RELATING GRAPH ENERGY WITH VERTEX-DEGREE-BASED ENERGIES Текст научной статьи по специальности «Математика»

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RELATING GRAPH ENERGY WITH VERTEX-DEGREE-BASED ENERGIES

Ivan Gutman u>

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University of Kragujevac, Faculty of Science, p

Kragujevac, Republic of Serbia, e-mail: gutman@kg.ac.rs,

ORCID iD: https://orcid.org/0000-0001-9681-1550

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DOI: 10.5937/vojtehg68-28083; https://doi.org/10.5937/vojtehg68-28083

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FIELD: Mathematics (Mathematics Subject Classification: primary 05C50, ^

secondary 05C07) -g

ARTICLE TYPE: Original Scientific Paper x

Abstract:

Introduction/purpose: The paper presents numerous vertex-degree-based graph invariants considered in the literature. A matrix can be associated to >, each ofthese invariants. By means of these matrices, the respective vertex-degree-based graph energies are defined as the sum of the absolute values of the eigenvalues. ^

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Results: The article determines the conditions under which the considered gr graph energies are greater or smaller than the ordinary graph energy g (based on the adjacency matrix). e

Conclusion: The results of the paper contribute to the theory of graph energies as well as to the theory of vertex-degree-based graph invariants.

Keywords: energy (of a graph), vertex-degree-based graph invariant, tma vertex-degree-based graph energy. Gu

Introduction

This paper is concerned with simple graphs, i.e. with graphs without multiple, directed, or weighted edges, and without loops. Let G be such a

graph with n vertices, labeled as vl, v2,..., vn. Two vertices connected by an edge are said to be adjacent. The degree of the vertex v., denoted by

deg(v.), is the number of the first neighbors of v..

The energy of a graph G was defined in 1978 as (Gutman, 1978), (Li et al, 2012)

i=1

where A,---,An are the eigenvalues of the adjacency matrix of G.

Recall that the adjacency matrix A(G) is a symmetrix square matrix of the go order n, whose (/j)-entry is

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A(G)j =

1 if v and vj are adjacent 0 if vt and vj are not adjacent 0 if i = j

In the mathematical (Cruz et al, 2015), (Das et al, 2018), (Furtula et

o al, 2013), (Liu et al, 2019), (Rada & Cruz, 2014), (Zhong & Xu, 2014) and

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chemical (Todeschini & Consonni, 2009) literature, several dozens of uj vertex-degree-based graph invariants (usually referred to as "topological

indices") have been introduced and extensively studied. Their general formula is

VDBI = VDBI (G) = X F (deg(Vi ),deg(Vj))

1<i< j <n

where F(x, y) is some function with the property F(x,y) = F(y, x). In particular,

If F(x, y) = x + y, then VDBI = first Zagreb index;

o if F(x, y) = xy, then VDBI = second Zagreb index;

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if F(x, y) =| x - y |, then VDBI = Albertson index,

if f ( y)=2

f \

xy — + —

v y x y

2

then VDBI = extended index;

if F(x, y) = (x - y) , then VDBI = sigma index, if F(x, y) = 1 / sjxy , then VDBI = Randic index; if F(x, y) = 2 / (x + y), then VDBI = harmonic index; if F(x, y) = V(x + y - 2)/(xy), then VDBI = ABC index;

if F(x, y) = <Jxy , then VDBI = reciprocal Randic index;

if F(x,y) = 1/^/x + y , then VDBI = sum-connectivity index;

if F(x, y) = yjx + y , then VDBI = reciprocal sum-connectivity index;

if F ( x, y ) = x2 + y2, then VDBI = forgotten index;

if F(x, y) = 2^/xy / (x + y), then VDBI = geometric-arithmetic index;

if F(x, y) = (x + y) / (2yjxy ), then VDBI = arithmetic-geometric index; and

if F ( x, y) = xy / ( x + y), then VDBI = inverse sum indeg index.

There are several more such graph invariants; see in (Das et al, 2018), (Kulli, 2020), where also bibliographic data can be found.

For each function F(x, y) and each graph G, a symmetric square matrix O = O(G) of the order n can be defined, whose (/j)-entry is

O(GY

Recall that if vi and vj are adjacent, then

deg(v.) > 1, deg(vy) > 1.

The respective vertex-degree-based graph energy (of the graph G) is equal to the sum of absolute values of the eigenvalues of O = O(G).

We will denote it by EF = EF (G).

For some of the above given functions F(x, y), the condition 0 < F(x, y) < 1 holds for all x > 1, y > 1. Such are the functions pertaining to the Randic, harmonic, sum-connectivity, and geometric-arithmetic indices. For some of the above given functions, F(x, y) > 1

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F (deg(vi), deg(vy) j if vt and vj are adjacent 0 if vt and vj are not adjacent |

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holds for all x > 1, y > 1. Such are those related to the first and second Zagreb, extended, forgotten, and arithmetic-geometric indices, as well as for some reciprocal and inverse indices. For such functions, we prove the following:

Theorem 1.

(a) If 0 < F(x, y) < 1 holds for all x > 1, y > 1, and if G is a bipartite graph, then EF (G) < E(G).

(b) If F(x,y) > 1 holds for all x > 1, y > 1, and if G is a bipartite graph, then EF (G) > E(G).

The equality cases will be considered later.

In order to prove Theorem 1, we need some preparations.

Preliminary considerations

Let

k >0

P( x) = I

be a polynomial with all zeros real. Then its energy satisfies (Mateljevic et al, 2010)

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O >

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E (P) = 1J

TT J vL

n ^0 x

I (-1)

V k >o

k c x 2 k 2 kA

2

+

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I (-1)

V k >o

kc x 2 k+1 k+1A

If the zeros of P(x) are symmetric w.r.t. x=0, i.e., if c2k+1 = 0 for all k > 0 , then

E (P) = .?. J I (-1)

7T » V

n 0 x k >0

kC2 kx 2k

As well known, a graph is bipartite if and only if it does not contain cycles of odd size. The characteristic polynomial of a bipartite graph is of the form

№, x) = I

C2kx

n-2k

k>0

Analogously, the characteristic polynomial of O = O(G) conforms to the relation

LO CM

( F ) n-2 k 2 k A

E(G) = 1J ^i^ln X (-1)kC2kx2k (1)

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ft 0 x k>0

Proving Theorem 1

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The respective energies are then

2 | dx

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0 k>0

and

Ep (G) = 1J ^ln X (-1) kc2 F} x2k (2) 5

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We apply the Sachs coefficient theorem (Cvetkovic et al, 2010), 2 (Gutman, 2017a). Recall that a Sachs graph is a graph consisting of □> vertices of degree one and/or two, i.e., all its components are isolated

edges and/or cycles. &

The application of the Sachs theorem to the coefficients of <F (G, x)

yields: |

c2F} = X (-1)'(')29(') w(s)

seS2 k (G)

where s is a Sachs graph and S2k (G) is the set of all (2k)-vertex Sachs

graphs that are as subgraphs contained in the graph G, and where p(s) = number of components of s, q(s) =number of cyclic components of s, and w(s) = weight of s.

The weight of s is equal to the product of the weights of all edges contained in the cycles of s, times the product of the squares of the weights of the isolated edges of s. The weight of a particular edge is equal to the respective element of the matrix O(G). For the proof of Theorem 1(a), it is only important that w(s) < 1.

Let G be a bipartite graph, and let the Sachs graph 5 e S2k (G)

contain a isolated edges, p cycles of the size 4/+2, and y cycles of the go size 4/.Then,

| p(s) = a + Ifi +1/,

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and

£ 2k = 2a +1 (4, + 2)£+1 (4,)/.

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o Therefore,

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o implying

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p(s) + k = 2 a + I(i +1)4+1iy a + I/

>o and

p(s) + k = I y (mod 2).

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In view of the above, the contribution of the Sachs graph s to the

term

(-1)p (s )2?(s) W(s) is:

LAS positive if s contains no cycles of size divisible by 4, cd negative if s contains an odd number of cycles of a size divisible by 4, I and

positive if s contains an even number of cycles of a size divisible by 4.

i Suppose first that Iy is zero or even. Then, the contribution of the

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^ Sachs graph 5 eS2k(G) to EF{G), Eq. (2), is positive, and because of w(s) < 1, it is not greater than the respective contribution of s to E(G), Eq. (1). In this case, EF(G) < E(G), with equality if all non-zero elements of O(G) are equal to unity.

There remains a case when Iy is an odd integer. Then, s has at

i

least one cycle whose size is divisible by 4. Let, for the sake of simplicity, this be a single 12-membered cycle, whose edges are

e1, e2,e3,e4, e5,e6, e7, e8, e9, e10, e11, e12. Then, in addition to s, there exist

two more Sachs graphs 5 5 '' e S2k (G ) in which instead of the 12-membered cycle, there are 6 isolated edges, e1, e3, e5, e7, e9, e11 and p; e2,e4,e6,e8,e10,e12, respectively. The total contribution of the three

Sachs graphs 5, 5 5 '' e S2k (G) is then

12 6 6

-2 H w(ei ) + n w(e2 H)2 +n w(e2/ )2 i=1 i=1 i=1 plus the (necessarily positive) contribution coming from the other

(mutually identical) fragments of 5, 5 ', 5 ''. The above expression is equal

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to £

f 6 6 \2 2 n w(e2 n-1)-n w(e2 i)

V i=1 i=1

which is non-negative. Because of w(e) < 1, this term is also less than or equal to unity.

Thus, also in this case, the joint contribution of the Sachs graphs 5, 5 ', 5 '' to Ef (G) is positive but not greater than their contribution to

E (G ).

This completes the proof of Theorem 1(a).

The proof of Theorem 1(b) is analogous. Note that the special case of Theorem 1(b), pertaining to extended energy, was earlier communicated in (Gutman, 2017b).

Discussion

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If the graph G is not bipartite, then it contains odd cycles. Then, of course, some Sachs graphs also contain odd cycles. Consequently,

( F )

some of the coefficients c2k+1 and c2k+1 are non-zero. Besides, the sign

of the coefficients c2k and c2k} cannot be predicted in the general case.

For these reasons, it is not easy to extend Theorem 1 to non-bipartite graphs, and we leave this for some later moment or some more skilled colleague.

From the definitions of E (G) and EF (G), it is evident that the

equality EF (G) = E (G) will hold if all non-zero elements of the matrix

O(G) are equal to unity. Whether this is an "if and only if" condition remains a (difficult) open problem.

In the case of regular graphs, for which deg(v.) = r, /'=1,2,...,n ,

the relation between E(G) and EF(G) is significantly simplified.

(Y

=3 Ef (G) = E(G) holds for the extended, geometric-arithmetic, and ° arithmetic energies. In addition, EF(G) = 2rE(G) holds for the first Zagreb energy, EF (G) = r2E(G) for the second Zagreb energy,

References

Ef (G) = 2r2E(G) for the forgotten energy, EF (G) =1E(G) for the

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^ Randic and harmonic energies, etc. Interestingly but evidently, the Albertson and sigma energies of regular graphs are equal to zero. On the other hand, for the class of stepwise irregular graphs (Gutman, 2018), the Albertson and sigma matrices coincide with the adjacency matrix, and for such graphs the Albertson and sigma energies are equal to the o ordinary graph energy.

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o Cruz, R., Perez, T. & Rada, J. 2015. Extremal values of vertex-degree-

^ based topological indices over graphs. Journal of Applied Mathematics and Computing, 48(1-2), pp.395-406. Available at: https://doi.org/10.1007/s12190-& 014-0809-y.

Cvetkovic, D., Rowlinson, P. & Simic, K. 2010. An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press. ISBN: 9780521134088.

Das, K.C., Gutman, I., Milovanovic, I., Milovanovic, E. & Furtula, B. 2018. Degree-based energies of graphs. Linear Algebra and its Applications, 554, pp.185-204. Available at: https://doi.org/10.1016Zj.laa.2018.05.027.

Furtula, B., Gutman, I. & Dehmer, M. 2013. On structure-sensitivity of degree-based topological indices. Applied Mathematics and Computation, 219(17), pp.8973-8978. Available at: https://doi.org/10.1016/j.amc.2013.03.072.

Gutman, I. 1978. The energy of a graph. Berichte der MathematischStatistischen Sektion im Forschungszentrum Graz, 103, pp.1-22.

Gutman, I. 2017a. Selected Theorems in Chemical Graph Theory. Kragujevac: University of Kragujevac.

Gutman, I. 2017b. Relation between energy and extended energy of a graph. Internat/onal Journal of Appl/ed Graph Theory,1(1), pp.42-48. ^

Gutman, I. 2018. Stepwise irregular graphs. Applied Mathematics and

7

Computation, 325, pp.234-238. Available at: £

https://doi.org/10.1016/j.amc.2017.12.045.

Kulli, V.R. 2020. Graph indices. In: Pal, M., Samanta, S. & Pal, A. (Eds.), p Handbook of Research of Advanced Applications of Graph Theory in Modern Society, pp.66-91. Hershey, USA: IGI Global. Available at: https://doi.org/10.4018/978-1-5225-9380-5.ch003.

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Li, X., Shi, Y. & Gutman, I. 2012. Introduction. In: Graph Energy, pp.1-9. -S

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New York, NY: Springer Science and Business Media LLC. Available at: https://doi.org/10.1007/978-1-4614-4220-2_1.

Liu, M., Xu, K. & Zhang, X.-D. 2019. Extremal graphs for vertex-degree-based invariants with given degree sequences. Discrete Applied Mathematics, 255, pp.267-277. Available at: https://doi.org/10.1016/j.dam.2018.07.026. £

Mateljevic, M., Bozin, V. & Gutman, I. 2010. Energy of a polynomial and the Coulson integral formula. Journal of Mathematical Chemistry, 48(4), pp.10621068. Available at: https://doi.org/10.1007/s10910-010-9725-z.

Rada, J. & Cruz, R. 2014. Vertex-degree-based topological indices over graphs. MATCH Communications in Mathematical and in Computer Chemistry, 72(3), pp.603-616 [online]. Available at:

http://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/match72n3_603- g

616.pdf [Accessed: 15 August 2020].

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Todeschini, R. & Consonni, V. 2009. Molecular Descriptors for "û Chemoinformatics. Weinheim: Wiley-VCH. ISBN: 978-3-527-31852-0. 01

Zhong, L. & Xu, K. 2014. Inequalities between vertex-degree-based topological indices. MATCH Communications in Mathematical and in Computer Chemistry, 71(3), pp.627-642 [online]. Available at: http://match.pmf.kg.ac.rs/electronic_versions/Match71/n3/match71n3_627-642.pdf [Accessed: 15 August 2020].

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ВЗАИМОСВЯЗЬ ЭНЕРГИИ ГРАФОВ С ЭНЕРГИЕЙ СТЕПЕНИ УЗЛОВ

Иван Гутман

Крагуевацкий университет, Естественно-математический факультет,

г. Крагуевац, Республика Сербия

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА;

27.29.19 Краевые задачи и задачи на собственные значения для обыкновенных дифференциальных уравнений и систем уравнений

ВИД СТАТЬИ: оригинальная научная статья

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^ Резюме:

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Введение/цель: На основании анализа существующей литературы, в статье представлены многочисленные инварианты графов, £ зависимые от степени узлов. К каждому из этих инвариантов

о подключается соответствующая матрица, с помощью которой

считывается энергия графа, как сумма абсолютных величин собственных значений данных матриц.

Результаты: В статье определены условия, при которых Ш вычисленные энергии графа были больше или меньше средней

з энергии графа (на основании матрицы смежности).

о Выводы: Результаты данной статьи вносят вклад в теорию

^ энергии графов, а также в теорию инвариантов графов,

0 основанных на степени узлов.

1 Ключевые слова: энергия (графа); инварианты, зависящие от

ш степени узлов; энергия, зависящая от степени узлов.

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а: <

РЕЛАЦШЕ ИЗМЕЪУ ЕНЕРГШЕ ГРАФА И ЕНЕРГША ЗАСНОВАНИХ НА СТЕПЕНИМА ЧВОРОВА

^ Иван Гутман

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Универзитет у Крагу]евцу, Природно-математички факултет,

^ Крагу]евац, Република Срби]а

>о ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад Сажетак:

Увод/циъ: У раду су приказане броjне, у литератури постоjеfiе, графовске инварианте зависне од степена чворова. Овим инвар^антама придружуjу се одговараjуhе матрице, преко щих се израчунава енерг^а као збир апсолутних вредности сопствених вредности ових матрица.

Резултати: Одре^ени су услови под щима су испитиване веЬе, односно мак>е енерг^е од обичне енерг^е графа (засноване на матрици суседства).

Закъучак: Рад доприноси теории графовских енерг^а, као и теории графовских инвар^анти зависних од степена чворова.

Къучне речи: енерг^а (графа), инварианте зависне од степена чворова, енерг^е зависне од степена чворова.

Paper received on / Дата получения работы / Датум приема чланка: 21.08.2020. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 28.08.2020.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 30.08.2020.

© 2020 The Author. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).

© 2020 Автор. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «Creative Commons» (http://creativecommons.org/licenses/by/3.0/rs/).

© 2020 Аутор. Обjавио Воjнотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). Ово jе чланак отвореног приступа и дистрибуира се у складу са Creative Commons licencom (http://creativecommons.org/licenses/by/3.0/rs/).

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