G)

d(x) d(y)

2

seems to have been never mentioned in the literature.1 It is easy to show that

IRM(G) = F (G) 2M2(G) . (2.1)

2.2. Reformulated Zagreb index

In 2004, Milicevic et al. [24] defined the “reformulated Zagreb indices” in whichvertex degrees were replaced by edge degrees. These are just the ordinary Zagrebindices, Eqs. (1.1), (1.2), of the line graph L(G) of the underlying graph G. It isimmediate to show that

M1(L(G)) =X

xy2E(G)

d(x) + d(y) 2

2

which leads to

M1(L(G)) = 4m 2M1(G) + 2M2(G) + F (G)

an expression reported in [33].1The quantity IRM was considered by the authors of [17], but was not included into their publica-

tion because of the occurrence of the “disturbing” term F in Eq. (2.1).


2.3. Third Zagreb index

In a recent paper [27], several degree–based topological indices were considered,and one of them named “third Zagreb index”. It was defined as

X

xy2E(G)

d(x) + d(y)

2.

In fact, this quantity is equal to F (G) + 2M2(G).

2.4. Generalized first Zagreb index

Several authors (e.g., [21, 22, 29, 14, 25, 3, 20, 23]) came to the obvious idea togeneralize the first Zagreb index, Eq. (1.1), as

M(p)1 (G) =

X

x2V (G)

d(x)p

with p being a positive real–number (not necessarily an integer). Evidently, our F -index is the special case of M (p)

1 for p = 3. Several properties of the generalizedfirst Zagreb index were shown to hold irrespective of the value of the exponent p,thus holding also for the F -index. This, in particular, is the case of graphs (belongingto some specified class, e.g., trees, unicyclic graphs, bicyclic graphs, . . . ), extremalw.r.t. M (p)

1 . In what follows, such properties will not be considered.Let x 2 V (G) and let f(x) be any function of the vertex x. Then the following

identity is obeyed [7]:

X

x2V (G)

f(x) =X

xy2E(G)

f(x)

d(x)+

f(y)

d(y)

. (2.2)

Special cases of (2.2) for f(x) = d(x)2 and f(x) = d(x)3 are

M1(G) =X

xy2E(G)

d(x) + d(y)

(2.3)

andF (G) =

X

xy2E(G)

d(x)2 + d(y)2

(2.4)

and these relations will be frequently used in the subsequent considerations.


3. Coindices

In 2006, bearing in mind Eqs. (1.2) and (2.3), Doslic [6] put forward the conceptof first and second Zagreb coindices, defined as

M1 = M1(G) =X

xy 62E(G)

d(x) + d(y)

(3.1)

andM2 = M2(G) =

X

xy 62E(G)

d(x) d(y) (3.2)

respectively. In formulas (3.1) and (3.2) it is assumed that x 6= y.The Zagreb coindices of a graph G and of its complement G can be expressed in

terms of the Zagreb indices of G. The respective formulas are collected in the survey[16].

In full analogy with Eqs. (3.1) and (3.2), relying on Eq. (2.4), we can now definethe F -coindex as

F (G) =X

xy 62E(G)

d(x)2 + d(y)2

. (3.3)

Theorem 3.1. Let G be a graph with n vertices and m edges. Let G be thecomplement of G. Then

F (G) = n(n 1)3 6m(n 1)2 + 3(n 1)M1(G) F (G), (3.4)

F (G) = (n 1)M1(G) F (G), (3.5)

F (G) = 2m(n 1)2 2(n 1)M1(G) + F (G) . (3.6)

PROOF. If the degree of the vertex x in G is d, then the degree of the same vertexin G is n 1 d. Bearing this in mind, from Eq. (1.3) we get

F (G) =X

x2V (G)

n 1 d(x)

3

=X

x2V (G)

(n 1)3 3(n 1)2 d(x) + 3(n 1) d(x)2 d(x)3

and Eq. (3.4) follows from (1.1), (1.3), and the fact the the sum of vertex degrees isequal to 2m.


Denote for brevity d(x)2 + d(y)2 by x,y

. Then in view of Eqs. (2.4), (3.3), and(1.1),

X

x2V

X

y2VF (x, y) =

X

xy2Ex,y

+X

xy 62Ex,y

+X

x2Vx,x

= F (G) + F (G) + 2M1(G) .

On the other hand,X

x2V

X

y2V

d(x)2 + d(y)2

= n

X

x2Vd(x)2 + n

X

y2Vd(y)2 = 2nM1(G) .

Therefore,F (G) + F (G) + 2M1(G) = 2nM1(G)

and Eq. (3.5) follows.In order to arrive at the Eq. (3.6), combine (3.4) and (3.5). By (3.5),

F (G) = (n 1)M1(G) F (G)

whereas F (G) can be expressed by means of (3.4). Eq. (3.6) is then obtained byusing the following relation

M1(G) = n(n 1)2 4m(n 1) +M1(G)

from [16].

Remark 3.1. In [16] it was proven that M1(G) = M1(G). From Theorem3.1 we see that an analogous identity for the forgotten topological index, namelyF (G) = F (G), does not hold.

4. Identities for the F -index

Using the same notation as in [13], we denote by G

(H) the number of distinctsubgraphs of the graph G that are isomorphic to H . In particular, we are interestedin

G

(S3) and G

(S4), where Sn

stands for the n-vertex star.

Theorem 4.1. Let G be a graph with n vertices and m edges, and let G

(S3)and

G

(S4) be as specified above. Then

F (G) = 6G

(S3) + 6G

(S4) + 2m. (4.1)


PROOF. Note thatG

(Sp

) =X

x2V (G)

d(x)

p 1

,

which implies

G

(S3) =X

x2V (G)

d(x)2

=

1

2

M1(G) 2m

G

(S4) =X

x2V (G)

d(x)3

=

1

6

F (G) 3M1(G) + 4m

.

Therefore,G

(S3) + G

(S4) =1

6F (G) 1

3m,

which directly leads to Eq. (4.1).

Theorem 4.2. Let G be a graph with n vertices and m edges. Then

F (G) =X

xy2E(G)

d(x) d(y)

2+ 2M2(G) . (4.2)

If, in addition, the graph G is triangle–free, then

F (G) =X

xy2E(G)

d(x) d(y)

2 2M1(G) + 4m+X

i=1

X

j=1

A3

ij

, (4.3)

where A is the adjacency matrix of G.

PROOF. Eq. (4.2) is a straightforward consequence of (1.2) and (2.4) and Eq.(4.3) is obtained from (4.2) by substituting into it the result of Lemma 4.1.

Lemma 4.1. Let G be a triangle–free graph of order n and let A be its adjacencymatrix. Then

nX

i=1

nX

j=1

A3

ij

= 2M1(G) + 2M2(G) 4m. (4.4)

PROOF. Let x and y be adjacent vertices of the graph G and xy the edge con-necting them.

As well known,A3

ij

is equal to the number of walks of length 3 in the graphG, starting at vertex i and ending at vertex j. We first determine the number of walksof length 3, which go over the edge xy.


For the sake of brevity, denote d(x) 1 and d(y) 1 by p and q, respectively.Let the first neighbors of the vertex x be y and x1, x2, . . . , xp . Let the first neighborsof the vertex y be x and y1, y2, . . . , yq . Because G is triangle–free, the verticesx1, x2, . . . , xp, y1, y2, . . . , yq are distinct.

The walks of length 3 that go over the edge xy can be classified as indicated inthe following table:

type countxi

xyyj

& yj

yxxi

pq + pqxi

xyx & xyxxi

p+ pyj

yxy & yxyyj

q + qxx

i

xy & yyj

yx p+ qxyxx

i

& yxyyj

p+ qxyxyx & yxyxy 1 + 1

Thus, the total count of such walks is

(pq + pq) + (p+ p) + (q + q) + (p+ q) + (p+ q) + (1 + 1)

= 2pq + 4(p+ q) + 2

= 2d(x) 1

d(y) 1

+ 4

d(x) + d(y) 2

+ 2

= 2d(x) d(y) + 2d(x) + d(y)

4,

which after summation over all edges of G and by taking into account Eqs. (1.2) and(2.3) yields the relation (4.4).

5. Bounds for the F -index

In [10], the following bounds for the forgotten topological index were estab-lished:2

F (G) 1

2mM1(G), (5.1)

F (G) 1

mM1(G)2 2M2(G), (5.2)

F (G) 2M2(G) +m(n 1)2 . (5.3)

Equality in (5.1) and (5.2) is attained if and only if the graph G is regular. Equalityin (5.3) holds if and only if G = S

n

.2In [10], there is a printing error in the proof and formulation of inequality (5.3).


An improvement of (5.1), namely

F (G) 2m

nM1(G) (5.4)

is obtained from (1.3), by using the Chebyshev inequality:

X

x2V (G)

d(x)3 =X

x2V (G)

d(x) d(x)2 1

n

0

@X

x2V (G)

d(x)

1

A

0

@X

x2V (G)

d(x)2

1

A .

Equality in (5.4) holds also for regular graphs.

Elphic and Reti [8] have recently shown that M2 m(2mn+1). By combiningthis result with (5.3), we get

F (G) m(n2 6n+ 4m+ 6)

with equality if G = Sn

.Let and be the smallest and greatest degree of the graph G. From another

inequality in [8], namely M2 m2m n+ 1 ( 1)(n 1)

, we get

F (G) m(n 2)2 + 4m 2( 1)(n 1)

with equality if G = Sn

.Let a1, a2, . . . , an be non-negative real numbers, such that

a1 + a2 + · · ·+ an

= 1.

Then according to a result by Motzkin and Straus [25], for any graph G of order nand clique number !,

X

ij2E(G)

ai

aj

! 1

2!. (5.5)

If we set ai

= d(i)2/M1(G) , i = 1, 2, . . . , n, then the conditions required by


the Motzkin–Straus theorem are satisfied. Starting with Eq. (2.4), we have

F (G) =X

ij2E(G)

d(i)2 + d(j)2

= M1(G)X

ij2E(G)

ai

+ aj

= M1(G)X

ij2E(G)

ai

aj

1

ai

+1

aj

= M1(G)2X

ij2E(G)

ai

aj

1

d(i)2+

1

d(j)2

2M1(G)2X

ij2E(G)

ai

aj

which by the Motzkin–Straus inequality (5.5) yields

F (G) ! 1

!M1(G)2 .

For triangle–free graphs, ! = 2, and then

F (G) 1

2M1(G)2 .

REFERENCES

[1] H. Abdo, D. Dimitrov, The total irregularity of graphs under graph operations, MiskolcMath. Notes 15 (2014), 3–17.

[2] M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219–225.

[3] V. Andova, M. Petrusevski, Variable Zagreb indices and Karamatas inequality,MATCH Commun. Math. Comput. Chem. 65 (2011), 685–690.

[4] F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992), 45–54.


[5] K. C. Das, I. Gutman, Some properties of the second Zagreb index, MATCH Commun.Math. Comput. Chem. 52 (2004), 103–112.

[6] T. Doslic, Vertex–weighted Wiener polynomials for composite graphs, Ars Math. Con-temp. 1 (2008), 66–80.

[7] T. Doslic, T. Reti, D. Vukicevic, On the vertex degree indices of connected graphs,Chem. Phys. Lett. 512 (2011), 283–286.

[8] C. Elphick, T. Reti, On the relations between the Zagreb indices, clique numbers andwalks in graphs, MATCH Commun. Math. Comput. Chem. 74 (2015), 19–34.

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Faculty of ScienceUniversity of KragujevacP. O. Box 6034000 KragujevacSerbiae-mail: [email protected], [email protected]

Department of MathematicsUniversity of MontenegroP. O. Box 21181000 PodgoricaMontenegroe-mail: [email protected], [email protected]

Department of Medicinal ChemistryTbilisi State Medical UniversityGE–0177 TbilisiGeorgiae-mail: [email protected]



WALK COUNTS AND THE SPECTRAL RADIUS OF GRAPHS

DRAGAN STEVANOVIC

(Presented at the 5th Meeting, held on June 26, 2015)

A b s t r a c t. We develop a new method that uses walk counts for comparing spectralradii of graphs similar in a precisely defined fashion. The method is applied to the caseswhere a path-like or a star-like structure is coalesced to a graph, in order to prove weakinequality in the conjectured inequality of Belardo, Li Marzi and Simic, and to resolve theBrualdi-Solheid problem for the classes of graphs consisting of rooted products with the samerooted graph.

AMS Mathematics Subject Classification (2000): 05C50.Key Words: adjacency matrix, spectral radius, walk counts.

1. Introduction

Study of the spectral radius of adjacency matrix of graphs has been a centralresearch theme in spectral graph theory since its inception in the 1950s [3] to thisday. Numerous results on the spectral radius have been surveyed by Cvetkovic andRowlinson [6] in 1990 and in a recent research monograph of the author [14].

Graphs mostly considered in the literature are simple graphs, due to the fact thattheir adjacency matrix is real and symmetric, so that its eigenvectors can be chosen toprovide an orthonormal basis for Rn [8]. A simple graph G = (V,E) consists of thevertex set V with n = |V | vertices and the edge set E

V2

with m = |E| edges.

34 D. Stevanovic

The adjacency matrix A(G) of the simple graph G is the nn matrix, indexed by V ,defined by

A(G)uv =

(1, if uv 2 E,

0, if uv /2 E.

Let us denote the eigenvalues of A(G) by

1(G) 2(G) · · · n(G),

and the corresponding orthonormal eigenvectors by

x1(G), x2(G), . . . , xn(G),

so thatA(G)xi(G) = i(G)xi(G), i = 1, . . . , n. (1.1)

and for i, j = 1, . . . , n,

xTi (G)xj(G) =

(1, if i = j,

0, if i 6= j.(1.2)

In the sequel we will drop the parameter G when the graph is clear from the context.The eigenvalues and the orthonormality of eigenvectors provide spectral decom-

position of the adjacency matrix [14]:

A =

nX

i=1

ixixTi . (1.3)

The eigenvalues of A are also the roots of its characteristic polynomial

PG() = det(I A). (1.4)

By the Perron-Frobenius theorem [8, Chap. XIII], when the graph G is connected, itsadjacency matrix A is irreducible, so that its largest eigenvalue 1 is also the spectralradius of A. In addition, 1 is a simple eigenvalue with a positive eigenvector x1.

Most of the research on the spectral radius of graphs deals with the Brualdi-Solheid’s general question [2] that asks to characterize graphs with extremal valuesof the spectral radius in a given class of graphs (where extremal usually means max-imal). The basic ingredient in tackling such extremal problems is the ability to com-pare spectral radii of different candidate graphs. Two well-developed techniques aremostly used in the literature for such comparisons.

Walk counts and the spectral radius of graphs 35

The first technique relies on the classical characterization of the largest eigen-value 1 in terms of the Rayleigh quotient of A [8]:

1 = max

y 6=0

yTAy

yT y=

2

Puv2E

yuyvPu2V

y2u, (1.5)

with the maximum attained for and only for y = x1. From here it is easy to comparespectral radii of two graphs, where one of them is obtained by a small modificationof the other one:

a) If pq /2 E then

1(G+ pq) xT1 A(G+ pq)x1

xT1 x1

=

xT1 Ax1

xT1 x1+

2x1,px1,q

xT1 x1> 1,

due to positivity of x1 (and hence of x1,px1,q).

b) If pq 2 E, pr /2 E and x1,q x1,r, then [13]

1(G pq + pr) xT1 A(G pq + pr)x1

xT1 x1

=

xT1 Ax1

xT1 x1+

2x1,p(x1,r x1,q)

xT1 x1> 1.

The equality cannot hold above as in such case one would have that x1 is alsothe principal eigenvector of G pq + pr and that x1,q = x1,r, which wouldthen imply contradictory statement x1,r = 0, by considering the eigenvalueequation (1.1) in both G and G pq + pr at the vertex s.

c) If pq, rs 2 E, pr, qs /2 S and (x1,p x1,s)(x1,r x1,q) 0, then [7]

1(G pq rs+ pr + qs) xT1 A(G pq rs+ pr + qs)x1

xT1 x1

=

xT1 Ax1

xT1 x1+

(x1,p x1,s)(x1,r x1,q)

xT1 x1

1.

36 D. Stevanovic

The second technique relies on the fact that the value of the characteristic polyno-mial PG(y) is positive whenever y > 1. Thus, if one can show that for two graphsG and H holds

(8y > 1(G))PG(y) < PH(y)

then PH(y) cannot have real roots that are greater than or equal to 1(G), so that itmust hold 1(G) > 1(H).

Illustrative examples of the use of the first technique may be found in [10], andthose of the use of the second technique both in [10] and [1].

Our goal here is to propose yet another technique for comparing spectral radiiof two graphs, based on the comparisons of closed walk counts in these graphs. Wehave used comparisons of closed walk counts earlier to compare the Estrada indicesof trees [11]. The technique presented in Section 2. is a comprehensive upgrade ofthe approach used in [11], applied to the spectral radius instead of the Estrada index.In Section 3. we show that the vertices of a path, in the rooted product of a pathand another graph, have unimodal closed walk counts. This result helps to showcasefruitfulness of the walk count technique in Section 4., where we give new proofs ofthe well-known 1979 lemmas of Li and Feng [12], and prove weak inequality in theconjectured inequality of Belardo, Li Marzi and Simic [1].

2. A walk count technique

Let G = (V,E) be a simple, connected graph with the adjacency matrix A, theeigenvalues 1 > 2 · · · n and the orthonormal eigenvectors x1, x2, . . . , xn.We assume that G is nontrivial, i.e., that it contains at least one edge. A sequenceW : u = u0, u1, . . . , uk = v of vertices from V such that uiui+1 2 E is called awalk between u and v in G of length k. A walk W is closed if u = v. The followingclassical result relates the adjacency matrix of a graph to its walk counts:

Theorem 2.1 ([14]). The number of walks of length k, k 0, between the ver-tices u and v in G is equal to (Ak

)u,v.

From the spectral decomposition (1.3) and the orthonormality of eigenvectors (1.2)we now have

Ak=

nX

i=1

ki xix

Ti . (2.6)

For k 0, let Nk denote the number of all walks of length k in G, and let Mk denote


the number of all closed walks of length k in G. From (2.6) we have

Nk =

X

u2V

X

v2V(Ak

)u,v =

nX

i=1

ki

X

u2Vxi,u

!2

, (2.7)

Mk =

X

u2V(Ak

)u,u =

nX

i=1

ki

X

u2Vx2i,u

!=

nX

i=1

ki . (2.8)

Lemma 2.1. For a connected graph G we have

1 = lim

k!1kpNk. (2.9)

If G is not bipartite, then also

1 = lim

k!1kpMk, (2.10)

while if G is bipartite, then1 = lim

k!12kpM2k. (2.11)

The first equality above is taken from [4].

PROOF. All three equalities rely on the Perron-Frobenius theorem [8, ChapterXIII], which implies that 1 |i| for each i = 2, . . . , n, and that the entries of x1in a connected graph G with at least one edge are strictly positive.

The distinction between bipartite and nonbipartite graphs stems from the fact thatif G is bipartite, then the spectrum of G is symmetric with respect to zero [4]. In suchcase, n = 1 is also a simple eigenvalue of G, and if V = V 0 [V 00, V 0 \V 00

= ;,represents a bipartition of G, then the eigenvector corresponding to n satisfies

xn,u =

(x1,u, if u 2 V 0,

x1,u, if u 2 V 00.

Therefore,

2k0+1pN2k0+1 = 1

2k0+1

vuut X

u2V

x1,u

!2

X

u2V

xn,u

!2

+

n1X

i=2

i

1

2k0+1 X

u2V

xi,u

!2

= 12k0+1

vuut2

X

u2V 0

x1,u

! X

u2V 00

x1,u

!+

n1X

i=2

i

1

2k0+1 X

u2V

xi,u

!2

,

38 D. Stevanovic

2k0pN2k0

= 12k0

vuut X

u2V

x1,u

!2

+

X

u2V

xn,u

!2

+

n1X

i=2

i

1

2k0 X

u2V

xi,u

!2

= 12k0

vuut2

X

u2V 0

x1,u

!2

+ 2

X

u2V 00

x1,u

!2

+

n1X

i=2

i

1

2k0 X

u2V

xi,u

!2

.

Eq. (2.9) follows from here, as both X

u2V 0

x1,u

! X

u2V 00

x1,u

!and

X

u2V 0

x1,u

!2

+

X

u2V 00

x1,u

!2

are positive constants, and for each i = 2, . . . , n 1 holds |i/1| < 1, while theterm

Pu2V xi,u

2 does not depend on k.For the closed walks we have M2k0+1 = 0 for k0 0, while

2k0p

M2k0 = 12k0

vuut2 +

n1X

i=2

i

1

2k0

,

from where (2.11) follows, due to |i/1| < 1 for each i = 2, . . . , n 1.On the other hand, if G is not bipartite, then n > 1, so that

kp

Nk = 1k

vuut X

u2Vx1,u

!2

+

nX

i=2

i

1

k X

u2Vxi,u

!2

,

kp

Mk = 1k

vuut1 +

nX

i=2

i

1

k

.

From here both (2.9) and (2.10) follow, sinceP

u2V x1,u2 is a positive constant and

for each i = 2, . . . , n, |i/1| < 1, while the termP

u2V xi,u2 does not depend

on k.

Our first new result is a simple lemma stating that a connected graph with morewalks of arbitrarily large lengths also has the larger spectral radius.

Lemma 2.2. Let G1 and G2 be connected graphs such that for an infinite se-quence of indices k0 < k1 < . . . holds

(8i 0) Nki(G1) Nki(G2). (2.12)

Then 1(G1) 1(G2).


PROOF. From Lemma 2.1 we get

lim

k!1

1(G1)

1(G2)

k

sNk(G2)

Nk(G1)= 1,

which implies

(8" > 0)(9k0)(8k k0)1(G1)

1(G2)> (1 ") k

sNk(G1)

Nk(G2).

The condition (2.12), with i0 taken to be the smallest index such that ki0 k0, nowimplies

(8" > 0)(9i0)(8i i0)1(G1)

1(G2)> 1 ".

However, since 1(G1) and 1(G2) are the constants that do not depend on i, theprevious expression actually means that

(8" > 0)

1(G1)

1(G2)> 1 ",

which is equivalent to 1(G1) 1(G2).

Remark 2.1. In order for previous lemma to imply that 1(G1) is strictly largerthan 1(G2), instead of (2.12) one would need to prove that

(9" > 0)(8i0)(9i i0) Nki(G1) 1 +

"

1 "

ki

Nki(G2),

which is not always feasible.We will, thus, allow our forthcoming results to include equality as a feasible

case. When applied to graphs in a certain class, this essentially means that, whilethese lemmas provide characterization of the extremal value of the spectral radiusof graphs in that class, they cannot provide characterization of all graphs with theextremal spectral radius. Instead, the lemmas will provide just one example of suchextremal graph. In many classes the extremal graph is unique, so that the lemmaswill necessarily pinpoint it, but they cannot be used to prove that there are no otherextremal graphs.

It is obvious from Lemma 2.1 that the previous result can be stated in the termsof closed walk counts as well. We restrict ourselves here to closed walks of evenlength simply to avoid the trouble of considering whether the graphs in question arebipartite or not.

40 D. Stevanovic

Lemma 2.3. Let G1 and G2 be connected graphs such that for an infinite se-quence of indices k0 < k1 < · · · holds

(8i 0) M2ki(G1) M2ki(G2). (2.13)

Then 1(G1) 1(G2).

Let us now define a graph operation that will be the basis for our comparisontechnique.

Definition 2.1. Let F and G be the graphs with disjoint vertex sets V (F ) andV (G). For p 2 N, let u1, . . . , up be distinct vertices from V (F ), and let v1, . . . , vpbe distinct vertices from V (G). Assume, in addition, that there is no pair (i, j),i 6= j, such that both uiuj is an edge of F and vivj is an edge of G. The multiplecoalescence of F and G with respect to the vertex lists u1, . . . , up and v1, . . . , vp,denoted by

F (u1 = v1, . . . , up = vp)G,

is the graph obtained from the union of F and G by identifying the vertices ui and vifor each i = 1, . . . , p.

The multiple coalescence is a generalization of the standard coalescence of twovertex-disjoint graphs, which is obtained by identifying a single pair of vertices, onefrom each graph [5]. Fig. 1 shows an example of multiple coalescence of the graphsF and G, with respect to the selected vertices u1, u2, u3 and v1, v2, v3.

The above assumption that for any i 6= j it is not allowed that both uiuj is anedge of F and vivj is an edge of G, serves to prevent the creation of multiple edgesin the multiple coalescence. This assumption is needed later, as our goal will be tohave each walk in the multiple coalescence clearly separated in smaller parts whoseall edges will belong to only one of its constituents. In such setting, the verticesv1, . . . , vp may be considered as the entrance points for a walk coming from F toenter G (and vice versa).

Our main tool is the following lemma.

Lemma 2.4. Let F and G be graphs with disjoint vertex sets V (F ) and V (G).For p 2 N, choose distinct vertices u1, . . . , up 2 V (F ), and make two separatechoices of distinct vertices v1, . . . , vp 2 V (G) and w1, . . . , wp 2 V (G). Let Gv andGw be the multiple coalescences

Gv= F (u1 = v1, . . . , up = vp)G,

Gw= F (u1 = w1, . . . , up = wp)G,


Figure 1. An example of multiple coalescence of two graphs

such that both Gv and Gw are connected.Let A be the adjacency matrix of G. If for each 1 i, j p (including the case

i = j) and for each k 1 holds

(Ak)vi,vj (Ak

)wi,wj , (2.14)

then1(G

v) 1(G

w).

Note that in the above lemma, while we request that vi 6= vj and wi 6= wj for alli 6= j, the possibility that vi = wj for some i and j is allowed.

PROOF. Let us first count the closed walks of length 2k in Gv. From the fact thatF and G, as constituents of Gv, do not have common edges, we see that the numberof closed walks in Gv, whose all edges belong to the same constituent, is equal toM2k(F ) +M2k(G).

The remaining closed walks in Gv contain edges from both F and G.

W : W0,W1, . . . ,W2l1,

for some l 2 N, such that the edges of the even-indexed subwalks W0, . . . , W2l2 allbelong to F , while the edges of the odd-indexed subwalks W1, . . . , W2l1 all belongto G. As a walk can enter from F to G only through one of the entrance points,

42 D. Stevanovic

we also see that the endpoints of the even-indexed subwalks belong to u1, . . . , up,while the endpoints of the odd-indexed subwalks belong to v1, . . . , vp. Thus, let(i0, . . . , i2l1) denote the 2l-tuple of indices such that

the walk W2j goes from ui2j to ui2j+1(= vi2j+1) in F , while

the walk W2j+1 goes from vi2j+1 to vi2j+2(= ui2j+2) in G,

for j = 0, . . . , l 1. (The addition above is modulo 2l, so that i2l = i0.)In addition, let kj denote the length of the walk Wj for j = 0, . . . , 2l 1. The

4l-tuple(i0, . . . , i2l1; k0, . . . , k2l1)

is called the signature of the closed walk W . Due to the fact that the walk W isclosed, its signatures are rotationally equivalent in the sense that the above signatureis identical to the signature

(i2p, . . . , i2l1, i0, . . . , i2p1; k2p, . . . , k2l1, k0, . . . , k2p1)

for each p = 1, . . . , l 1. In order to assign a unique signature to W , we mayassume its signature is chosen to be lexicographically minimal among all rotationallyequivalent signatures.

Now, let B be the adjacency matrix of F . Then for any feasible signature

(i0, . . . , i2l1; k0, . . . , k2l1)

the number of closed walks in Gv with that signature is equal to

l1Y

j=0

(Bki2j)ui2j ,ui2j+1

l1Y

j=0

(Aki2j+1)vi2j+1 ,vi2j+2

.

The argument is identical for closed walks of length 2k in Gw: the number ofclosed walks, whose all edges belong to the same constituent of Gw, is equal to

M2k(F ) +M2k(G),

while the number of closed walks with the feasible signature (i0, . . . , i2l1;

k0, . . . , k2l1) is equal to

l1Y

j=0

(Bki2j)ui2j ,ui2j+1

l1Y

j=0

(Aki2j+1)wi2j+1 ,wi2j+2

.


From the condition (2.14) we now see that for any feasible signature the numberof closed walks with that signature in Gv is larger than or equal to the number of suchclosed walks in Gw. Summing over all feasible signatures we obtain that

M2k(Gv) M2k(G

w),

and, thus, from Lemma 2.3 we conclude that 1(Gv) 1(G

w).

The usefulness of the above lemma is clearly visible: in order to obtain an in-equality between the spectral radii of the multiple coalescences Gv and Gw it isenough to count just the walks in the G-part of the coalescences–the walk counts inthe F -part have no influence, since the entrance points to F are the same in both Gv

and Gw.

Remark 2.2. Let 1 > 2 · · · n and x1, x2, . . . , xn denote the eigenval-ues and the corresponding orthonormal eigenvectors of the adjacency matrix A of aconnected graph G. Recall that

(Ak)vi,vj =

nX

p=1

kpxp,vixp,vj ,

(Ak)wi,wj =

nX

p=1

kpxp,wixp,wj .

Since 1 has the largest absolute value among all eigenvalues and a positive eigen-vector, the most important summands in the above expressions, especially for largervalues of k, become k

1x1,vix1,vj and k1x1,wix1,wj . It is, thus, tempting to think that

the condition (2.14) in Lemma 2.4 might be replaced by a simpler condition

x1,vix1,vj x1,wix1,wj .

This, however, cannot be done, as shown by the following example. Let u be anarbitrary vertex of the complete graph K50, and let G be the graph shown in Fig. 2.Although

0.41712 x1,a < x1,b 0.45699,

we still have that

49.00123 1(K50(u = a)G) > 1(K50(u = b)G) 49.00083.

The reason for such behavior lies simply in the fact that the degree of a is largerthan the degree of b. Note that the degree of a vertex represents, at the same time, also

44 D. Stevanovic

Figure 2. The vertex a has smaller principal eigenvector component than the vertex b,but there are more closed walks of even lengths up to 12 that start at a than at b

the number of closed walks of length two starting from that vertex. When coalescedwith K50, which has substantially more closed walks than G, the spectral radius ofthe coalescence is roughly determined by the spectral radius of the larger K50, buttends to be fine tuned by the shorter (i.e., the shortest) closed walks in G, of whichthere are more that start at a than those that start at b.

3. On closed walk counts in rooted products of paths and stars

In order to be able to apply Lemma 2.4 we need to exhibit sufficiently manygraphs satisfying (2.14). Paths are among the simplest such graphs. The followinglemma appeared in the authors’ earlier paper with Ilic:

Lemma 3.1 ([11]). Let A be the adjacency matrix of the path Pn on vertices1, . . . , n. Then for every k 0 holds

(Ak)1,1 (Ak

)2,2 · · · (Ak)dn/2e,dn/2e (3.15)

and(Ak

)1,2 (Ak)2,3 · · · (Ak

)bn/2c,bn/2c+1. (3.16)

We reprint here the proof of this lemma from [11], as it serves as the basis for theproof of a more general lemma that follows.

PROOF. We prove slightly more than stated in (3.15) and (3.16): that each diag-onal of Ak, parallel to the main diagonal, is unimodal. Due to the automorphism ofthe path Pn given by ↵ : i ! n + 1 i for i = 1, . . . , n, it is enough to prove thateach of these diagonals is nondecreasing up to its middle entry.

We proceed by induction on k and prove that for all 2 i, j n such thati+ j n+ 1 holds

(Ak)i1,j1 (Ak

)i,j . (3.17)


This is trivial for k = 0 and k = 1, as each diagonal of A0= I and A1

= A is eitherall-zero or all-one. Suppose now that (3.17) has been proved for some k 1. Theexpression Ak+1

= Ak ·A then yields

(Ak+1)i1,j1 = (Ak

)i1,j2 + (Ak)i1,j ,

(Ak+1)i,j = (Ak

)i,j1 + (Ak)i,j+1.

(To avoid dealing separately with the endpoints 1 and n of the path Pn, we simplyassume that (Ak

)i1,0 = 0 and (Ak)i,n+1 = 0 in the above equations.) We have

(Ak)i1,j2 (Ak

)i,j1

from the inductive hypothesis (and the nonnegativity of (Ak)i,j1). If i + j + 1

n+ 1, then(Ak

)i1,j (Ak)i,j+1

also follows from the inductive hypothesis. For i+ j+1 = n+2, from the automor-phism ↵ : i ! n+ 1 i and the symmetry of Ak we have

(Ak)i1,j = (Ak

)n+1j,n+2i = (Ak)i,j+1.

This proves (3.17).

We will now extend this lemma to the rooted products of a path by another graph.

Definition 3.1 ([9]). Let H be a labeled graph on n vertices, and let G1, . . . , Gn

be a sequence of n rooted graphs. The rooted product of H by G1, . . . , Gn, denotedas H[G1, . . . , Gn], is the graph obtained by identifying the root of Gi with the i-thvertex of H for i = 1, . . . , n. In the case when all the rooted graphs Gi, i = 1, . . . , n,are isomorphic to a rooted graph G, we denote H[G, . . . , G| z

n

] simply as H[G,n].

Lemma 3.2. Let n be a positive integer and let G be an arbitrary rooted graph.Denote by G1, . . . , Gn the copies of G, and for any vertex u of G, denote by ui thecorresponding vertex in the copy Gi, i = 1, . . . , n. If A is the adjacency matrix ofthe rooted product Pn[G,n], then for any two (not necessarily different) vertices uand v of G and for every k 0 holds

(Ak)u1,v1 (Ak

)u2,v2 · · · (Ak)udn/2e,vdn/2e (3.18)

and(Ak

)u1,v2 (Ak)u2,v3 · · · (Ak

)ubN/2c,vbN/2c+1. (3.19)

46 D. Stevanovic

PROOF. Let r denote the root vertex of G, so that r1, . . . , rn then also denote thevertices of Pn in the rooted product Pn[G,n].

The number of k-walks between ui and vi whose edges fully belong to Gi is,obviously, equal to the number of k-walks between u and v in G. If a k-walk W be-tween ui and vi contains other edges of Pn[G,n], then let W 0 denote longest subwalkof W such that W 0 is a closed walk that starts and ends at ri: simply, the first edgeof W 0 is the first edge of W that does not belong to Gi, and the last edge of W 0 is thelast edge of W that does not belong to Gi. It is easy to see then that the number ofk-walks between ui and vi in Pn[G,n] is governed by the numbers of walks betweenu and v in G, and the numbers of closed walks (of lengths k and less) that start andend at ri in Pn[G,n]. In particular, the chain of inequalities (3.18) follows from

(Ak)r1,r1 (Ak

)r2,r2 · · · (Ak)rdn/2e,rdn/2e . (3.20)

Similarly, the number of k-walks between ui in the copy Gi and vi+1 in the copy Gi+1

is governed by the numbers of walks between u and r in G (that get mapped to walksbetween ui and ri in Gi), the numbers of walks between r and v in G (that get mappedto walks between ri+1 and vi+1 in Gi+1), and the numbers of walks between ri andri+1 in Pn[G,n]. Thus, the chain of inequalities (3.19) follows from

(Ak)r1,r2 (Ak

)r2,r3 · · · (Ak)rbN/2c,rbN/2c+1

. (3.21)

Similarly as in the proof of Lemma 3.1, (3.18) and (3.19) are the special cases ofthe inequalities

(Ak)ui1,vj1 (Ak

)ui,vj , 2 i, j n, i+ j n+ 1, (3.22)

which are, from the argument above, corollaries of the inequalities

(Ak0)ri1,rj1 (Ak0

)ri,rj , k0 k, 2 i, j n, i+ j n+ 1. (3.23)

We will now prove (3.22) by induction on k. This is trivial for k = 0, as A0= I .

Suppose, therefore, that (3.22) has been proved for all values of k0 up to somek 0. We will now prove that (3.23) holds for k0 = k+1, from which the correctnessof (3.22) for k0 = k + 1 follows as well. (Actually, from the above discussion it iseasy to see that the correctness of (3.22) for k0 = k + 1 follows already from theinductive hypothesis if at least one of u, v is not r. Therefore, one only needs toprove (3.23) for k0 = k + 1.)

Let N(r) denote the set of neighbors of the root r in the graph G. Then

(Ak+1)ri1,rj1 = (Ak

)ri1,rj2 + (Ak)ri1,rj +

X

u2N(r)

(Ak)ri1,uj1 ,

(Ak+1)ri,rj = (Ak

)ri,rj1 + (Ak)ri,rj+1 +

X

u2N(r)

(Ak)ri,uj .


The inequalities(Ak

)ri1,rj2 (Ak)ri,rj1

and(Ak

)ri1,uj1 (Ak)ri,uj

hold by the inductive hypothesis. (We now again assume that (Ak)ri1,r0 = 0 and

(Ak)ri,rn+1 = 0 to avoid dealing separately with the end vertices of the path Pn.)If i+ j + 1 n+ 1, then

(Ak)ri1,rj (Ak

)ri,rj+1

also holds by the inductive hypothesis. For i+j+1 = n+2, from the automorphism : ri ! rn+1i of Pn[G,n] and the symmetry of Ak we have

(Ak)ri1,rj = (Ak

)rn+1j ,rn+2i = (Ak)ri,rj+1 .

This proves (3.23), and consequently (3.22).

In order to be able to prove the conjecture of Belardo, Li Marzi and Simic [1], weneed to consider a slight extension of the previous lemma as well.

Lemma 3.3. Let n be a positive integer and let G be an arbitrary rooted graphwith the root r. Let P+

n [G,n] denote the graph obtained from the rooted prod-uct Pn[G,n] by adding two new pendant vertices r0 and rn+1 and the edges r0r1and rnrn+1 to it. If A is the adjacency matrix of P+

n [G,n], then for any two (notnecessarily different) vertices u and v of G and for every k 0 holds

(Ak)u1,v1 (Ak

)u2,v2 · · · (Ak)udn/2e,vdn/2e (3.24)

and(Ak

)u1,v2 (Ak)u2,v3 · · · (Ak

)ubN/2c,vbN/2c+1. (3.25)

PROOF. The proof of this lemma is fully analogous to the proof of Lemma 3.2,with the difference that now the terms (Ak

)ri1,r0 and (Ak)ri,rn+1 are no longer con-

sidered to be identically equal to 0. We will, therefore, indicate here only the dif-ferences that the introduction of the pendant vertices r0 and rn+1 produces in theproof.

In the proof of (3.22) and (3.23) by induction on k, the basis remains trivial andcan be extended to both k = 0 and k = 1, as the values Aui1,vj1 and Aui,vj arenonzero (and equal to 1) if and only if either i = j and u and v are adjacent in G or|i j| = 1 and both u and v are equal to r.

48 D. Stevanovic

Since we assume 2 i, j n, i + j n + 1 in (3.22), the only difference inthe proof of the inductive step lies in encountering the case j = 2, where we need toadditionally prove that

(Ak)ri1,r0 (Ak

)ri,r1 .

This, however, follows immediately from the fact that r1r0 has to be the last edge inany walk from ri1 to r0, so that

(Ak)ri1,r0 = (Ak1

)ri1,r1

and the inequality

(Ak)ri,r1 = (Ak1

)ri1,r1 + (Ak1)ri+1,r1 +

X

u2N(r)

(Ak1)ui,r1 (Ak1

)ri1,r1 .

Hence (3.23), and consequently (3.22), holds again, which implies the chains of in-equalities (3.24) and (3.25).

Stars form another, even simpler class of graphs that satisfy (2.14). Let c be thecenter, and l1, . . . , ln1 the leaves of the star Sn, n 2. The inequality

(Ak)li,li (Ak

)c,c (3.26)

for i 2 1, . . . , n1 follows easily by induction on k. For k = 0 we have (A0)li,li =

(A0)c,c = 1. Assuming that the inequality (3.26) has been proved up to some k 0,

we then have

(Ak+1)li,li = (Ak

)c,li n1X

j=1

(Ak)c,lj = (Ak+1

)c,c,

simply by observing that any walk that stars at li must use the edge lic first.Inequality (3.26) can also be extended to the rooted products of a star by another

graph.

Lemma 3.4. For n 2, let c be the center and l an arbitrary leaf of the star Sn.Let G be an arbitrary rooted graph. Denote by Gc the copy of G in Sn[G,n] whoseroot is identified with c, and by Gl the copy of G in Sn[G,n] whose root is identifiedwith l. For any vertex u of G, let uc and ul denote the corresponding vertices in Gc

and Gl, respectively. If A is the adjacency matrix of the rooted product Sn[G,n],then for any two (not necessarily different) vertices u and v of G and for every k 0

holds(Ak

)ul,vl (Ak)uc,vc . (3.27)


PROOF. Let r denote the root vertex of G, so that rc and rl become identifiedwith c and l, respectively, in Sn[G,n]. Following the argument from the proof ofLemma 3.2, inequality (3.27) for arbitrary u and v will follow from

(Ak)rl,rl (Ak

)rc,rc . (3.28)

We prove (3.27) by induction on k. This is trivial for k = 0, as A0= I .

Suppose, therefore, that (3.27) has been proved for all values of k0 up to somek 0. We prove that (3.28) holds for k0 = k + 1, from which the correctnessof (3.27) for k0 = k + 1 follows as well. Let N(r) denote the set of neighbors of theroot r in the graph G. Then

(Ak+1)rl,rl = (Ak

)rc,rl +

X

u2N(r)

(Ak)ul,rl ,

(Ak+1)rc,rc (Ak

)rl,rc +

X

u2N(r)

(Ak)uc,rc ,

where in the second expression we have deliberately disregarded k-walks betweenroots of other copies of G and rc. From the inductive hypothesis we have

(Ak)ul,rl (Ak

)uc,rc

for any vertex u 2 N(r). Together with the fact that Ak is symmetric, this proves(3.28), and consequently (3.27).

4. Spectral radii of certain multiple coalescences

As our simplest examples of the use of Lemma 2.4 and the walk count lemmasfrom the previous section, we first provide new proofs for the useful and well-cited1979 lemmas of Li and Feng [12]. Note, however, that the original lemmas claimthe strict inequality between the spectral radii, and that we actually prove the weakinequality here, due to reasons explained in Remark 2.1 on page 39.

Lemma 4.1 ([12]). Let u be a vertex of a connected graph G and for positiveintegers p and q, let Gu

p,q denote the graph obtained from G by adding two pendantspaths of lengths p and q at u. If p q 1, then

1(Gup,q) 1(G

up+1,q1).

Lemma 4.2 ([12]). Let u and v be two adjacent vertices of a connected graph Gand for positive integers p and q, let Gu,v

p,q denote the graph obtained from G byadding pendant paths of length p at u and q at v. If p q 1, then

1(Gu,vp,q ) 1(G

u,vp+1,q1).

50 D. Stevanovic

The proof of Lemma 4.1 follows by observing that Gup,q is a coalescence of the

graph G and the path Pp+q+1 by identifying the vertex u from G and the vertex q+1

of Pp+q+1. (Here the vertices of Pp+q+1 are enumerated with 1, . . . , q + 1, . . . , p +q+1, starting from the endpoint of Pq+1 toward u, and then continuing from u towardthe endpoint of Pp+1.) The inequality

1(Gup,q) = 1(G(u = q + 1)Pp+q+1) 1(G(u = q)Pp+q+1) = 1(G

up+1,q1)

then follows from (3.15) and Lemma 2.4.The proof of Lemma 4.2 further follows by observing that the graphs Gu,v

p,q andGu,v

p+1,q1 are multiple coalescences of the edge-deleted graph G uv and the pathPp+q+2:

Gu,vp,q

=

G uv(u = q + 2, v = q + 1)Pp+q+2,

Gu,vp+1,q1

=

G uv(u = q + 1, v = q)Pp+q+2.

Lemma 2.4 requires that for k 1

(Ak)q+2,q+2 (Ak

)q+1,q+1,

(Ak)q+1,q+1 (Ak

)q,q,

(Ak)q+2,q+1 (Ak

)q+1,q,

which are the special cases of (3.15) and (3.16), with

(Ak)q+2,q+2 = (Ak

)q+1,q+1

in the case p = q due to the automorphism of the path P2q+2.Next, we improve these lemmas by showing their analogs when, instead of a path,

the rooted product of a path gets attached to the basis graph.

Lemma 4.3. Let G be a rooted graph, H a connected graph, and p and q twopositive integers. For a vertex u of H , suppose that H contains a rooted subgraph G0,with u as its root, that is isomorphic to the rooted graph G.

Let Hu,Gp,q denote the graph obtained from H by identifying the rooted subgraph G0

with the (q+1)-st copy of G in the rooted product Pp+q+1[G, p+ q+1] (see Fig. 3).If p q 1, then

1(Hu,Gp,q ) 1(H

u,Gp+1,q1).

Lemma 4.4. Let G be a rooted graph, H a connected graph, and p and q twopositive integers. For two adjacent vertices u and v of H , suppose that H contains


two vertex-disjoint rooted subgraphs G0, with a root u, and G00, with a root v, bothisomorphic to the rooted graph G.

Let Hu,v,Gp,q denote the graph obtained from H by identifying the rooted sub-

graph G0 with the (q+2)-nd copy of G and the rooted subgraph G00 with the (q+1)-stcopy of G in the rooted product Pp+q+2[G, p+ q+2] (see Fig. 3). If p q 1, then

1(Hu,v,Gp,q ) 1(H

u,v,Gp+1,q1).

Figure 3. The graphs Hu,Gp,q and Hu,v,G

p,q

Both of these lemmas follow directly from Lemmas 2.4 and 3.2 by observing thatboth Hu,G

p,q and Hu,v,Gp,q are multiple coalescences.

If H 0 is the graph obtained from H by deleting the edges of G0, then Hu,Gp,q is the

multiple coalescence of H 0 and Pp+q+1[G, p + q + 1], obtained by identifying thecorresponding vertices of G0 in H 0 and the (q+1)-st copy of G in Pp+q+1[G, p+q+1].

If H 00 is the graph obtained from H by deleting the edges of G0 and G00, thenHu,v,G

p,q is the multiple coalescence of H 00 and Pp+q+2[G, p + q + 2], obtained byidentifying the corresponding vertices of G0 and the (q + 2)-nd copy of G in

Pp+q+2[G, p+ q + 2],

and by identifying the corresponding vertices of G00 and the (q + 1)-st copy of Gin Pp+q+2[G, p+ q + 2].

52 D. Stevanovic

In addition, note that the conditions that H has to contain rooted subgraphs iso-morphic to G can be easily removed from the last two lemmas: if H does not containa complete copy of G rooted at u as its subgraph, then we can form H 0 from H byadding the necessary number of isolated vertices, and then apply Lemmas 2.4 and 3.2to the multiple coalescence of H 0 and Pp+q+1[G, p+ q + 1], where the new isolatedvertices are identified with the vertices of the (q + 1)-st copy of G that do not orig-inally appear in H . Similar argument holds in the case of adjacent vertices u and vand the two vertex-disjoint copies of G needed in H . In the extreme case, we canjust identify the vertex u (or u and v) of H with the root(s) of the copies of G in therooted product, and apply Lemmas 2.4 and 3.2 to obtain the following two lemmas:

Lemma 4.5. Let G be a rooted graph with the root r, p and q two positive inte-gers, and let rq and rq+1 denote the roots of the q-th and the (q + 1)-st copies of G,respectively, in the rooted product Pp+q+1[G, p+ q + 1].

If p q 1, then for any connected graph H and any vertex u of H holds

1(H(u = rq+1)Pp+q+1[G, p+ q + 1])

1(H(u = rq)Pp+q+1[G, p+ q + 1]).

Lemma 4.6. Let G be a rooted graph with the root r, p and q two positive inte-gers, and let rq1, rq and rq+1 denote the roots of the (q1)-st, q-th and the (q+1)-stcopies of G, respectively, in the rooted product Pp+q+2[G, p+ q + 2].

If p q 1, then for any connected graph H and any two adjacent vertices uand v of H holds

1(H(u = rq+2, v = rq+1)Pp+q+2[G, p+ q + 2])

1(H(u = rq+1, v = rq)Pp+q+2[G, p+ q + 2]).

The use of Lemma 3.3 instead of Lemma 3.2 further allows us to state Lemmas4.3–4.6 in terms of multiple coalescences with P+

p+q+1[G, p+q+1] and P+p+q+2[G, p+

q+2] as well. This leads to the observation that the weak inequality in the 2009 con-jecture of Belardo, Li Marzi and Simic [1] becomes merely a corollary of Lemmas 2.4and 3.3:

Conjecture 4.1 ([1]). Let G be a rooted graph having r as its root, with deg(r) 2. Denote by G(l,m), with l,m 0, the graph obtained from G by identify-ing r with two pendant vertices of P+

[K1,2, l] and P+[K1,2,m] (see Fig. 4).

If G is not the star K1,2 and l m 1 then

1(G(l,m)) > 1(G(l + 1,m 1)). (4.29)


Figure 4. The graph G(l,m) (reprinted from [1]). Large black vertices denote co-cliques of order 2, so that the degrees of the vertices 1, . . . ,m, . . . , l are all equalto

The key here is to observe that the graph G(l,m) is another instance of a mul-tiple coalescence. Let s1, . . . , s2 be distinct neighbors of r in G, and let G be theedge-deleted subgraph

G= G rs1 · · · rs2.

Next, let um+1 be the root of the (m+ 1)-st copy of K1,2 in the graph

P+[K1,2, l +m+ 1]

(counting the copies of K1,2 backwards from the m-end in Fig. 4), and lettm+1,1, . . . , tm+1,2 denote the leaves adjacent to um+1 in P+

[K1,2, l+m+1].The graph G(l,m) from the conjecture above is then a multiple coalescence

G(l,m)

=

G(r = um+1, s1 = tm+1,1, . . . , s2 = tm+1,2)H(m+ 1 + l),

for which the application of Lemmas 2.4 and 3.3 yields the weak inequality in (4.29).The combination of Lemmas 2.4 and 3.4 yield the following lemmas on multiple

coalescence with rooted products of a star by another graph.

Lemma 4.7. For n 2, let c be the center and l an arbitrary leaf of the star Sn.Let G be a rooted graph and let H be a connected graph. For a vertex u of H ,suppose that H contains a rooted subgraph G0, with u as its root, that is isomorphicto the rooted graph G.

Let H l be the multiple coalescence of H and Sn[G,n], obtained by identifyingthe rooted subgraph G0 with a copy of G rooted at l in Sn[G,n], and let Hc be the

54 D. Stevanovic

multiple coalescence of H and Sn[G,n], obtained by identifying the rooted subgraphG0 with a copy of G rooted at c in Sn[G,n]. Then

1(Hc) 1(H

l).

Lemma 4.8. For n 2, let c be the center and l an arbitrary leaf of the star Sn.Let G be a rooted graph with the root r, and let rc and rl denote the roots of copiesof G rooted at c and l, respectively, in the rooted product Sn[G,n]. Let H be aconnected graph and u an arbitrary vertex of H . Then

1(H(u = rc)Sn[G,n]) 1(H(u = rl)Sn[G,n]).

Lemmas 4.5 and 4.8 enable us to solve the Brualdi-Solheid problem for theclasses of graphs consisting of rooted products with the same rooted graph G.

Theorem 4.1. Let G be an arbitrary rooted graph. If T is a tree on n vertices,then

1(Pn[G,n]) 1(T [G,n]) 1(Sn[G,n]). (4.30)

PROOF. If T is not the path Pn, then let u be a vertex of T with deg(u) 3 andthe largest eccentricity (= the maximum distance from u to any other vertex of T ).The vertex u cannot lie on a path between any two vertices of degrees at least three, asthen one of them would have eccentricity larger than u. This shows all other verticesof T with degree at least three belong to only one of the deg(u) subtrees of T u.Consequently, the remaining deg(u) 1 2 subtrees of T u represent pendantpaths of T attached at u. Let P 0 and P 00 be two such pendant paths of lengths pand q, respectively, and let T be the tree obtained by deleting the vertices of thesepaths (other than u) from T . Let v1, . . . , vq+1 denote the first q + 1 vertices of thepath Pp+q+1 of length p+ q, counting from one of the endpoints. Tree T can then berepresented as a multiple coalescence

T =

T(u = vq+1)Pp+q+1,

and from Lemma 4.5 we then obtain that

1(T [G,n]) = 1(T(u = vq+1)Pp+q+1[G,n])

1(T(u = vq)Pp+q+1[G,n])

... 1(T

(u = v1)Pp+q+1[G,n]).


The degree of u in tree T 0= T

(u = v1)Pp+q+1 has, however, decreased by one.Repeating the above procedure as long as the tree contains vertices of degree at leastthree, we eventually obtain that 1(T [G,n]) 1(Pn[G,n]).

With respect to the right-hand side inequality in (4.30), let u be a vertex of Twith d = deg(u) 2 and the largest eccentricity. Let v1, . . . , vd be the neighborsof u in T . The vertex u cannot lie on a path between any two other vertices of degreeat least two, as one of them would then have eccentricity larger than u. If T is notthe star Sn, then exactly one neighbor of u, say v1, has degree at least two, while theremaining neighbors v2, . . . , vd all have degree one. Let

T 0= T uv2 · · · uvd + v1v2 + · · ·+ v1vd.

Further, let T be the tree obtained from T by deleting vertices u, v2, . . . , vd. If cand l are the center and an arbitrary leaf of the star Sd+1, then both T and T 0 can berepresented as multiple coalescences:

T =

T(v1 = l)Sd+1,

T 0 =

T(v1 = c)Sd+1.

From Lemma 4.8 we then obtain that

1(T [G,n]) = 1(T(v1 = l)Sd+1[G,n])

1(T(v1 = c)Sd+1[G,n]) = 1(T

0[G,n]).

The degree of u in T 0 is, however, equal to one. Repeating the above procedure aslong as the tree contains at least two vertices of degree at least two, we eventuallyobtain that 1(T [G,n]) 1(Sn[G,n]).

Theorem 4.2. Let G be an arbitrary rooted graph. If H is a connected graph onn vertices, then

1(Pn[G,n]) 1(H[G,n]) < 1(Kn[G,n]),

where Kn denotes the complete graph on n vertices.

PROOF. From the fact that Kn[G,n] contains H[G,n] as a proper subgraph forany H 6

=

Kn, we immediately see that 1(H[G,n]) < 1(Kn[G,n]), as the spec-tral radius of a connected graph strictly increases with the addition of edges (seeitem a) on page 35). From the same reason, if T is an arbitrary spanning treeof H , then 1(T [G,n]) 1(H[G,n]). From the previous theorem, we then have1(Pn[G,n]) 1(T [G,n]) 1(H[G,n]).

56 D. Stevanovic

5. Conclusion

We have developed a new method for comparing spectral radii of adjacency ma-trices of graphs, that applies to graphs that can be represented as multiple coales-cences of the same basis graph with different smaller subgraphs. The method, basedon Lemma 2.4, works by comparing walk counts in the smaller subgraphs in order toimply inequality between spectral radii for the whole graphs. We have further devel-oped a number of walk count lemmas for cases when smaller subgraphs are rootedproducts of paths or stars by another graph. Most of the results in this manuscriptare named lemmas, as we expect them to become useful ingredients in the proofs offurther results. Examples of such results here include the proof of weak inequalityin the 2009 conjecture of Belardo, Li Marzi and Simic [1], and the solution of theBrualdi-Solheid problem for the classes of graphs consisting of rooted products withthe same rooted graph.

REFERENCES

[1] F. Belardo, E. M. Li Marzi, S. Simic, Bidegreed trees with small index, MATCH Com-mun. Math. Comput. Chem. 61 (2009), 503–515.

[2] R. A. Brualdi, E. S. Solheid, On the spectral radius of complementary acyclic matricesof zeros and ones, SIAM J. Algebra. Discrete Method. 7 (1986), 265–272.

[3] L. Collatz, U. Sinogowitz, Spektren endlicher Graphen (in German), Abh. Math.Semin. Univ. Hamb. 21 (1957), 63–77.

[4] D. M. Cvetkovic, Graphs and their spectra (Grafovi i njihovi spektri) (Ph.D. Thesis),Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 354–356 (1971), 1–50.

[5] D. M. Cvetkovic, M. Doob, H. Sachs, Spectra of graphs—theory and application, Aca-demic Press, New York, 1980.

[6] D. Cvetkovic, P. Rowlinson, The largest eigenvalue of a graph: A survey, Linear Mul-tilinear Algebra 28 (1990), 3–33.

[7] D. Cvetkovic, P. Rowlinson, S. Simic, Eigenspaces of graphs, Cambridge UniversityPress, Cambridge, 1997.

[8] F. R. Gantmacher, The theory of matrices, Vol. II, Chelsea Publishing Company, NewYork, 1959.

[9] C. D. Godsil, B. D. McKay, A new graph product and its spectrum, Bull. Austral. Math.Soc. 18 (1978) 21–28.


[10] P. Hansen, D. Stevanovic, On bags and bugs, Discrete Appl. Math. 156 (2008), 986–997.

[11] A. Ilic, D. Stevanovic, The Estrada index of chemical trees, J. Math. Chem. 47 (2010),305–314.

[12] Q. Li, K. E. Feng, On the largest eigenvalue of graphs (in Chinese), Acta Math. Appl.Sinica 2 (1979), 167–175.

[13] P. R. Rowlinson, More on graph perturbations, Bull. Lond. Math. Soc. 22 (1990), 209–216.

[14] D. Stevanovic, Spectral radius of graphs, Academic Press, Amsterdam, 2015.

Mathematical InstituteSerbian Academy of Science and ArtsKnez Mihajlova 3611000 BelgradeSerbiae-mail: [email protected]



FIXED POINT OF MAPPINGS OF PEROV TYPE FOR w-CONEDISTANCE

MARIJA CVETKOVIC, VLADIMIR RAKOCEVIC


A b s t r a c t. In this paper, we investigate fixed points of mappings of Perov type for w-cone distance. Many results of Ciric, Lakzian and Rakocevic, Suzuki and Takahashi, Abbasand Rhoades, Pathak and Shahzad and Raja and Veazpour are generalized. Our results couldnot be obtained by Du’s scalarization method because, in our case, contractive constant isreplaced by an operator.

AMS Mathematics Subject Classification (2000): 47H10; 54H25.Key Words: Perov’s theorem, w-cone distance, cone metric space.

1. Introduction and preliminaries

Kada, Suzuki and Takahashi [12] introduced w-distance in 1996. and indicatedthat it is more general concept than metric. They gave examples of w-distance andimproved Caristi’s fixed point theorem [3], Eklands variationals principle [8] and thenonconvex minimization theorem according to Takahashi [18].

Definition 1.1 ([12]). Let X be a metric space with metric d. Then a functionp : X X ! [0,1) is called a w-distance on X if the following are satisfied:

(1) p(x, z) p(x, y) + p(y, z), for any x, y, z 2 X;

60 M. Cvetkovic, V. Rakocevic

(2) for any x 2 X , p(x, ·) : X ! [0,1) is lower semi-continuous;(3) for any " > 0, there exists > 0 such that p(z, x) and p(z, y) imply

d(x, y) ".

Example 1.1. If (X, k·k) is a normed space, then w(x, y) = kxk+kyk, x, y 2 X ,is an w-distance on X . Obviously, every metric d is an w-distance.

Many papers considering different fixed point conditions with w-distance wererecently published.

Kurepa [13] in 1934 has initiated the idea of more general concept of metricspace that was later introduced by Zabreiko [20] as K-metric space and by Huangand Zhang [10] as cone metric space. There were published many results concerningfixed point theorems on both normal and non-normal cone metric spaces in the senseof Huang and Zhang (see e.g., [1, 11, 20]).

Du [7] has noticed that some fixed point theorems on cone metric spaces could beeasily derived for similar results on metric space by scalarization. His scalarizationmethod establishes many equivalences between fixed point results on metric and conemetric spaces, but it is important to mention that there exists many exceptions wherethe scalarization method isn’t applicable. The results presented in this article couldnot be derived from analogous results on metric spaces by Du’s scalarization methodsince the contractive condition contains an operator instead of a constant.

In 2013 Ciric, Lakzian and Rakocevic [5] generalized w-distance concept of Kadaet al. to the tvs-cone metric space where the underlying cone is in topological vectorspace instead of Banach space as in [10]. Therefore, they generalized many resultsincluding [1], [9], [15] and [19] and established some unsolved problems.

Let us remark that Perov [16] studied Banach contraction principle on a general-ized metric space. He replaced the contractive constant with a matrix with nonnega-tive entries and spectral radius less than 1. His generalized metric space is a specialcase of a normal cone metric space. In this article, we investigate fixed points ofmappings of Perov type for w-cone distance but we include a bounded linear opera-tor instead of a contractive constant.

For the convenience of the reader, we give some basic definitions and propertiesrelated to cone metric spaces and w-cone distance that are presented in [1, 5, 10, 12].

Let E be a real Banach space and 2 E the zero vector. A subset P of E iscalled a cone if:

(i) P is closed, nonempty and P 6= ;(ii) a, b 2 R, a, b 0, and x, y 2 P imply ax+ by 2 P ;(iii) P \ (P ) = .

Given a cone P E, the partial ordering with respect to P is defined by x y if

Fixed point of mappings of Perov type for w-cone distance 61

and only if y x 2 P . We shall write x y to indicate that x y but x 6= y, whilex y denotes y x 2 intP (interior of P ).

The cone P in a real Banach space E is called normal if

inf

kx+ yk : x, y 2 P and kxk = kyk = 1

> 0 (1.1)

or, equivalently, if there is a number K > 0 such that for all x, y 2 P ,

x y implies kxk K kyk . (1.2)

The least positive number satisfying (1.2) is called the normal constant of P .The cone P is called solid if int(P ) 6= ;. Further on, in the case that P is a

non-normal, we will assume that P is a solid cone.

Definition 1.2 ([10]). Let X be a nonempty set, and let P be a cone on a realordered Banach space E. Suppose that the mapping d : X X 7! E satisfies:

(d1) d(x, y) for all x, y 2 X and d(x, y) = if and only if x = y;(d2) d(x, y) = d(y, x) for all x, y 2 X;(d3) d(x, y) d(x, z) + d(z, y) for all x, y, z 2 X .Then d is called a cone metric on X and (X, d) is called a cone metric space.

Example 1.2. Let E = C1[0, 1] with kxk = kxk1 + kx0k1 and P = x 2 E :

x(t) 0 on [0, 1]. This cone is not normal. Consider, for example,

xn(t) =1 sinnt

n+ 2

and yn(t) =1 + sinnt

n+ 2

.

Since, kxnk = kynk = 1 and kxn + ynk =

2n+2 ! 0, it follows by (1.1) that P is a

non-normal cone.

Let xn be a sequence in X and x 2 X . If for every c in E with c, thereis n0 such that for all n > n0, d(xn, x) c, then it is said that xn converges to x,and we denote this by limn!1 xn = x, or xn ! x, n ! 1. If for every c in E with c, there is n0 such that for all n,m > n0, d(xn, xm) c, then xn is calleda Cauchy sequence in X . If every Cauchy sequence is convergent in X , then X iscalled a complete cone metric space.

The following properties are often used (particulary when dealing with cone met-ric spaces in which the cone need not to be normal):

(p1) If u v and v w then u w.(p2) If a b+ c for each c 2 intP then a b.(p3) If xn yn for each n 2 N, and lim

n!1xn = x, lim

n!1yn = y, then

x y.


We write B(E) for the set of all bounded linear operators on E and L (E) forthe set of all linear operators on E. B(E) is a Banach algebra, and if A 2 B(E) let

r(A) = lim

n!1kAnk1/n = inf

nkAnk1/n

be the spectral radius of A. Let us remark that if r(A) < 1, then the series1Pn=0

An is

absolutely convergent, I A is invertible in B(E) and1X

n=0

An= (I A)

1.

Furthermore, if kAk < 1, then I A is invertible and

k(I A)

1k 1

1 kAk .

If A,B 2 B(E) and AB = BA, then r(A+ B) r(A) + r(B) and r(AB) r(A)r(B).

2. Main results

We start with two auxiliary results.

Lemma 2.1. Let E be a Banach space and A 2 B(E) a bounded linear opera-tor. If r(A) < 1, then

r(I A)

1 1

1 r(A)

.

PROOF. If r(A) < 1, then, as previously stated, (I A)

1=

1Pn=0

An, and

r(I A)

1= r

I +A(I A)

1 1 + r(A)r

(I A)

1.

Therefore,r(I A)

1 1

1 r(A)

.

Lemma 2.2 ([4]). Let E be Banach space, P E cone in E and A : E 7! E alinear operator. The following conditions are equivalent:

(i) A is increasing, i.e., x y implies A(x) A(y).(ii) A is positive, i.e., A(P ) P .


(p4) If θ d(xn, x) bn and bn → θ, then xn → x.(p5) If c ∈ intP , θ an and an → θ, n→∞, then there exists n0 such that for

any n > n0 we have an c.From the property (p5) it follows that the sequence xn converges to some x ∈

X if d(xn, x) → θ as n → ∞ and xn is a Cauchy sequence if d(xn, xm) → θas n,m → ∞. In the situation with a non-normal cone, we have only one part ofLemmas 1 and 4 from [10]. Also, in this case the fact that d(xn, yn) → d(x, y), ifxn → x and yn → y, is not applicable.

In the definition of w-cone distance, Ciric at al. assumed that E is a real Haus-dorff topological vector space (tvs for short) but instead we will, as for cone metricspace, assume that E is a Banach space.

A mapping T : X 7→ X is a continuous mapping on X if for any x ∈ X and asequence xn ⊆ X such that lim

n→∞xn = x, it follows lim

n→∞Txn = Tx.

Function G : X → P is lower semi-continuous at x ∈ X if for any ε θ, thereis n0 ∈ N such that

G(x) G(xn) + ε, for all n ≥ n0, (1.3)

whenever xn is a sequence in X and xn → x, n→∞.

Definition 1.3 ([5]). Let (X, d) be a cone metric space. Then a function p :X×X → P is called aw-cone distance onX if the following conditions are satisfied:

(w1) p(x, z) p(x, y) + p(y, z), for any x, y, z ∈ X;(w2) For any x ∈ X , p(x, ·) : X → P is lower semi-continuous;(w3) For any ε inE with θ ε, there is δ inE with θ δ, such that p(z, x) δ

and p(z, y) δ imply d(x, y) ε.

It is important to mention that every cone metric is w-cone distance and thereexist w-cone distances such that underlying cone is not normal.

Lemma 1.1 ([5]). Let (X, d) be a tvs-cone metric space and let p be a w-conedistance on X . Let xn and yn be sequences in X , let αn with θ αn, andβn with θ βn, be sequences in E converging to θ, and x, y, z ∈ X . Then:

(i) If p(xn, y) αn and p(xn, z) βn for any n ∈ N, then y = z. In particular,if p(x, y) = θ and p(x, z) = θ, then y = z.

(ii) If p(xn, yn) αn and p(xn, z) βn for any n ∈ N, then yn convergesto z.

(iii) If p(xn, xm) αn for any n,m ∈ N with m > n, then xn is a Cauchysequence.

(iv) If p(y, xn) αn for any n ∈ N, then xn is a Cauchy sequence.


PROOF. If A is a monotonically increasing and p 2 P , it follows p andA(p) A() = . Thus, A(p) 2 P , and A(P ) P .

To prove the other implication, let us assume that A(P ) P and x, y 2 E aresuch that x y. Now, y x 2 P , and so A(y x) 2 P .

Thus, finally A(x) A(y).

In the following theorem, which extends and improves Theorem 2 of [12] andTheorem 1 of [17], we give an estimation for a wcone distance p(xn, z) of anapproximate value xn and a fixed point z.

Theorem 2.1. Let (X, d) be a complete cone metric space with w-cone distance pon X . Suppose that for some increasing operator A 2 B(E), r(A) < 1, a mappingT : X ! X satisfies the following condition:

p(Tx, T 2x) A (p(x, Tx)) , for all x 2 X . (2.4)

Assume that either of the following holds:

(i) If y 6= Ty, there exists c 2 int(P ), c 6= , such that

c p(x, y) + p(x, Tx), for all x 2 X;

(ii) T is continuous.

Then, there exists z 2 X , such that z = Tz and

p(Tnx, z) An(I A)

1(p(x, Tx)) , for n 2 N, (2.5)

where z = lim

n!1Tnx.

Moreover, if y = Ty for some y 2 X , then p(y, y) = .

PROOF. Let x 2 X be arbitrary and define a sequence xn by x0 = x, xn =

Tnx, for any n 2 N. Then from (2.4) we have, for any n 2 N,

p(xn, xn+1) = p(Txn1, Txn)

A(p(xn1, xn)) · · · An(p(x, Tx)), (2.6)

since A is an increasing operator. Thus, if m > n, then from (w1) of Definition 1.3


and (2.6),

p(xn, xm) m1X

i=n

p(xi, xi+1)

m1X

i=n

Ai(p(x, Tx))

1X

i=n

Ai(p(x, Tx))

= An(I A)

1(p(x, Tx)). (2.7)

However An(I A)

1(p(x, Tx)) ! , n ! 1, so xn is a Cauchy sequence

in X by Lemma 1.1 and, because X is a complete, xn converges to some z 2 X .We will prove that z is a fixed point of T by estimating p(xn, z). Since xn ! z,

as n ! 1, from the lower semi-continuity of w distance, we have that for any " ,there is n0 2 N such that for any m n0

p(xn, z) p(xn, xm) + ".

Therefore, for an arbitrary n 2 N, if we choose m > maxn, n0, then, from (2.7),it follows that the inequality

p(xn, z) An(I A)

1(p(x, Tx)) + "

holds for any " , i.e., (2.5) holds for any n 2 N.Let us assume that (i) is satisfied and that Tz 6= z. Then, there exists c ,

c 6= , such that

c p(x, z) + p(x, Tx), for all x 2 X. (2.8)

Obviously, An(I A)

1(p(x, Tx)) ! and An

(p(x, Tx)) ! as n ! 1, so,from the definition of convergence and (p5), there exists n1 2 N such that

An(I A)

1(p(x, Tx)) c

3

and An(p(x, Tx)) c

3

,

for any n n1. The last observation contradicts to (2.8) since, for any n n1

inequalities

c p(xn, z) + p(xn, xn+1)

An(I A)

1(p(x, Tx)) +An

(p(x, Tx))

2c

3

,


imply that c/3 0, i.e., c = . But, we have already assumed that c 6= , henceTz = z in this case.

Otherwise, if T is a continuous, then, since xn+1 = Txn ! Tz, n ! 1, by (i)of Lemma 1.1, we may conclude that Tz = z.

It remains to prove that if Ty = y, then p(y, y) = . Obviously,

p(y, y) = p(Ty, T 2y) A(p(y, Ty)).

The operator (I A)

1=

1Pn=0

An is an increasing linear operator and the last in-

equality gives us p(y, y) (I A)

1() = , i.e., p(y, y) = .

Example 2.1. Let X = E, where E and P are defined as in Example 1.2. Let usdefine cone metric d : X X 7! E for any f, g 2 X by

d(f, g) =

(f + g, f 6= g,

0, f = g.

If T : X 7! X is defined by T (f) = f/2, f 2 X , then

d(Tf, T 2f) A(d(f, Tf)), f 2 X,

where A : E 7! E, is a bounded linear operator defined by A(f) = f/2, f 2 E.Clearly, r(A) = kAk = 1/2 and T is a continuous, thus all the assumptions from

Theorem 2.1 are satisfied. Hence, T has a fixed point f = 0 2 X and it is evidentlyan unique fixed point of T .

Corollary 2.1. Let (X, d) be a complete cone metric space with wcone distancep on X and A 2 B(E) an increasing operator with spectral radius less than 1/2.Suppose that the mapping T : X ! X satisfies either (i) or (ii) of Theorem 2.1 and

p(Tx, T 2x) A(p(x, T 2x)), for all x 2 X .

Then, there exists z 2 X , such that z = Tz and if y = Ty, then p(y, y) = .

PROOF. If x 2 X is arbitrary, then

p(Tx, T 2x) A(p(x, T 2x)) A(p(x, Tx) + p(Tx, T 2x)).

Hence,p(Tx, T 2x) A(I A)

1(p(x, Tx)).


Observe that

r(A(I A)

1) r(A)r

(I A)

1

r(A)

1 r(A)

< 1,

and the condition (2.4) is satisfied. All the conclusions of this corollary followsdirectly from Theorem 2.1.

If T : X ! X and F (T ) is a set of all fixed points of T , then T has a propertyP if F (T ) = F (Tn

) for each n 2 N. The following theorem extends and improvesTheorem 2 of [1] and Theorem 12 of [5] for cone metric space.

Theorem 2.2. Let (X, d) be a complete cone metric space with w-cone distancep on X . Suppose T : X ! X satisfies the condition (2.4) for an increasing operatorA 2 B(E). If r(A) < 1, then T has property P .

PROOF. Obviously, F (T ) F (Tn), n 2 N, so it remains to show that Tz = z

for any z 2 F (Tn) and arbitrary n > 1.

Remark that

p(T iz, T i+1z) = p(T kn+iz, T kn+i+1z) Akn+i(p(z, Tz)), k, i 2 N,

allows us to determine that, because Akn+i(p(z, Tz)) ! 0, as k ! 1, when r(A) <

1, p(T iz, T i+1z) = , i 2 N, and, furthermore, Tz = Tnz = z.

Instead of observing contractive conditions on X , we observe only T -orbit O(x,1)

of an arbitrary element x 2 X where O(x,1) = Tnx | n 2 N0.Function G : X ! P is a T -orbitally lower semi-continuous at x if for any " ,there is n0 2 N such that (1.3) holds whenever xn O(x;1) and xn ! x,n ! 1.

The following theorems implies some results of [9], [15], [10] and [19].

Theorem 2.3. Let (X, d) be a complete cone metric space with w-cone distancep on X and A 2 B(E) an increasing operator with spectral radius less than 1.Suppose that T : X ! X and there exists an x 2 X such that

p(Ty, T 2y) A(p(y, Ty)), for all y 2 O(x,1).

Then,(i) lim

n!1Tnx = z exists and

p(Tnx, z) An(I A)

1(p(x, Tx)) , n 2 N;


(ii) p(z, Tz) = if and only if G(x) = p(x, Tx) is T -orbitally lower semi-continuous at z.

PROOF. (i) First observation easily follows from the proof of Theorem 2.1.(ii) If p(z, Tz) = then G is obviously T -orbitally lower semi-continuous at z.

Otherwise, choose " arbitrary. There exists n1 2 N such that

An(p(x, Tx)) "

2

for any n n1, and n2 2 N such that

G(z) G(Tnx) +"

2

, n n2.

Then, for n maxn1, n2,

p(z, Tz) p(Tnx, Tn+1x) +"

2

An(p(z, Tx)) +

"

2

".

The last inequality holds for any " , and by (p2), p(z, Tz) = .

Theorem 2.4. Let (X, d) be a complete cone metric space with w-cone distancep on X and A 2 B(E) an increasing operator with spectral radius less than 1.Suppose that T : X ! X is a p-contractive mapping of Perov type, i.e.,

p(Tx, Ty) A (p(x, y)) , for all x, y 2 X .

Then, T has a unique fixed point z 2 X , and p(z, z) = .

PROOF. ¿From the proof of Theorem 2.1 we get that Tnx ! z as n ! 1,Tz = z and p(z, z) = .

If Ty = y, then

p(y, z) = p(Ty, Tz) A(p(y, z)) =) p(y, z) (I A)

1() = ,

thus p(y, z) = and p(z, z) = imply, by (i) of Lemma 1.1, that y = z.

Example 2.2. Let X be C[0, 1], set of real continuous functions on a closedinterval [0, 1] with a norm kxk = max

t2=[0,1]|x(t)|, x 2 X , and P X a cone defined

withx 2 P , x(t) 0 for all t 2 [0, 1].


Then, d(x, y) = |x y|, x, y 2 X , is a cone metric on X , where |x|(t) = |x(t)|,t 2 [0, 1], and p = d. If f 2 X is chosen arbitrary, then for 0 < L < 1, define amapping T : X 7! X ,

(Tx) (t) = f(t) +

Z t

0Lx(

ps) ds, t 2 [0, 1].

Remark that T is a continuous mapping.Let us define a bounded linear operator A 2 B(X),

(Ax) (t) =

Z t

0Lx(

ps) ds, t 2 [0, 1].

Zima proved in [21] that the spectral radius of operator A is L/2, thus less than 1

and, evidently, A is an increasing operator. Then, easily follows,

p(Tx, T 2x) A(p(x, Tx)), x 2 X

Hence, we may apply Theorem 2.1 and conclude that there exists g 2 X such thatTg = g.

Remark 2.1. Theorem 2.1 does not imply uniqueness of fixed point but it is notdifficult to show that T has an unique fixed point in X . Let us suppose that Th = h.Then

d(g, h)(t) = d(Tg, Th)(t) =

Z t

0L(g(

ps) h(

ps)) ds

Ltd(g, h)(t),

and Lt < 1 implies d(g, h)(t) = 0 for any t 2 [0, 1], i.e., g = h.

Remark 2.2. Let us remark that De Pascale and De Pascale [6] used K-normedspace to prove that Lou’s fixed point theorem [14] in a space of continuous functionsis equivalent to the Banach contraction principle with contractive constant replacedby bounded linear operator with spectral radius < 1. Observe that in [6] cone is nor-mal, but we have investigated the case when cone is not normal [4]. It is interestingto to investigate possibility of extending Lou’s theorem in the case when cone is notnormal.

Let us notice that the condition A(P ) P is unnecessary in some cases.

Example 2.3. Let

A =

2

664

12 1

4 0

14 1

2 0

0 0

12

3

775 ,


X =

8<

:

2

4x11

x3

3

5 | x1, x3 2 R

9=

; and T : X 7! X a mapping defined by

T

0

@

2

4x11

x3

3

5

1

A=

2

64

12(x1 + 1)

1

13(x3 + 2)

3

75 .

Set kxk = max|x1|, 1, |x3|, x 2 X , and d(x, y) =

2

4|x1 y1|

0

|x3 y3|

3

5, x, y 2 X , and

p = d.It is easy to show that kAk =

34 and, consequently, r(A) < 1. Also,

p(Tx, T 2x) A(p(x, Tx)), x 2 X.

Clearly, A(P ) * P , and (1, 1, 1) is a fixed point of T in X .

We state the similar results when cone metric space (X, d) is normal by replacingthe condition r(A) < 1 and excluding the condition that the operator A is increasing,i.e., not demanding the condition A(P ) P .

Theorem 2.5. Let (X, d) be a complete normal cone metric space with normalconstant K and w-cone distance p on X . Suppose that for some operator A 2 B(E),KkAk < 1, a mapping T : X ! X satisfies the following condition:

p(Tx, T 2x) A (p(x, Tx)) , for all x 2 X .

Assume that either of the following holds:

(i) If y 6= Ty, there exists c > 0, such that

c < kp(x, y)k+ kp(x, Tx)k, for all x 2 X;

(ii) T is continuous.

Then, there exists z 2 X , such that z = Tz and if y = Ty for some y 2 X , thenp(y, y) = .

PROOF. Let x 2 X be an arbitrary and let us define a sequence xn, x0 = x,xn = Tnx, for any n 2 N . Then,

kp(xn, xn+1)k KkAkkp(xn1, xn)k · · · (KkAk)nkp(x, Tx)k. (2.9)


Thus, if m > n, then from (w1) of Definition 1.3 and (2.6),

kp(xn, xm)k m1X

i=n

(KkA)

ikp(x, Tx)k

1X

i=n

(KkAk)ikp(x, Tx)k

=

(KkAk)n

1KkAkkp(x, Tx)k. (2.10)

However,(KkAk)n

1KkAkkp(x, Tx)k ! 0, n ! 1,

so xn is a Cauchy sequence in X and, because X is complete, xn converges tosome z 2 X .

¿From the lower semi-continuity of w distance, we have that for any " , thereis n0 2 N such that for any m n0

p(xn, z) p(xn, xm) + ".

Moreover, for arbitrary n 2 N, if we choose m > n, then from (2.10) it follows thatthe inequality

kp(xn, z)k (KkAk)n

1KkAkkp(x, Tx)k+Kk"k

holds for any " , so, for " := "/n, n 2 N,

kp(xn, z)k (KkAk)n

1KkAkkp(x, Tx)k.

Let us assume that (i) is satisfied and that Tz 6= z. Then, there exists c > 0 such that

c < kp(x, z)k+ kp(x, Tx)k, for all x 2 X. (2.11)

Then,

c <(KkAk)n

1KkAkkp(x, Tx)k+ (KkAk)nkp(x, Tx)k

for any n 2 N and that is impossible since (2.11) holds.Otherwise, if T is continuous, then, since xn+1 = Txn ! Tz, n ! 1, it

follows Tz = z.It remains to prove that if Ty = y, then p(y, y) = . Obviously,

kp(y, y)k = kp(Tny, Tn+1y)k (KkAk)nkp(y, Ty)k, n 2 N,

implies kp(y, y)k = 0.


Corollary 2.2. Let (X, d) be a complete normal cone metric space with normalconstant K, w-cone distance p on X and A 2 B(E) an operator, KkAk < 1/2.Suppose that the mapping T : X ! X satisfies either (i) or (ii) of Theorem 2.1 and

p(Tx, T 2x) A(p(x, T 2x)), for all x 2 X .

Then, there exists z 2 X , such that z = Tz and if y = Ty, then p(y, y) = .

PROOF. If x 2 X is arbitrary, then

kp(Tx, T 2x)k KkAkkp(x, Tx)k+KkAkkp(Tx, T 2x)k.

Hence,

kp(Tx, T 2x)k KkAk1KkAk kp(x, Tx) .

and, since KkAk/(1KkAk) < 1 it directly follows by Theorem 2.5.

Theorem 2.6. Let (X, d) be a complete normal cone metric space with normalconstant K and w-cone distance p on X . Suppose T : X ! X satisfies the condition(2.4) for an operator A 2 B(E). If KkAk < 1, then T has property P .

PROOF. As in the proof of previously stated theorem, it follows that, for anyz 2 F (Tn

) and i 2 N,

kp(T iz, T i+1z)k (KkAk)kn+ikp(z, Tz)k ! 0, k ! 1,

thus Tz = Tnz = z.

The proofs of the following theorems follows similarly as in the case when conemetric space is not normal.

Theorem 2.7. Let (X, d) be a complete normal cone metric space with nor-mal constant K, w-cone distance p on X , A 2 B(E) an operator such that KkAk <1. Suppose T : X ! X and there exists an x 2 X such that

p(Ty, T 2y) A(p(y, Ty)), for all y 2 O(x,1).

Then, (i) lim

n!1Tnx = z exists and

kp(Tnx, z)k (KkAk)n

1KkAk kp(x, Tx)k for n 2 N;

(ii) p(z, Tz) = if and only if G(x) = p(x, Tx) is T -orbitally lower semi-continuous at z.


Theorem 2.8. Let (X, d) be a complete normal cone metric space with normalconstant K and w-cone distance p on X and A 2 B(E) an with such that KkAk <1. Suppose that T : X ! X is a p-contractive mapping of Perov type, i.e.,

p(Tx, Ty) A (p(x, y)) , for all x, y 2 X .

Then, T has a unique fixed point z 2 X , and p(z, z) = .

Acknowledgements. The authors are supported By Grant No. 174025 of theMinistry of Science, Technology and Development, Republic of Serbia.

REFERENCES

[1] M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl.Math. Lett. 22 (2009), 511–515.

[2] I. Beg, A. Azam, M. Arshad, Common fixed points for maps on topological vec-tor space valued cone metric spaces, Int. J. Math. Math. Sci. (2009), 8 pages,doi:10.1155/2009/560264.

[3] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans.Am. Math. Soc. 215 (1976), 241–251.

[4] M. Cvetkovic, V. Rakocevic, Exstensions of Perov theorem, Carpathian J. Math. 31

(2015), 181–188.

[5] Lj. Ciric, H. Lakzian, V. Rakocevic Fixed point theorems for w-cone distance contrac-tion mappings in tvs-cone metric spaces, Fixed Point Theory Appl., 2012, 2012:3

[6] E. de Pascale, L. de Pascale, Fixed points for some non-obviously contractive operators,Proc. Amer. Math. Soc. 130 (2002), 3249–3254.

[7] W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal.72 (2010), 2259–2261.

[8] I. Ekelend, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474.

[9] T. L. Hicks, B. E. Rhoades, A Banach type fixed-point theorem, Math. Jpon. 24 (1979),327–330.

[10] L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractivemappings, J. Math. Anal. Appl. 332 (2) (2007). 1468–1476.


[11] S. Jankovic, Z. Kadelburg, S. Radenovic, On cone metric spaces, a survey, NonlinearAnal. 74 (2011), 2591–2601.

[12] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed pointtheorems in complete metric spaces, Math. Jpon. 44 (1996), 381-591.

[13] Dj. Kurepa, Tableaux ramifis d’ensembles, Espaces pseudo-distancis, C. R. Math.Acad. Sci. Paris 198 (1934), 1563–1565.

[14] B. Lou, Fixed points for operators in a space of continuous functions and applications,Proc. Amer. Math. Soc. 127 (1999), 2259–2264.

[15] H. K. Pathak, N. Shahzad, Fixed point results for generalized quasicontraction map-pings in abstract metric spaces, Nonlinear Anal. 71 (2009,) 6068–6076.

[16] A. I. Perov, On Cauchy problem for a system of ordinary diferential equations, (inRussian), Priblizhen. Metody Reshen. Difer. Uravn. 2 (1964), 115–134.

[17] T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. J.44 (1997), 61–72.

[18] W. Takahashi, A convexity in metric space and nonexpansive mappings, I. Kodai Math.Sem. Pep. 22 (1970), 142–149.

[19] S. M. Veazpour, P. Raja, Some Extensions of Banach’s contraction principle in completecone metric spaces, Fixed Point Theory and Applications, 2008, 2008:768294

[20] P. P. Zabreiko, K-metric and K-normed spaces: survey, Collect. Math. 48 (1997), 825–859.

[21] M. Zima, A certain fixed point theorem and its applications to integral-functional equa-tions, Bull. Austral. Math. Soc. 46 (1992), 179–186.

University of NisFaculty of Science and MathematicsVisegradska 3318000 NisSerbiae-mail: [email protected]

University of NisFaculty of Science and MathematicsVisegradska 3318000 NisSerbiae-mail: [email protected]

Bulletin T.CXLVIII de l’Academie serbe des sciences et des arts − 2015

Classe des Sciences mathematiques et naturellesSciences mathematiques, No 40

DISCRIMINANTLY SEPARABLE POLYNOMIALS: AN OVERVIEW

VLADIMIR DRAGOVIC


A b s t r a c t. The concept of discriminantly separable polynomials has been introduced

by the author some years ago. We review the basic notions and several applications to dif-

ferent areas of mathematics and mechanics which arose in the meantime. Some of the results

were obtained jointly with Dr. Katarina Kukic, a former author’s PhD student.

AMS Mathematics Subject Classification (2000): Primary 37K60; Secondary 14N05,37K20.

Key Words: Quad-equations, integrability, Kowalevski top, elliptic curves, n-valued

groups, discriminantly separable polynomials.

1. Introduction

The concept of discriminantly separable polynomials has been introduced by the

author some years ago [14]. We review the basic notions and indicate several rela-

tionships and applications to different areas of mathematics and mechanics, which

arose in the meantime. Some of the results were obtained jointly with Dr. Katarina

Kukic, a former author’s PhD student.

Although purely algebraic in nature, the concept of discriminantly separable

polynomials emerged within author’s attempt to develop a novel approach to the clas-

sical, celebrated Kowalevski top and a geometrization of the Kowalevski integration

76 V. Dragovic

procedure from [33]. Thus, one direction of applications goes toward continuous in-

tegrable systems and classical mechanics. Geometric applications are related to the

fact that the equations of pencils of conics in appropriate coordinates induce discrim-

inantly separable polynomials. Algebraic and algebro-geometric connections lead

to so-called Buchstaber-Novikov n-valued groups. Beside continuous integrable sys-

tems, discrete integrable systems, namely integrable quad-graphs appear to be closely

related to dicriminantly separable polynomials. Moreover, there is a full parallelism

between a classification of discriminantly separable polynomials and a well-known

ABS classification [2] of quad-graphs. The results presented in this short overview

are obtained in [14], [34], [35], [21], [22], [23], [24], [25], [15], [16].

2. Discriminantly separable polynomials-definition and basic notions

Before giving a formal definition of the discriminantly separable polynomials, let

us recall the equations of a pencil of conics. Denote such an equation as F(w, x1, x2)= 0, where w is the pencil parameter; x1 and x2 are the Darboux coordinates. The

choice of that classical, but mainly forgotten notion of the Darboux coordinates, in-

stead of usual projective coordinates appear to be a subtle point and ”an educated

guess” which has had important consequences to the development of the theory of

discriminantly separable polynomials. These Darboux coordinates [13], (see also

[19]), should not be confused with a well-known Darboux coordinates from simplec-

tic geometry, [3]. We recall some of the details: given two conics C1 and C2 in a

general position by their tangential equations

C1 : a0w21 + a2w

22 + a4w

23 + 2a3w2w3 + 2a5w1w3 + 2a1w1w2 = 0;

C2 :w22 − 4w1w3 = 0.

(2.1)

Then the conics of this general pencil C(s) := C1 + sC2 have four common tangent

lines. Denote the matrix M :

M(s, z1, z2, z3) =

⎡

⎢

⎢

⎣

0 z1 z2 z3z1 a0 a1 a5 − 2sz2 a1 a2 + s a3z3 a5 − 2s a3 a4

⎤

⎥

⎥

⎦

. (2.2)

The coordinate equations of the conics of the pencil are

F (s, z1, z2, z3) := detM(s, z1, z2, z3) = 0,

which determines a quadratic polynomial in the pencil parameter s, namely

F := H +Ks+ Ls2,

Discriminantly separable polynomials: an overview 77

with H , K , and L being quadratic expressions in (z1, z2, z3).Assume the standard projective coordinates (z1 : z2 : z3) in the plane, and

choose, without loss of generality, a rational parametrization of the conic C2 by

(1, ℓ, ℓ2). The tangent line to the conic C2 through a point of the conic with the

parameter ℓ0 is given by the equation

tC2(ℓ0) : z1ℓ

20 − 2z2ℓ0 + z3 = 0.

For a given point P outside the conic in the plane with the coordinates P = (z1, z2, z3),there are two corresponding solutions x1 and x2 of the equation quadratic in ℓ

z1ℓ2 − 2z2ℓ+ z3 = 0. (2.3)

The two solutions correspond to two tangent lines to the conic C2 from the point P .

We will define the pair (x1, x2) as the Darboux coordinates of the point P . One finds

immediately the converse formulae z1 = 1, z2 = (x1 + x2)/2, z3 = x1x2.Changing the variables in the polynomial F from the projective coordinates (z1 :

z2 : z3) to the Darboux coordinates, we rewrite its expression in the form

F(s, x1, x2) = L(x1, x2)s2 +K(x1, x2)s+H(x1, x2).

The key algebraic property of the pencil polynomial written in this form, as a quadratic

polynomial in each of the three variables s, x1, x2 is: all three of its discriminants are

expressed as products of two polynomials in one variable each:

Dw(F)(x1, x2) = P (x1)P (x2), Dxi(F)(w, xj) = J(w)P (xj), i, j = 1, 2,

where J and P are polynomials of degree 3 and 4 respectively, and the elliptic curves

Γ1 : y2 = P (x), Γ2 : y

2 = J(s)

appear to be isomorphic (see Proposition 1 of [14]). Here, and below, we denote by

DxiF(xj , xk), the discriminant of F considered as a quadratic polynomial in xi.As a geometric interpretation of F (s, x1, x2) = 0 we may say that the point P in

the plane, with the Darboux coordinates with respect to C2 equal to (x1, x2) belongs

to two conics of the pencil, with the pencil parameters equal to s1 and s2, such that

F(si, x1, x2) = 0, i = 1, 2.

Now we provide a general definition of the discriminantly separable polynomials.

With Pnm denote the polynomials of m variables of the degree n in each variable.

78 V. Dragovic

Definition 2.1 ([14]). A polynomial F (x1, . . . , xn) is discriminantly separable

if there exist polynomials fi(xi) such that for every i = 1, . . . , n

DxiF (x1, . . . , xi, . . . , xn) =

∏

j =i

fj(xj).

It is symmetrically discriminantly separable if f2 = f3 = · · · = fn, while it is

strongly discriminantly separable if f1 = f2 = f3 = · · · = fn. It is weakly discrimi-

nantly separable if there exist polynomials f ji (xi) such that for every i = 1, . . . , n,

DxiF (x1, . . . , xi, . . . , xn) =

∏

j =i

f ij(xj).

2.1. Two-valued groups

The idea of n-valued groups, on a local level, goes back to Buchstaber and

Novikov (see [9]), to their 1971 study of characteristic classes of vector bundles.

That concept was significantly developed further by Buchstaber and his collaborators

([11] and references therein). An n-valued group on X can be defined as a map:

m : X ×X → (X)n,

m(x, y) = x ∗ y = [z1, . . . , zn],

where (X)n denotes the symmetric n-th power of X and zi coordinates therein. Such

a map should satisfy the following axioms. Associativity: the condition of equality

of two n2-sets[x ∗ (y ∗ z)1, . . . , x ∗ (y ∗ z)n],[(x ∗ y)1 ∗ z, . . . , (x ∗ y)n ∗ z]

for all triplets (x, y, z) ∈ X3. Similarly, an element e ∈ X is a unit if

e ∗ x = x ∗ e = [x, . . . , x],

for all x ∈ X. A map inv : X → X is an inverse if it satisfies

e ∈ inv(x) ∗ x, e ∈ x ∗ inv(x),

for all x ∈ X. Buchstaber says that m defines an n-valued group structure

(X,m, e, inv) if it is associative, with a unit and an inverse.

An n-valued group X acts on a set Y if there is a mapping

φ : X × Y → (Y )n,

φ(x, y) = x y,

such that the two n2-multisubsets x1 (x2 y), (x1 ∗ x2) y of Y are equal for all

x1, x2 ∈ X, y ∈ Y . It is also assumed e y = [y, . . . , y] for all y ∈ Y .


Example 2.1 (A two-valued group structure on Z+, [10]). Let us consider the set

of nonnegative integers Z+ and define a mapping

m : Z+ × Z+ → (Z+)2,

m(x, y) = [x+ y, |x− y|].

This mapping provides a structure of a two-valued group on Z+ with the unit e = 0and the inverse equal to the identity inv(x) = x.

In [10], the algebraic action of this group on CP1 was studied and it was shown

that in the irreducible case all such actions are generated by the Euler-Chasles corre-

spondences.

There is another 2-valued group and its action on CP1 which is also closely

related to the Euler-Chasles correspondence and to the Great Poncelet Theorem.

This action is intimately related to the Kowalevski fundamental equation and to the

Kowalevski change of variables as well.

Let us consider one more simple example.

Example 2.2. Two-valued group p2 is defined by the relation

m2 : C× C → (C)2,

x ∗2 y = [(√x+

√y)2, (

√x−

√y)2]

(2.4)

The product x ∗2 y is given by the solutions of the polynomial equation

p2(z, x, y) = 0,

in z, where

p2(z, x, y) = (x+ y + z)2 − 4(xy + yz + zx).

The polynomial p2(z, x, y) is discriminantly separable:

Dz(p2)(x, y) = P (x)P (y), Dx(p2)(y, z) = P (y)P (z), Dy(p2)(x, z) = P (x)P (z),

where P (x) = 2x.

The polynomial p2 as a discriminantly separable, generates a case of generalized

Kowalevski system of differential equations from [14].

2.2. 2-valued group on CP1 and the Kowalevski top

It appears that the general equation of pencil of conics corresponds to an action of

a two valued group. We use this correspondence to provide a novel interpretation of

’the mysterious Kowalevski change of variables’ (the adjective being borrowed from

80 V. Dragovic

[4]). This line of thoughts may be seen as a further development of the ideas of Weil

and Jurdjevic (see [38], [29], [30]). It turned out that the associativity condition for

this action is equivalent to the Great Poncelet Theorem for a triangle, see [14].

The general pencil equation F(s, x1, x2) = 0 is related to two elliptic curves

Γ1 : y2 = P (x), Γ2 : t2 = J(s), where the polynomials P, J are of degree four and

three respectively. These two elliptic curves are isomorphic. Rewrite the cubic one

Γ2 in the canonical form Γ2 : t2 = J ′(s) = 4s3 − g2s − g3. Let ψ : Γ2 → Γ1 be a

birational morphism between the curves induced by a fractional-linear transformation

ψ which maps the three zeros of J ′ and ∞ to the four zeros of the polynomial P .

The curve Γ2 as a cubic has a group structure with the neutral element at infinity.

With the subgroup Z2, it defines the standard two-valued group structure on CP1 (see

[8]):

s1 ∗c s2 =

[

−s1 − s2 +

(

t1 − t22(s1 − s2)

)2

,−s1 − s2 +

(

t1 + t22(s1 − s2)

)2]

, (2.5)

where ti = J ′(si), i = 1, 2.

Theorem 2.1 ([14]). The general pencil equation after fractional-linear trans-

formations

F(s, ψ−1(x1), ψ−1(x2)) = 0

induces the two valued coset group structure (Γ2,Z2) defined by the relation (2.5).

A proof is given in [14].

2.3. Review of the fundamental steps of the Kowalevski integration procedure

The Kowalevski top [33] is a celebrated example of a heavy rigid body which rotates

about a fixed point, under the conditions I1 = I2 = 2I3, I3 = 1, Y0 = Z0 = 0 (see

subsection 2.1). More about the theory of motion of heavy rigid-bodies one may find

in, for example, [3], [28], [5], [18], [27], [17]. Denote by c = mgX0, where m is the

mass of the top, and denote by (p, q, r) the vector of angular velocity Ω. Then the

equations of motion take the following form, see [33], [28], [35], [34], [26]:

2p = qr, Γ1 = rΓ2 − qΓ3,

2q = −pr − cΓ3, Γ2 = pΓ3 − rΓ1,

r = cΓ2, Γ3 = qΓ1 − pΓ2.

(2.6)

The system (2.6) as any other system of equations of a heavy rigid body, has three

well known first integrals of motion. In this particular case there is also an additional,


fourth, first integral, discovered by Kowalevski

2(p2 + q2) + r2 = 2cΓ1 + 6l1,

2(pΓ1 + qΓ2) + rΓ3 = 2l,

Γ21 + Γ2

2 + Γ23 = 1,

(

(p+ iq)2 + Γ1 + iΓ2

) (

(p− iq)2 + Γ1 − iΓ2

)

= k2.

(2.7)

A significance of the Kowalevski top is that the additional first integral is of fourth

degree in momenta.

By using the change of variables

x1 = p+ iq, e1 = x21 + c(Γ1 + iΓ2),

x2 = p− iq, e2 = x22 + c(Γ1 − iΓ2),(2.8)

the first integrals (2.7) transform into

r2 = E + e1 + e2,

rcΓ3 = G− x2e1 − x1e2,

c2Γ23 = F + x22e1 + x21e2,

e1e2 = k2,

(2.9)

with E = 6l1 − (x1 + x2)2, F = 2cl + x1x2(x1 + x2), G = c2 − k2 − x21x22. One

easily gets

(E + e1 + e2)(F + x22e1 + x21e2)− (G− x2e1 − x1e2)2 = 0,

which has an equivalent form

e1P (x2) + e2P (x1) +R1(x1, x2) + k2(x1 − x2)2 = 0, (2.10)

where the polynomial P is

P (xi) = x2iE + 2x1F +G = −x4i + 6l1x2i + 4lcxi + c2 − k2, i = 1, 2,

and

R1(x1, x2) = EG− F 2

= −6l1x2

1x2

2 − (c2 − k2)(x1 + x2)2 − 4lc(x1 + x2)x1x2 + 6l1(c

2 − k2)− 4l2c2.

A remarkable and not obvious property of P is its dependence on only one vari-

able. Let

R(x1, x2) = Ex1x2 + F (x1 + x2) +G.

82 V. Dragovic

From (2.10), following Kowalevski, one gets

(√

P (x1)e2 ±√

P (x2)e1)2 = −(x1 − x2)

2k2 ± 2k√

P (x1)P (x2)−R1(x1, x2). (2.11)

After a few transformations, (2.11) can be written in the form

[

√e1

√

P (x2)

x1 − x2±

√e2

√

P (x1)

x1 − x2

]2

= (w1 ± k)(w2 ∓ k), (2.12)

where w1, w2 are the solutions of an equation, quadratic in s:

Q(s, x1, x2) = (x1 − x2)2s2 − 2R(x1, x2)s−R1(x1, x2) = 0. (2.13)

The quadratic equation (2.13) is known as the Kowalevski fundamental equa-

tion. As it has been observed in [14], the discriminant separability condition for

Q(s, x1, x2) is satisfied

Ds(Q)(x1, x2) = 4P (x1)P (x2),

Dx1(Q)(s, x2) = −8J(s)P (x2), Dx2

(Q)(s, x1) = −8J(s)P (x1),

with

J(s) = s3 + 3l1s2 + s(c2 − k2) + 3l1(c

2 − k2)− 2l2c2.

The equations of motion (2.6) in new variables (x1, x2, e1, e2, r,Γ3) take the form:

2x1 = −if1, e1 = −me1,

2x2 = if2, e2 = me2.(2.14)

Two additional differential equations for r and Γ3 can be easily derived. Here m = irand f1 = rx1 + cΓ3, f2 = rx2 + cΓ3. The following formulas hold:

f21 = P (x1) + e1(x1 − x2)

2, f22 = P (x2) + e2(x1 − x2)

2. (2.15)

Further steps of the integration procedure are presented in [33], see for example

[23].

Theorem 2.2 ([14]). The Kowalevski fundamental equation coincides with the

point pencil equation generated by the conics given by their tangential equations

C1 : − 2w21 + 3l1w

22 + 2(c2 − k2)w2

3 − 4clw2w3 = 0;

C2 :w22 − 4w1w3 = 0.

(2.16)

The Kowalevski variables w, x1, x2 get a novel geometric interpretation in this set-

tings: they are the pencil parameter, and the Darboux coordinates with respect to the

conic C2, respectively.


The Kowalevski case is extracted from the general case of pencil of conics by the

conditions a1 = 0, a5 = 0, a0 = −2. The last relation is nothing but a normaliza-

tion condition, provided a0 = 0. The Kowalevski parameters l1, l, c can be expressed

by the formulas

l1 =a23, l = ±

1

2

√

−a4 +√

a4 + 4a23, c = ∓a3

√

−a4 +√

a4 + 4a23

,

with an additional condition that l and c are real. For the sake of historic clarity, we

observe that Kowalevski in [33], didn’t use the relation (2.13), but an equivalent one.

The equivalence is obtained by putting w = 2s − l1.

The success of the mechanism of the Kowalevski change of variables is based on

the following consequence of the discriminant separability property of the polyno-

mial F = Q:

dx1√

P (x1)+

dx2√

P (x2)=

dw1√

J(w1),

dx1√

P (x1)−

dx2√

P (x2)=

dw2√

J(w2).

(2.17)

The Kowalevski change of variables (see equations (2.17)) can be seen as an

infinitesimal of the correspondence which maps a pair of points (M1,M2) to a pair

of points (S1, S2). Both pairs belong to a P1 as a factor of an appropriate elliptic

curve. A geometric interpretation of this mapping is the correspondence which maps

two tangents to the conic C to the pair of conics from the pencil which contain the

intersection point of the two lines.

Theorem 2.3 ([14]). The Kowalevski change of variables is equivalent to an

infinitesimal of the action of the two valued coset group (Γ2,Z2) on P1 as a factor

of the elliptic curve. Up to a fractional-linear transformation, it is equivalent to the

operation of the two valued group (Γ2,Z2).

Now, the Kotter trick (see [32], [14]) can be applied to the following commutative

diagram.

Proposition 2.1 ([14]). The Kowalevski integration procedure may be coded in

84 V. Dragovic

the following commutative diagram:

C4 Γ1 × Γ1 × C Γ2 × Γ2 × C

Γ1 × Γ1 × C× C CP1 × CP

1 × C

C× C CP1 × CP

1 × C

CP2 CP

2 × C/ ∼

iΓ1

×iΓ1

×m

iΓ1

×iΓ1

×id×id

$

ia×ia×m

p1×p1×id

ψ−1×ψ−1×id

p1×p1×id

ϕ1×ϕ2

ψ−1×ψ−1×id

m2

mc×τc

f

The mappings are defined as follows

iΓ1

: x ,→ (x,√

P (x)),

m : (x, y) ,→ x · y,ia : x ,→ (x, 1),

p1 : (x, y) ,→ x,

mc : (x, y) ,→ x ∗c y,τc : x ,→ (

√x,−

√x),

ϕ1 : (x1, x2, e1, e2) ,→√e1

√

P (x2)

x1 − x2

,

ϕ2 : (x1, x2, e1, e2) ,→√e2

√

P (x1)

x1 − x2

,

f : ((s1, s2, 1), (k,−k)) ,→ [(γ−1(s1) + k)(γ−1(s2)− k), (γ−1(s2) + k)(γ−1(s1)− k)].


3. Systems of the Kowalevski type. Definition

Following [21, 23, 24], we present a class of integrable systems, which general-

ize the Kowalevski top. Instead of the Kowalevski fundamental equation (see formula

(2.13)), the starting point here would be an arbitrary discriminantly separable poly-

nomial of degree two in each of three variables.

Given a discriminantly separable polynomial of the second degree in each of three

variables

F(x1, x2, s) := A(x1, x2)s2 +B(x1, x2)s+ C(x1, x2), (3.1)

such that

Ds(F)(x1, x2) = B2 − 4AC = 4P (x1)P (x2),

andDx1

(F)(s, x2) = 4P (x2)J(s),

Dx2(F)(s, x1) = 4P (x1)J(s).

Suppose, that a given system in variables x1, x2, e1, e2, r, γ3, after some transfor-

mations reduces to

2x1 = −if1, e1 = −me1,

2x2 = if2, e2 = me2,(3.2)

where

f21 = P (x1) + e1A(x1, x2), f2

2 = P (x2) + e2A(x1, x2). (3.3)

Suppose additionally, that the first integrals of the initial system reduce to a rela-

tion

P (x2)e1 + P (x1)e2 = C(x1, x2)− e1e2A(x1, x2). (3.4)

The equations for r and Γ3 are not specified for the moment and m is a function

of system’s variables.

If a system satisfies the above assumptions we will call it a system of the Kowalev-

ski type. As it has been pointed out in the previous subsection, see formulae (2.10,

2.13, 2.14,2.15), the Kowalevski top is an example of the systems of the Kowalevski

type.

The following theorem is quite general, and concerns all the systems of the

Kowalevski type. It explains in full a subtle mechanism of a quite miraculous jump

in genus, from one to two, in integration procedure, which has been observed in the

Kowalevski top, and now it is going to be established as a characteristic property of

the whole new class of systems.

86 V. Dragovic

Theorem 3.1. Given a system which reduces to (3.2, 3.3, 3.4). Then the system

is linearized on the Jacobian of the curve

y2 = J(z)(z − k)(z + k),

where J is a polynomial factor of the discriminant of F as a polynomial in x1 and kis a constant such that

e1e2 = k2.

The last Theorem basically formalizes the original considerations of Kowalevski,

in a slightly more general context of the discriminantly separable polynomials. A

proof is presented in [24].

3.1. An example of systems of the Kowalevski type

In this subsection we present the Sokolov system given in [37] as an example of

systems of the Kowalevski type, see [23], [24]. Consider [37] the Hamiltonian

H = M21 +M2

2 + 2M23 + 2c1γ1 + 2c2(γ2M3 − γ3M2) (3.5)

on e(3) with the Lie-Poisson brackets

Mi,Mj = ϵijkMk, Mi, γj = ϵijkγk, γi, γj = 0, (3.6)

where ϵijk is the totally skew-symmetric tensor. In [31], an explicit map between the

integrable system on e(3) with the Hamiltonian (3.5) and the Kowalevski top on so(4)has been found. The separation of variables for the system (3.5) was performed. The

aim of this section is to show that the system can be seen as an element of the class

of the systems of the Kowalevski type, [23], [24].

The Lie-Poisson brackets (3.6) have two Casimirs:

γ21 + γ22 + γ23 = a,

γ1M1 + γ2M2 + γ3M3 = b.

As in [31] and [33], one can introduce new variables z1 = M1+iM2, z2 = M1−iM2

and

e1 = z21 − 2c1(γ1 + iγ2)− c22a− c2(2γ2M3 − 2γ3M2 + 2i(γ3M1 − γ1M3)),

e2 = z22 − 2c1(γ1 − iγ2)− c22a− c2(2γ2M3 − 2γ3M2 + 2i(γ1M3 − γ3M1)).

The second first integral of motion of the system (3.5) can be written as

e1e2 = k2. (3.7)


The equations of motion for new variables zi, ei can be written in the form of (3.2)

and (3.3). This is in a full accordance with our definition of the systems of Kowalevski

type. It is easy to prove that:

e1 = −4iM3e1, e2 = 4iM3e2,

and−z1

2 = P (z1) + e1(z1 − z2)2,

−z22 = P (z2) + e2(z1 − z2)

2,(3.8)

where P is a polynomial of degree four:

P (z) = −z4 + 2Hz2 − 8c1bz − k2 + 4ac21 − 2c22(2b2 −Ha) + c42a. (3.9)

The biquadratic form and the separated variables were defined [31]:

F (z1, z2) = −1

2

(

P (z1) + P (z2) + (z21 − z22)2)

,

s1,2 =F (z1, z2)±

√

P (z1)P (z2)

2(z1 − z2)2,

(3.10)

such that

s1 =

√

P5(s1)

s1 − s2, s2 =

√

P5(s2)

s2 − s1, P5(s) = P3(s)P2(s),

with

P3(s) = s(4s2 + 4sH +H2 − k2 + 4c21a+ 2c22(Ha− 2b2) + c42a2) + 4c21b

2,

P2(s) = 4s2 + 4(H + c22a)s+H2 − k2 + 2c22ha+ c42a2.

To verify that, we still need to show that a relation of the form of (3.4) is satisfied and

to relate it with a corresponding discriminantly separable polynomial in the form of

(3.1). Starting from the equations

z1 = −2M3(M1 − iM2) + 2c2(γ1M2 − γ2M1) + 2c1γ3

and

z2 = −2M3(M1 + iM2) + 2c2(γ1M2 − γ2M1) + 2c1γ3,

one can prove that

z1 · z2 = −(

F (z1, z2) + (H + c22a(z1 − z2)2))

,

88 V. Dragovic

where F (z1, z2) is given by (3.10). After equating the square of z1z2 from previous

relation and z12 · z22 with zi2 given by (3.8) we get

(z1 − z2)2[2F (z1, z2)(H + c22a) + (z1 − z22)

4(H + c22a)2 − P (z1)e2

− P (z2)e1 − e1e2(z1 − z2)2] + F 2(z1, z2)− P (z1)P (z2) = 0.

(3.11)

Denote by C(z1, z2) a biquadratic polynomial such that F 2(z1, z2)−P (z1)P (z2) =(z1 − z2)2C(z1, z2). Then we can rewrite relation (3.11) in the form of (3.4):

P (z1)e2 + P (z2)e1 = C(z1, z2)− e1e2(z1 − z2)2, (3.12)

with

C(z1, z2) = C(z1, z2) + 2F (z1, z2)(H + c22a) + (H + c22a)2(z1 − z2)

2. (3.13)

Further integration procedure follows Theorem 3.1. The discriminantly separable

polynomial of three variables of degree two in each variable “plays role” of the

Kowalevski fundamental equation in this case: i

F (z1, z2, s) = (z1 − z2)2s2 + B(z1, z2)s + C(z1, z2), (3.14)

with

B(z1, z2) = F (z1, z2) + (H + c22a)(z1 − z2)2.

The discriminants of (3.14) as polynomials in s and in zi, for i = 1, 2 are

Ds(F )(z1, z2) = P (z1)P (z2),

Dz1(F )(s, z2) = 8J(s)P (z2), Dz2(Q)(s, z1) = 8J(s)P (z1),

where J is a polynomial of degree three

J = s3 + (H + 3ac22)s2 + (−4c22b

2 − 2k2 + 4ac21 + 4c42a2 + 4c22Ha)s

− 8c21b2 − 4c42ab

2 + 4c21a2c22 − k2c22a−Hk2 + 2aH2c22 − 4Hb2c22

+ 4Hc21a+ 4c42Ha2 + 2c62a3.

4. Classification of strongly discriminantly separable polynomials ofdegree two in three variables

In this section we present a classification from [22] of the strongly discriminantly

separable polynomials F(x1, x2, x3) ∈ C[x1, x2, x3] which are of degree two in


each of three variables. This classification is done modulo the group of the Mobius

transformations

x1 ,→ax1 + b

cx1 + d, x2 ,→

ax2 + b

cx2 + d, x3 ,→

ax3 + b

cx3 + d. (4.1)

Denote by

F(x1, x2, x3) =2

∑

i,j,k=0

aijkxi1x

j2x

k3 (4.2)

a strongly discriminantly separable polynomial with

DxiF(xj , xk) = P (xj)P (xk), (i, j, k) = c.p.(1, 2, 3). (4.3)

One gets a system of 75 equations of degree two with 27 unknowns aijk, by

plugging (4.2) into (4.3) for a given polynomial P (x) = Ax4+Bx3+Cx2+Dx+E.

Theorem 4.1. Given a nonzero polynomial P (x). The strongly discriminantly

separable polynomials F(x1, x2, x3) of degree two in each of the three variables

which satisfy (4.3), are exhausted modulo Mobius transformations, by the following

list coded by the structure of the roots of the polynomial P (x):

(A) If P has four simple zeros, it can be transformed to a canonical form PA(x) =(k2x2 − 1)(x2 − 1), and

FA =1

2(−k2x21 − k2x22 + 1 + k2x21x

22)x

23 + (1− k2)x1x2x3

+1

2(x21 + x22 − k2x21x

22 − 1),

(B) if P has two simple zeros and one double, it can be transformed to a canonical

form PB(x) = x2 − e2, e = 0, and

FB = x1x2x3 +e

2(x21 + x22 + x23 − e2),

(C) If P has two double zeros, and the canonical form PC(x) = x2, then

FC1= λx21x

22 + µx1x2x3 + νx23, µ2 − 4λν = 1,

FC2= λx21x

23 + µx1x2x3 + νx22, µ2 − 4λν = 1,

FC3= λx22x

23 + µx1x2x3 + νx21, µ2 − 4λν = 1,

FC4= λx21x

22x

23 + µx1x2x3 + ν, µ2 − 4λν = 1,

90 V. Dragovic

(D) if P has one simple and one triple zero, then the canonical form is PD(x) = x,

and,

FD = −1

2(x1x2 + x2x3 + x1x3) +

1

4(x21 + x22 + x23),

(E) if P has one quadruple zero, then the canonical form is PE(x) = 1, and

FE1= λ(x1 + x2 + x3)

2 + µ(x1 + x2 + x3) + ν, µ2 − 4λν = 1,

FE2= λ(x2 + x3 − x1)

2 + µ(x2 + x3 − x1) + ν, µ2 − 4λν = 1,

FE3= λ(x1 + x3 − x2)

2 + µ(x1 + x3 − x2) + ν, µ2 − 4λν = 1,

FE4= λ(x1 + x2 − x3)

2 + µ(x1 + x2 − x3) + ν, µ2 − 4λν = 1.

The proof from [22] is performed by a straightforward calculation and solving

the system of equations (4.3) for the canonical representatives of the polynomials

P. The correspondence between this classification and pencil of conics in the case

(A) is as follows: In the case of a general position, the conics of a pencil intersect

in four distinct points, and we code such situation with (1, 1, 1, 1). It corresponds

to the case above where the polynomial P has four simple zeros. In this case, the

family of strongly discriminantly separable polynomials corresponds to the equations

of the families constructed above of general pencils of conics. These families were

indicated in [14]. A corresponding pencil of conics is presented on Fig. 1.

Figure 1. Pencil with four simple points


Without loss of generality, we use C2 as the conic with respect to which the Dar-

boux coordinates are defined. The obtained families of polynomials in cases (A), (B),

and (D) are unique up to Mobius equivalence. Each of them is Mobius-equivalent to

a corresponding strongly discriminantly separable polynomial, and they represent the

equations of pencils of conics of the types (A) = (1, 1, 1, 1), (B) = (1, 1, 2), and

(D) = (1, 3). The pencils (1, 1, 2) consist of conics sharing two simple points and

one double point, while the pencils (1, 3) consist of conics having one common sim-

ple point and one common triple point. However, the situations in the cases (C) and

(E) are significantly different. Not only are uniqueness, up to Mobius equivalence, of

the families of the polynomials, is lost, but also such a transparent geometric correla-

tion with pencils of conics disappears. We will skip here the details of the connection

with pencils of conics in the cases (B) and (D), see [22], which are analog to (A).

We will discuss now the cases (C) and (E) and the lack of relationship to the pencils

of types (2, 2) and (4) respectively. Former pencils contain conics which share two

double points, see Fig. 2 (left), while later describe pencils of conics having one point

of order 4 in common, see Fig. 2 (right).

Figure 2. Pencil with two double points (left) and with one quadruple point (right)

This unexpected lack of corresponding pencils of conics in the cases (C) and (E)

can be understood better in the light of the following statement:

Proposition 4.1 (Corollary, [36], VIII, Ch. 1). A symmetric (2 − 2) algebraic

correspondence cut on C by tangents to C2 splits into a Mobius transformation and

its inverse if and only if C2 has double contact with C . If C2 coincides with C , then

the correspondence is the identity taken twice.

The two points of contact correspond to the fixed points of the Mobius transfor-

mation.

92 V. Dragovic

Suppose that the two fixed points of the Mobius transformation are the points

with the parameters equal to 0 and ∞. The Mobius transformation, denoted by w, is

of the form w(x) = ax. In a case of the pencil of conics of type (2, 2) with the inter-

section at two double points, according to the previous Proposition, the polynomial

Fs(x1, x2) := F(x1, x2, s) has to have the form

Fs(x1, x2) = (ax1 + bx2)(bx1 + ax2). (4.4)

For a fixed value of the parameter s, the polynomials FC1− FC4

, do not have the

form (4.4). Those polynomials do not correspond to the pencils of conics with two

double base points. A similar argument when the fixed points of w coincide, explains

the case (E).

5. From discriminantly separable polynomials to integrablequad-equations

The discriminantly separable polynomials appear to be related to discrete inte-

grable systems. We will show a relationship with integrable quad-equations, from

[22]. The theory of quad-graphs and quad-equations emerged in works of Adler,

Bobenko, Suris [1], [2], see also [6], [7].

x4

x3

x1

Q

x2

x3

x13

x2

x12

x23

x123

x x1

Figure 3. Quad-equation Q(x1, x2, x3, x4) = 0 on an elementary quadrilateral (left)

and 3D-consistency (right)

The quad-equations are defined on quadrilaterals and they have the form

Q(x1, x2, x3, x4) = 0. (5.1)

Here Q is a polynomial of degree one in each variable. Such a polynomial is said to be

multiaffine. So-called field variables xi are assigned to four vertices of a quadrilateral


as in a Figure 3 (left). The polynomial Q depends on the variables x1, . . . , x4 ∈ C,

but also depends on two additional parameters α,β ∈ C that are assigned to the edges

of a quadrilateral. The opposite edges carry the same parameter.

The equation (5.1) solved for each variable, gives the solution as a rational func-

tion of the other three variables. A solution (x1, x2, x3, x4) of the equation (5.1) is

said to be singular with respect to xi if it also satisfies the equation

Qxi(x1, x2, x3, x4) = 0.

Following [2] we adopt the idea of integrability as a consistency, see Figure 3

(right). We assign six quad-equations to the faces of the coordinate cube. The system

is 3D-consistent if the three values for x123 obtained from the equations on the right,

back, and top faces coincide for arbitrary initial data x, x1, x2, x3.

The discriminant-like operators are introduced in [2]

δx,y(Q) = QxQy −QQxy, δx(h) = h2x − 2hhxx, (5.2)

and one can make a descent from the faces to the edges and then to the vertices of

the cube: in that way, from a multiaffine polynomial Q(x1, x2, x3, x4) we pass to a

biquadratic polynomial h(xi, xj) := δxk,xl(Q(xi, xj , xk, xl)) and then, further, to a

polynomial P (xi) = δxj(h(xi, xj)) of degree up to four. Using the relative invariants

of polynomials under fractional linear transformations, the formulae that express Qthrough the biquadratic polynomials of three edges, were obtained in [2]:

2Qx1

Q=

h12x1h34 − h14x1

h23 + h23h34x3− h23x3

h34

h12h34 − h14h23. (5.3)

A biquadratic polynomial h(x, y) is said to be nondegenerate if no polynomial in

its equivalence class with respect to the fractional linear transformations, is divisible

by a factor of the form x − c or y − c, with c = const. A multiaffine function

Q(x1, x2, x3, x4) is said to be of type Q if all four of its accompanying biquadratic

polynomials hjk are nondegenerate. Otherwise, it is of type H . Previous notions

were introduced in [2].

Take an arbitrary strongly discriminantly separable polynomial

F(x1, x2,α)

of degree two in each of the three variables. To relate that polynomial to the cor-

responding quad-equations, one needs to provide a biquadratic polynomial h =h(x1, x2) and a multiaffine polynomial Q = Q(x1, x2, x3, x4).

94 V. Dragovic

The requirement that the discriminants of h(x1, x2) are independent on α, see

[1], [2], is fulfilled if as a biquadratic polynomials h(x1, x2) we select

h(x1, x2) :=F(x1, x2,α)√

P (α).

Proposition 5.1 ([22]). The biquadratic polynomials

hI(x1, x2) =FI(x1, x2,α)√

PI(α)(5.4)

satisfy

δx1(h) = PI(x2), δx2

(h) = PI(x1)

for I = A,B,C,D,E and polynomials PI ,FI from Theorem 4.1.

By using the formulae (5.3) and replacing the polynomials hij by hij , one gets

the quad-equations which correspond to representatives of discriminantly separable

polynomials from Theorem 4.1. These equations are re-parameterizations of the

quad-equations of type Q from the list obtained in [2].

For the quad-equations obtained from the biquadratic polynomials h(x1, x2), that

parameter α has a role symmetric to x1 and x2.

Another class of discrete integrable systems of a type similar to quad-graphs of a

geometric origin has been given in [20].

Acknowledgements. The research was partially supported by the Serbian Min-

istry of Education and Science, Project 174020 Geometry and Topology of Mani-

folds, Classical Mechanics and Integrable Dynamical Systems.

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graphs. The consistency approach, Commun. Math. Phys. 233 (2003), 513–543.

[2] V. E. Adler, A. I. Bobenko, Y. B. Suris, Discrete nonlinear hiperbolic equations. Clas-

sification of integrable cases, Funct. Anal. Appl 43, (2009) 3–21.

[3] V. I. Arnold, Mathematical methods of classical mechanics, Nauka, Moscow, 1974(Russian); English translation: Springer, 1988.

[4] M. Audin, Spinning Tops. An introduction to integrable systems, Cambridge studies inadvanced mathematics 51, 1999.


[5] A. Bilimovic, Rigid body dynamics, Mathematical Institute SANU, Special Editions,1955 (Serbian).

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The consistency approach, Lett. Math. Phys. 61, no. 3 (2002), 241–254.

[7] A. I. Bobenko, Yu. B. Suris, Integrable systems on quad-graphs. Int. Math. Res. Not.2002, no. 11, 573–611.

[8] V. M. Buchstaber, Functional equations, associated with addition theorems for elliptic

functions, and two-valued algebraic groups, Russian Math. Surv. 45 (1990), 213–214.

[9] V. M. Buchstaber, S. P. Novikov, Formal groups, power systems and Adams operators,Mat. Sb. (N. S) 84 (126) (1971), 81–118. (Russian)

[10] V. M. Buchstaber, A. P. Veselov, Integrable correspondences and algebraic represen-

tations of multivalued groups, Internat. Math. Res. Notices (1996), 381-400.

[11] V. Buchstaber, n-valued groups: theory and applications, Moscow Mathematical Jour-nal, 6 (2006), 57–84.

[12] V. M. Buchstaber, V. Dragovic, Two-valued groups, Kummer varieties and integrable

billiards, arXiv: 1011.2716

[13] G. Darboux, Principes de geometrie analytique, Gauthier-Villars, Paris, 1917, 519 p.

[14] V. Dragovic, Geometrization and generalization of the Kowalevski top, Comm. MathPhys. 298 (2010), 37–64.

[15] V. Dragovicc, Pencils of conics as a classification code, pp. 323–330 Trends in Math-ematics 2013 Geometric Methods in Physics XXX Workshop, Bialowieza, Poland,June 26 to July 2, 2011 Editors: P. Kielanowski, S. Twareque Ali, A. Odzijewicz,M. Schlichenmaier, Th. Voronov ISBN: 978-3-0348-0447-9 (Print) 978-3-0348-0448-6 (Online)

[16] V. Dragovic, Pencils of conics and biquadratics, and integrability, In: Topology, geom-etry, integrable systems, and mathematical physics, 117–140, Amer. Math. Soc. Transl.Ser. 2, 234, Amer. Math. Soc., Providence, RI, 2014.

[17] V. Dragovic, B. Gajic, Some recent generalizations of the classical rigid body systems,Arnold Math. Journal (accepted).

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[19] V. Dragovic, M. Radnovic, Poncelet porisms and beyond, Birkhauser/Springer, 2011.

[20] V. Dragovic, M. Radnovic, Billiard algebra, integrable line congruences, and doublereflection nets, J. Nonlinear Math. Phys. 19 (2012), no. 3, 1250019, 18 pp.

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[21] V. Dragovic, K. Kukic, New examples of systems of the Kowalevski type, Regul. ChaoticDyn. 16 (2011), no. 5, 484–495.

[22] V. Dragovic, K. Kukic, Discriminantly separable polynomials and quad-graphs, Jour-nal of Geometry and Mechanics 6, no. 3 (2014), 319–333.

[23] V. Dragovic, K. Kukic, Systems of the Kowalevski type and discriminantly separable

polynomials, Journal Regular and Chaotic Dynamics 19, No 2 (2014), 162–184.

[24] V. Dragovic, K. Kukic, The Sokolov case, integrable Kirchhoff elasticae, and genus

2 theta-functions via discriminantly separable polynomials, Proceedings of SteklovMathematical Institute 286 (2014), 224–239.

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namical systems, AIP Conf. Proc. 1634, 3 (2014); http://dx.doi.org/10.1063/1.490300.

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(1981), 11–80 (Russian).

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Mathematical InstituteSerbian Academy of Science and ArtKneza Mihaila 3611000 Belgrade, Serbia

The Department of Mathematical SciencesThe University of Texas at Dallas, USA

e-mail: [email protected]



LAPLACE TRANSFORM OF FUNCTIONS DEFINED ON A BOUNDEDINTERVAL

BOGOLJUB STANKOVIC

(Presented at the 8th Meeting, held on November 27, 2015)

A b s t r a c t. Laplace transform ˙L for functions belonging to L[0, b], 0 < b < 1 isdefined. This definition is given by using the idea of H. Komatsu [J. Fac. Sci. Univ. Tokyo,IA, 34 (1987), 805–820] and [Structure of solutions of differential equations (Katata/Kyoto,1995), pp. 227–252, World Sci. Publishing, River Edge, NJ, 1996]. for Laplace hyperfunc-tions. As an application of ˙L we solve an equation with fractional derivative and an integralequation of the first kind of convolution type.

AMS Mathematics Subject Classification (2000): 46F12.Key Words: Space of locally integrable functions, Laplace transform of functions be-

longing to L[0, b], 0 < b < 1.

1. Introduction

Laplace transform of numerical functions has been elaborated as a powerfulmathematical theory very useful in practice and many a time applied by engineers.Although it has been belived to have two important shortcomings. First, applicationof the Laplace transform (In short, LT) not only to functions, but to distributions, ul-tradistributions, and hyper-functions, calls for some growth conditions of them ([7],[9], [17], [20], [21], [22] and [23]). Secondly, there is no simple characterisation of

100 B. Stankovic

the functions which are LT of the numerical functions. Hence, we are not always surewhether or not an obtained function f(s) is the LT of a function g(t) of exponentialtype.

To overcome these difficulties mathematiciens defined LT of functions as classes([4], [6]) without a rich repercussions, or used algebraic approaches to the Heavisidecalculus ([12], [16]).

H. Komatsu [9], [10] overcome successfully all defects of the classical LT. Hedefined the LT of Laplace hyperfunctions and of hyperfunctions, as well, but in onedimensional case. Since it is a very abstract theory, it cannot be easily accepted bythe greater part of people working in applications.

In [20] a definition was developed of LT applicable to locally Bochner integrableBanach space valued functions with arbitrary growth at infinity, based on old ideas(cf. [4], [6]). For f L

loc

this LT coincides with the LT defined by Komatsu.The aim of the paper [18] was to define and to elaborate the LT of a subset of dis-

tributions which contains also the space Lloc

(R), distributions with compact supportsand tempered distributions. The methods of Komatsu can not be applied directly be-cause the space of distributions is not a flabby sheaf.

In this paper we define the LT, ˙L, of elements of the space L[a, b], 0 < b < 1;

the function f 2 L[0, b] if there existsbR

0

f() d in the sense of Lebesgue. This

definition is easy accesible to every mathematicien who works with classical LT;because ˙L is defined by the classical Laplace transform L of a class of functions. Insuch a way to prove some properties of ˙L we use properties of L.

2. Some spaces we use

Vector space L[0, b], 0 b < 1. The function f 2 L[0, b] if there existsbR

0

f() d in the sense of Lebesgue;bR

0

f() d = 0 , f(t) = 0 a.e. on [0, b].

Vector space of locally integrable functions. f 2 Lloc

[0,1) if for any T1

<T2

2 [0,1), f 2 L[T1

, T2

]. If f and g belong to Lloc

[0,1), then their convolution,f g 2 L

loc

[0,1), as well. The convolution is an operation commutative, associativeand distributive.

Vector space Lexp[0,1). The function f 2 Lexp

[0,1) if f 2 Lloc

[0,1) andsuch that for an s

0

> 0 there existsR10

|es0f()| d . With operation additionthis is a vector space on C. A function f 2 Lexp

[b,1) if f 2 Lexp

[0,1) andf(t) = 0, t < b, b < 1. Lexp

[0,1) is also a vector space.

Lemma 2.1. If f, g 2 Lexp

[0,1), then f g 2 Lexp

[0,1).

Laplace transform of functions defined on a bounded interval 101

Lemma 2.2. If f 2 Lexp

[0,1), and f /2 Lexp

[c,1), for any c > 0, but g 2Lexp

[b,1), then f g 2 Lexp

[b,1).

Proofs of these lemmas can be found in [7], p. 123 and p. 131, respectively.Vector space Lexp[0,1)/Lexp[b,1). In Lexp

[0,1) we define a two ele-ments relation: f g = f g 2 Lexp

[b,1), b > 0. Since Lexp

[b,1) is avector space, a subspace of Lexp

[0,1), the relation is an equivalence relation in ac-cordance with the vector space Lexp

[0,1). The equivalence classes are elements ofLb = Lexp

[0,1)/Lexp

[b,1). An element fb 2 Lb is defined by f + Lexp

[b,1),where f 2 Lexp

[0,1). In Lb is defined the addition and product by r 2 R.If fb, gb 2 Lb, and r 2 R, then

fb + gb = f + g + Lexp

[b,1), (2.1)

andrfb = (rf)b .

With this two operation Lb is also a vector space.

3. Properties of the space Lb

Lemma 3.1. Every function f 2 L[0, b] can be extended to a function f 2Lexp

[0, b]. The space Lb is algebraically isomorph to L[0, b].

PROOF. One extension of the function f 2 L[0, b] is the function ¯f(t) =

Lexp

[0,1) such that ¯f(t) = f(t), 0 t b.Let f

1

2 Lexp

[0,1). It defines the class fb = f1

+ Lexp

[b,1) 2 Lb. Then

(f1

(t) + Lexp

[b,1))

[0,b]= f(t) 2 L[0, b].

Here, ¯f

[0,b]is the restriction of ¯f on [0, b].

Conversely, let f 2 L[0, b]. It can be extended to ¯f 2 Lexp

[0,1) which definesthe class fb = ¯f + Lexp

[b,1).Finally, operation “

[0,b]” is an isomorphism of Lb , L[0, b].

Convolution in Lb. If fb, gb 2 Lb, fb = f + Lexp

[0,1), gb = g + Lexp

[0,1),then there exists

fb gb =tZ

0

f(t )g() d, t 0

and this convolution belongs to Lexp

[0,1) (see [7] Theorem 2, p. 123). Conse-quently,

fb gb = f g + Lexp

[b,1) = f g + Lexp

[b,1).

102 B. Stankovic

Lemma 3.2. If f, g 2 L[0, b] and ¯f, g be theirs extensions in Lexp

[0,1), thenf g , ¯f g + Lexp

[b,1) 2 Lb.

PROOF. By Lemma 2.1 we have

fb gb = f g + Lexp

[b,1) , f g. (3.1)

The next two theorems give the extensions of functions having the derivatives.

Theorem 3.1. If the function f has its derivative f 0(t) in every point of (a, b]

(the derivative in b means the left derivative) and if f 0(t) 2 L[0, b], then f 0

(t) can beextended on [0,1) so that f 0

(t) = f0(t). The function ¯f 0 extends f 0 on (0,1) and

¯f extends f on (0,1). Consequently f 0 , f0+ Lexp

[b,1).

PROOF. Let f(t) and f0(t) 2 Lexp

[0,1) extend f and f 0. If the function f hasthe left derivative in the point b, then it is well-known that there exists

lim

h!0

+

f(b) f(b h)

h= f 0

(b 0

+

)

andlim

t!bf(t) = f(b 0).

First we extended f in such a way that

¯f(t) = f(t) (0 < t < b) and lim

t!b+¯f(t) = f(b 0).

Then,lim

t!b¯f(t) = lim

t!b+f(t) = f(b 0).

The constructed ¯f is a continuous function on t = b. Now, we can extend f 0(t) on

[0,1) as

f 0(t) = f

0(t) = f 0

(t), 0 < t < b,

lim

t!b+f 0(t) = lim

t!bf0(t) = f 0

(b 0).

In t = b we havelim

t!bf 0(t) = lim

t!b+f0(t) = f 0

(b 0).

In such a way we have f 0(t) = f

0(t), t 0,

f 0 , f0(t) + Lexp

[b,1) (3.2)

and lim

t!0

+f(t) = f(0+) exists (see [7], p. 99).


Theorem 3.2. If the function f has n derivatives: f (1)

(t), . . . , f (n)(t), n 2,

in every point of the interval (a, b] and f (n)(t) 2 L[0, b], then also f, . . . , f (n1) 2

L[0, b] and f (n)(t) can be extended on [0,1) such that

f (n)(t) = f

(n)(t), t > 0

f (n)(t) , f

(n)(t) + Lexp

[b,1). (3.3)

and lim

t!0

+f (i)

(t) exist for every i = 0, 1, . . . , n 1.

Proof goes just in the same manner as for Theorem 3.1 (see [7], p. 100).

4. Fractional derivatives

CASE 0 < ↵ < 1, ↵ = [↵] + = 2 (0, 1). By definition, Riemann-Lionvillefractional derivative

0

D↵t f , 0 < ↵ < 1, f 2 L[0, b] is

0

D↵t f =

d

dt

↵

(1 ↵) f()

(t). (4.1)

Theorem 4.1. 1

If f has the derivative f (1)

(t) in every point of (0, b] and iff (1)

(t) 2 L[0, b], then there exists f(0+) and

0

D↵t f , d

dt

↵

(1 ↵) ¯f()

(t) + Lexp

[b,1)

,

↵

(1 ↵) ¯f (1)

()

(t) + f(0+)t↵

(1 ↵)+ Lexp

[b,1). (4.2)

(For the proof see Theorem 3.1 and [7], pp. 117–118.)

2

If the function

↵

(1↵) f()

(t) has the derivative in every point (0, b] andthis derivative belongs to L[0, b], then

0

D↵t f ,

↵

(1 ↵) f()

(1)

(t) + Lexp

[b,1)

,

↵

(1 ↵) ¯f()

(1)

+ Lexp

[b,1). (4.3)

The proof follows from Theorem 3.1.

CASE ↵ > 1, ↵ = [↵] + , 2 (0, 1).

104 B. Stankovic

Theorem 4.2. 1

If f (k)(t), k = 0, 1, . . . , n, n 2, exist in every point t 2

(0, b] and f (n) 2 L[0, b], n = [↵] + 1, then

0

D↵t f , D[↵]+1

(1 ) ¯f()

+ Lexp

[b,1)

,

(1 ) ¯f ([↵]+1)

()

(t) + f(0+)

t

(1 )

([↵])

+f (1)

(0

+

)

t

(1 )

([↵]1)

+ · · · (4.4)

+f ([↵])(0

+

)

t

(1 )+ Lexp

[b,1).

(The proof goes as the proof of Theorem 4.1, part 1

. To have0

D↵t f 2 L[0, b] we

must suppose: f(0) = · · · = f ([↵]1)

= 0.)

2

If the function F (t)

↵

(1↵) f()

(t) has n derivatives, n 2 in everypoint of interval (0, b] and n-th derivative belongs to L[0, b], then also i-th derivativeF (i), i = 0, 1, . . . , n 1, belongs to L[0, b] and F (n)

(t) can be extended in [0,1)

such that F (n)(t) = F

(n)(t), t > 0, and

0

D↵t f ,

↵

(1 ) f()

(n)

(t) + Lexp

[b,1). (4.5)

The proof goes as the proof of Theorem 4.1, part 2

.

Caputo fractional derivative. By definition for ↵ = [↵] + , and f 2 L[0, b],we have for Caputo fractional derivative cD↵

0

+f :

(

cD↵0

+f) (t) =

((1 )) f ([↵]+1)

()

(t)

,

((1 )) f ([↵]+1)

()

(t) + Lexp

[b,1)

,

((1 )) f ([↵]+1)

()

+ Lexp

[b,1).

Fractional integrals. We use Rimann-Liouville fraction integral

(

0

I↵t f) (t) =

↵1

(↵) f()

,

↵1

(↵) ¯f()

(t) + Lexp

[b,1).


5. Laplace transform L of elements of L[0, b] given by Laplace transform Lof elements of Lb

If f 2 L[0, b], then

˙Lf(s) , L ¯f+ LLexp

[b,1),

or˙LL[0, b] , LLexp

[0,1) /LL[b,1).The inverse operator of ˙L, operator ˙L1, is

˙L1

L

¯f

+ Lexp

[b,1)

(t) = L1

L

¯f

+ LLexp

[b,1)

[0,b].

Consequently, every function f 2 L[0, b] has its Laplace transform ˙Lf. Itis given by the class L ¯f + LLexp

[b,1), where ¯f is any extension of f inLexp

[0,1).To find ˙Lf we can use tables of the classical Laplace transform. If the function

F (s) is the Laplace transform of f(t) 2 Lexp

[0,1), then f(t)

[0,b]2 L[0, b] and

˙Ln

f(t)

[0,b

o

, F (s) + LLexpb,1) .

There exist functions, as it is F (t) = exp(t2), t > 0, which have not the Laplaceintegrale for any s, but F (t)

[0,b]2 L[0, b] has its Laplace transform ˙L. Let ¯F (t) =

F (t), 0 t 1, and ¯F (t) = e, t > 1, then

˙L

F (t)

[0,b]

=

Z 1

0

e

stF (t) dt+ LLexp

[1,1) .

As regards convolution defined in Lemma 3.2 we have

˙L (f g) , L ¯f · Lg+ LLexp

[b,1) .

Laplace transform of fractional derivatives. We analyzed the fractional deriva-tive

0

D↵t f , in previous pages, dividing ↵ in two cases: 0 < ↵ < 1 and 1 [↵]. To

find the Laplace transform of0

D↵t f we do the same.

CASE 0 < ↵ < 1. Starting from (4.2) (Theorem 4.1 gives the conditions that(4.2) is valid),

˙L0

D↵t f , L

↵

(1 ↵)

(s)L

¯f (1)

(s) + f(0+)s↵1

+ LLexp

[b,1) (s)

, s↵1

sL ¯f(s) f(0+)

+ f(0+)s↵1

+ LLexp

[b,1) (s)

, s↵L ¯f(s) + LLexp

[b,1) (s). (5.1)

106 B. Stankovic

We have only to prove that s↵ ¯f(t) 2 Lexp

[0,1). Since ¯f 2 Lexp

[0, b], then theintegral

R10

|es0tf(t)| dt converges for an s0

> 0.

Let 0 < " < s0

for s0

> 0. Then for s0

" > 0 we have

lim

!1!1!2>!1

Z !2

!1

s↵es0tf(t) dt lim

!1!1!2>!1

Z !2

!1

|s↵e"t||e(s0")tf(t)| dt

lim

!1!1!2>!1

Z !2

!1

e

(s0")t|f(t)| dt.

Let us now start with (4.3), then we have:

˙L0

D↵t f (s) = sL

(1 ) ¯f()

(s)

(1 ) f()

(0

+

) + LLexp

[b,1) (s). (5.2)

If [↵] 1, ↵ > 1, then we start with (4.4) (Theorem 3.1 gives the conditionsthat (4.4) is invalid)

0

D↵t f ,

(1 ) ¯f ([↵]+1)

()

(t) + f(0)

(1 )

([↵])

+f (1)

(0)

(1 )

(|↵|1)

+ · · ·+ f ([↵])(0)

t

(1 )

+ Lexp

[b,1) . (5.3)

To have the existence of ˙L of0

D↵t f we must have f(0) = · · · = f ([↵]1)

(0) = 0

(and f ([↵])(0) can be 6= 0), then

˙L0

D↵t f (s) , L

¯f ([↵]+1)

()

(1 )

(s) + f ([↵])(0)s1 (5.4)

+LLexp

[b,1) (s)

, s1s[↵]+1L ¯f f ([↵])s1

+ f ([↵])(0)s1

+ LLexp

[b,1)

, s↵L ¯f+ LLexp

[b,1) .


Let us start with (4.5), then (Theorem 4.1 gives the conditions that (4.5) is valid)

˙L0

D↵t f (s) , L

8

<

:

(1 ↵) f()

!

(n)

(t)

9

=

;

(s) + LLexp

[b,1) (s)

, sn

L

(1 ) f()

(s) (5.5)

n1

X

k=0

(1 ) f()

(k)

(0

+

)sk1

#

+ LLexp

[b,1)(s).

In this way we have ˙L0

D↵t f in both cases, 0 < ↵ < 1 and ↵ > 1.

Laplace transform of Caputo fractional derivative. Here

˙L(cD↵0

+f)

(s) , s↵

Lf

(s) f(0)s↵1 f (1)

(0)s↵2 · · ·

f ([↵])(0)s1

+ LLexp

[b,1)(s).

6. Applications of the Laplace transform of functions defined on the boundedinterval

Laplace transform is often used by processes observed in time, but in this casewith limitation which bring Laplace transform. The equation

y0(t) + y(t) = exp(t2), t > 0,

can not be treated by the Laplace transform because the function exp(t2), but it canbe treated by the Laplace transform defined on bounded interval [0, b], for any b > 0.

We show two different cases of application of Laplace transform defined on anbounded interval.

6.1. Laplace transform and equation with fractional derivatives

The procedure is the following. If we have an equation on a bounded interval[0, b], we construct the corresponding equation in Lb = Lexp

[0,1)/Lexp

[b,1), tak-ing care that

Lexp

[b,1) + Lexp

[b,1) = Lexp

[b,1),

which means: If f, g 2 Lexp

[b,1), then f + g 2 Lexp

[b,1). The solutions of theconstructed equation have to be of the form yb = y + Lexp

[b,1), where (y) is theextension of y 2 L[0, b] in Lexp

[0,1). Then the solution y of the equation on [0, b]is

y(t) = y(t)|[0,b] + Lexp

[b,1)|[0,b] = y(t)|

[0,b].

108 B. Stankovic

As an example we consider Bagley-Torvik equation ([2], [3], [13], [14], [15],[19], [24]):

D2

+AD3/20

+ +B

y(t) = f(t), 0 t b, A,B > 0, (6.1)

with initial conditionsy(0+) = y

0

, y0(0+) = y00

. (6.2)

The fractional derivative D3/20

+ is used to describe the damping force studing theforced motion of the rigid plate immersed in the Newtonian fluid.

First of all we have to construct the corresponding equation in Lb (see Section 2of this text). In Theorem 4.2 we have two possibilities, denoted by (a) and (b). Byequation (6.1) we have first to suppose that y(2)(t) 2 L[0, b], then

y(t) , y(t) + Lexp

[b,1);

D3/20

+ y(t) , D1+1

1/2

(1/2) y()

!

(t) + Lexp

[b,1)

,

1/2

(1/2) y()

!

(t) + y(0+)

1/2

(1/2)

!

(1)

+y(1)t1/2

(1/2)+ Lexp

[b,1); (6.3)

D2y(t) , D2y(t) + Lexp

[b,1).

Second, we have to add supposition that relation (6.3) are valid and that theyare in Lb That means that y(2) exists for t > 0 and y(0+) = 0. The correspondingequation in Lb is

D2y(t) +A

1/2

(1/2) y()

(t) +By(t) = g(t) + y(1)(0+)t1/2

(1/2)+Lexp

[b,1). (6.4)

Now, we can apply the classical Laplace transform to (6.3) (see (5.4)),

(s2 +As3/2 +B)(Ly)(s) = (Lq)(s) + y(0)s+ y(1)(0) + Lexp

[b,1)(s), (6.5)

where y(0) = 0.In (b) the supposition we have to take may be lees as in (a), but they are also less

visible.


The function

1/2

(1/2) y()

(2)

(t) has to belong to L[0, b] and instead of initialcondition (6.2) we will have

1/2

(1/2) y()

!

(0) = 0

,

1/2

(1/2) y()

!

(1)

(0) = 1

.

We return to equation (6.5)

(Ly)(s) = (Lg)(s) + y00

s2 +As3/2 +B+

(LLexp

[b,1))(s) + y00

s2 +As3/2 +B. (6.6)

Function1

s2 +As3/2 +B= (L( + ( ())))(s), (6.7)

where

(t) =1X

r=1

(1)

rr(t), r(t) =

A1/2

(1/2)+B(t)

!r

,

and (·)r denotes r time convolution (·) (·) · · · (·) (see [1]).

The solution yb = (y) + Lexp

[b,1) of equation (6.5) is

yb(t) = L1L[( + ( ())) (g() + y1(0+))](t)

= L1L[( + ( ()) Lexp

[b,1))](t)

Since

[ + ( ()) Lexp

[b,1)] (t) 2 Lexp

[b,1),

(see Lemma 2.2) we have

yb(t) =

+ ( ())) (g() + y10

))

(t) + Lexp

[b,1))](t).

The solution of equation (6.1) with initial conditions y(0+) = 0 and y(1)(0+) =y10

is

yb(t)|[a,b] =

+ ( ())) (g() + y10

))

(t).

110 B. Stankovic

6.2. Equation of the convolution type as an other examples

Solve an integral equation of the first kind of convolution typeZ t

0

K(t )X() d = G(t), 0 t b < 1. (6.8)

This equation (6.8) can be solved only for some special cases. So, for example, ifK(t) and G(t) have derivatives and K(0) 6= 0; also if K(t) = t↵, 0 < ↵ < 1, it isAbel singular equation. If the interval in (6.8) is the half axis, 0 t < 1, one canuse the classical L-transform to give a solution to equation (6.8).

In this paper we apply ˙L-transform for function belonging to L[0, b], b < 1 tosolve equation (6.8), as an application.

Proposition 6.1. If the equation (6.8) has a solution belonging to L[0, b], b < 1,and if there is no > 0 such that K(t) = 0, 0 t , then this solution is uniquein L[0, b].

PROOF. Let us suppose that there exist two solutions X1

and X2

of (6.8). Then

[K (X1

X2

)] (t) = 0, 0 t b. (6.9)

To equation (6.9) in Lb corresponds

¯K

¯X1

¯X2

(t) = Lexp

[b,1), 0 t b (6.10)

From (6.10) it follows that

K

X1

X2

= 0, 0 t b. (6.11)

By [7], p. 131, if

¯K

X1

X2

(t) = 0, 0 t b, then K1

(t) = 0, 0 t a,and (X

1

X2

)(t) = 0, 0 t c, where a+ c = b. By our supposition that there isno > 0 such that K(t) = 0, 0 t , it follows that X

1

(t) = X2

(t), 0 t b.

Proposition 6.2. If K(t) = 0, 0 t for no > 0 and if

L

¯G

(s) =g(s)

s1+"L

¯K

(s),

where g(s) is analytical in Re s > X1

0 and bounded for Re s X1

+ > X1

,then equation (6.8) has a solution

X(t) = L1

L ¯G/L ¯K

(t) = L1

g(s)

s1+", 0 t b.


PROOF. Let us suppose that

L

¯G

(s) =g(s)

sL ¯K(s). (6.12)

To the equation (6.8) corresponds in Lb the equation

¯K() ¯X()

(t) ¯G(t) = Lexp

[b,1), 0 < t. (6.13)

The L-transformation of equation (6.13) is

L

¯K() ¯X()

(s) L

¯G

(s) = LLexp

[b,1) (s), Re s > X1

,

or

L ¯K(s) · L ¯X(s) L ¯K(s) g(s)s1+"

= LLexp

[b,1) (s), Re s > X1

,

i.e.,

L

¯K

(s)

L ¯X(s) g(s)

s1+"

= LLexp

[b,1) (s), Re s > X1

. (6.14)

By Theorem 4, p. 263, from [4], L1

g(s)s1+" = G

1

(t), t 0, and G1

(t) 2 Lexp

[0,1).From (6.14) we have

¯K()

¯X()G1

()

(t)

[0,b]= 0, 0 t b.

By the cited theorem from [4] (p. 131) and supposition that K(t) = 0, 0 t , forno one > 0 it follows that X(t) = G

1

(t), 0 t b.

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Department of Mathematics and InformaticsFaculty of Natural Sciences and MathematicsUniversity of Novi SadTrg Dositeja Obradovica 4Novi Sad 21000, Serbiae-mail: [email protected]

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