On vertex and edge degree-based topological indices Текст научной статьи по специальности «Математика»

ОРИГИНАЛНИ НАУЧНИ РАДОВИ ОРИГИНАЛЬНЫЕ НАУЧНЫЕ СТАТЬИ ORIGINAL SCIENTIFIC PAPERS

On vertex and edge degree-based topological indices

Ivan Gutman

University of Kragujevac, Faculty of Science, Kragujevac, Republic of Serbia, e-mail: [email protected],

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

DOI: 10.5937/vojtehg71-45971;https://doi.org/10.5937/vojtehg71-45971

FIELD: mathematics (mathematics subject classification: primary 05C07, secondary 05C09)

ARTICLE TYPE: original scientific paper Abstract:

Introduction/purpose: The entire topological indices (TIent) are a class of

graph invariants depending on the degrees of vertices and edges. Some

general properties of these invariants are established.

Methods: Combinatorial graph theory is applied.

Results: A new general expression for TIent is obtained. For triangle-

free and quadrangle-free graphs, this expression can be significantly

simplified.

Conclusion: The paper contributes to the theory of vertex and edge degree-based graph invariants.

Key words: entire topological index, vertex and edge degree-based graph invariant, degree (of vertex), degree (of edge).

Introduction

In this paper, we are concerned with connected simple graphs. Let G be such a graph, and let V = V(G) and E = E(G) be its vertex and edge sets, respectively. The graph G has |V(G)| = n vertices and |E(G)| = m edges. Two vertices u, v of the graph are said to be adjacent, denoted as u ~ v, if u and v are the endpoints of an edge. The respective edge will then be denoted by uv. A vertex u and an edge e are said to be incident, denoted as u ~ e, if u is an endpoint of the edge e. Two edges e, f are said to be incident, denoted as e ~ f, if the edges e and f have a common vertex as the endpoint.

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"J The degree of a vertex u e V(G), denoted as du, is the number of vertices of G that are adjacent to u. The degree of an edge e e V(G), denoted by de, is the number of edges of G that are incident to e. If the endpoints of the edge e are the vertices u and v, then it is easy to see that

de — du I dv 2.

A graph is said to be regular of degree r if du — r holds for all u e V(G). ^ A regular graph of degree r with n vertices, has m — 1 nr edges. ^ The distance between two vertices u,v e V(G) (= length of the shortest g path connecting u and v) is denoted by d(u, v). If u and v are adjacent, then

u d(u, v) — 1.

For additional details of graph theory, see (Harary, 1969; Bondy & Murty, 1976).

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u In contemporary graph theory, especially in network theory and chemi-

cal graph theory, a large number of invariants of the form

TI — TI(G) — F(dx,dy) (1)

x,yeV(G)

x~y

are being considered. They are usually referred to as "'vertex-degree-based topological indices". In formula (1), the summation goes over all ^ pairs of adjacent vertices of the underlying graph G, i.e., over the pairs of ■y vertices at the unit distance, d(x, y) — 1. F stands for a real-valued func-x tion with the property F(x,y) — F(y,x) and F(x,y) > 0 for all values of ¡5 the variables x and y that the vertex degrees of the graph G may assume. Some best known and most studied invariants of this kind are the first Zagreb index (F — x+y), the second Zagreb index (F — xy), the Randic index (F — 1/^xy), the forgotten index (F — x2 + y2), the atom-bond-connectivity index (F — \/(x + y - 2)/(xy)), and the Sombor index (F — \fx2+y2). For details see (Todeschini & Consonni, 2000; Gutman, 2023).

Motivated by the success of both the mathematical theory and (mainly chemical) applications of the vertex-degree-based indices of type (1), their modified version

TIve — TIve(G)— Y, F (dx,de) (2)

xeV(G) , eeE(G)

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was put forward, first for F — x + y and F — xy (Kulli, 2016), and re-

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cently also for F — ^x2 + y2 (Kulli, 2022; Kulli & Gutman, 2022). The latter graph invariant is now called the KG-Sombor index, and is denoted

by KG = KG(G) (Kulli & Gutman, 2022; Kulli et al., 2022; Kosari et al., S

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2023; Madhumitha et al., 2024). In the case of the KG-Sombor index, it 1 has been shown (Kulli et al., 2022; Kulli & Gutman, 2022) that

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x,yeV(G)

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It is straightforward to state the generalization of the above formula: o

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Theorem 1. Let G be a simple graph. Then the invariant TIve, Eq. (2), can be expressed solely in terms of the vertex degrees of G, and satisfies the relation

TIve(G)= V \f(dx,dx + dy - 2) + F(dy,dx + dy - 2)1 . (3)

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Proof.The edge e in formula (2) has two endpoints, say x and y. Bearing this in mind, the summation in (2), for any particular edge e, must go over both x and y. This results in the two terms on the right-hand side of (3). Formula (3) follows now by taking into account that de = dx + dy - 2 for any o edge e = xy. □

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Entire topological indices u

Short time after the vertex- and edge-degree-based graph invariants of the type (2) were conceived (Kulli, 2016), the "entire" topological indices were put forward (Alwardi et al., 2018), first for the choices F = x + y (first Zagreb index) and F = xy (second Zagreb index). Eventually, entire indices were considered for the forgotten index (F = x2 + y2), (Bharali et al., 2020), the Randic index (F = 1/^xy), (Saleh & Cangul, 2021), and quite recently for the Sombor index (F = a/x2 + y2), (Movahedi & Akhbari, 2023).

The general form of these indices is

TIent = TIent(G)= Y, F(dx, dy) . (4)

x,yeV(G)UE(G)

x~y

Before stating Theorem 2, we recall some basic facts on line graphs (Harary, 1969; Bondy & Murty, 1976).

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The line graph L(G) of the graph G is defined so that its vertex set is E(G), and two vertices of L(G) are adjacent if the respective edges of G are incident. Thus the line graph of the graph G has |E(G)| = m vertices

and m(L(G)) = Zuzv(c) edges.

Theorem 2. Let G be a simple graph and let L(G) be its line graph. Then the invariant TIent, Eq. (4), can be expressed solely in terms of the vertex

Su degrees of G and L(G), and satisfies the relation

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o Tient = Tient (G) = Y [F (d*>dy )+ F (d*>d* + dV - 2) +

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+ F (dy ,dx + dy - 2)1 + Y F(dx, dy ). (5)

œ,yeV(L(G))

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Proof. It is evident that the summation on the right-hand side of (4) can be divided into three parts, namely for

(a) x e V(G) and y e V(G)

(b) x e V(G) and y e E(G) or vice versa;

3 (c) x e E(G) and y e E(G)

y In view of Eq. (1), the component of TIent pertaining to (a) is equal to

TI, i.e.,

O J] F (dx,dy) (6)

x,yeV(G)

o x~y

whereas by Eqs. (2) and (3), the component pertaining to (b) is equal to TIve, i.e.,

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x~y

F(dx,dx + dy - 2)+ F(dy,dx + dy - 2) . (7)

The component of TIent, corresponding to the choice of pairs of incident edges, namely (c), is equal to the TI-index of the line graph of the graph

G:

E F (dx,dy) (8)

x,yeV(L(G))

x~y

Combining (6)-(8), we arrive at

Tient(G) = TI (G) + Tive(G) + TI (L(G))

from which Eq. (5) directly follows. □

Special cases of expressions (6) and (8) were recognized by the authors

of the earlier papers on entire topological indices (Alwardi etal., 2018; Bhar-ali et al., 2020; Saleh & Cangul, 2021; Movahedi & Akhbari, 2023), but no one of them was aware of formula (7).

Theorem 2, and in particular formula (7), are stated here for the first time.

x,yeV(G)

+ F(dy, dx + dy - 2)

+

+ Y F(du + dw - 2,dv + dw - 2) (9)

u,veV((G))

d(u,v)=2

where w is the (unique) vertex lying between the vertices u and v.

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Corollary 1. If G is a regular graph of the degree r, with n vertices and m ° edges, then

TIent(G) = 1 nr [F(r, r) + 2F(r, 2r - 2) + (r - 1) F(2r - 2, 2r - 2)] . g

Proof. In formula (5), all vertex degrees of G are equal to r, whereas all vertex degrees of L(G) are equal to 2r - 2. The first summation in (5) goes over m = 2nr terms, whereas the second summation goes over

m(L(G)) = 2nr(r - 1) terms. □

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In the case of triangle-free and quadrangle-free graphs (such are the trees, hexagonal systems, fullerene graphs, nanotubes, etc.), formula (5) can be simplified. Namely, the entire topological indices TIent of triangle- ^ free and quadrangle-free graphs can be expressed in terms of vertex degrees of the underlying graph G, without any reference to its line graph L(G).

Corollary 2. If G is a triangle-free and quadrangle-free graph, then Eq. (9) holds:

TIent = TIent(G) = V F(dx,dy)+ F(dx, dx + dy - 2) +

"J Proof. If the graph G does not contain triangles and quadrangles, then two vertices at distance 2, say u and v, have a unique vertex between them, say w. Then uw and vw form a pair of incident edges, resulting in

£ Y F(dx,dy)= Y F(du + dw - 2,dv + dw - 2). (10)

x,yeV(L(G)) u,veV((G))

o x~y d(u,v)=2

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Su Substituting (10) back into (5) yields (9). □

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0 References

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^ Alwardi, A., Alqesmah, A., Rangarajan, R. & Cangul, I.N. 2018. Entire Zagreb

1 indices of graphs. Discrete Mathematics, Algorithms and Applications, 10(03), p. íá 1850037. Available at: https://doi.org/10.1142/S1793830918500374. >- Bharali, A., Doley, A. & Buragohain, J. 2020. Entire forgotten topological index <c of graphs. Proyecciones (Antofagasta), 39(4), pp. 1019-1032. Available at:

https://doi.org/10.22199/issn.0717-6279-2020-04-0064.

Bondy, J.A. & Murty, U.S.R. 1976. Graph theory with applications. Macmillan Press. ISBN: 0-444-19451-7. Sô Gutman, I. 2023. On the spectral radius of VDB graph matrices. Vojnotehnicki

^ glasnik/Military Technical Courier, 71(1), pp. 1-8. Available at: ^ https://doi.org/10.5937/vojtehg71-41411.

Harary, F. 1969. Graph Theory. Boca Raton: CRC Press. ISBN: E 9780429493768.

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¡5 Kosari, S., Dehgardi, N. & Khan, A. 2023. Lower bound on the KG-Sombor

index. Communications in Combinatorics and Optimization, 8(4), pp. 751-757. o Available at: https://doi.org/10.22049/CC0.2023.28666.1662.

Kulli, V.R. 2016. On K Banhatti indices of graphs. Journal of Computer and Mathematical Sciences, 7(4), pp. 213-218. ISSN 0976-5727 (Print), ISSN 23198133 (Online).

Kulli, V.R. 2022. KG Sombor indices of certain chemical drugs. International Journal of Engineering Sciences & Research Technology, 11(6), pp. 27-35 [online]. Available at: https://www.ijesrt.com/index.php/J-ijesrt/article/view/48 [Accessed: 10 August 2023].

Kulli, V.R. & Gutman, I. 2022. Sombor and KG-Sombor Indices of Benzenoid Systems and Phenylenes. Annals of Pure and Applied Mathematics, 26(2), pp. 49-53. Available at: https://doi.org/10.22457/apam.v26n2a01883.

Kulli, V.R., Harish, N., Chaluvaraju, B. & Gutman, I. 2022. Mathematical properties of KG Sombor index. Bulletin of International Mathematical Virtual Institute, 12(2), pp. 379-386[online]. Available at: http://www.imvibl.org/buletin/bulletin_im vi_12_2_22/bulletin_imvi_12_2_22_379_386.pdf [Accessed: 10 August 2023].

860

with self loops. Communications in Combinatorics and Optimization. in press.

Резюме:

Введение/цель: Все топологические индексы (TIent) представляют собой класс инвариантов графа, зависящих от степеней расположения вершин и ребер. Установлены некоторые общие свойства этих инвариантов.

Методы: В данной статье применяется комбинаторная теория графов.

Результаты: Получено новое обобщенное выражение для TIent. Для графов без треугольников и четырехугольников это выражение может быть значительно упрощено.

Выводы: Данная статья вносит вклад в теорию инвариантов графов, зависящих от степеней вершин и ребер.

Ключевые слова: полный топологический индекс, инварианты графов, зависящих от степеней вершин и ребер, степень (вершины), степень (ребра).

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(Л Ф О ТЗ

Madhumitha, K., D'Souza, S. & Nayak, S. 2024. KG Sombor energy of graphs

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Movahedi, F. & Akhbari, M.H. 2023. Entire Sombor index of graphs. Iranian ^ Journal of Mathematical Chemistry, 14(1), pp. 33-45. Available at: https://doi.org/10.22052/IJMC.2022.248350.1663.

Saleh, A. & Cangul, I.N. 2021. On the entire Randic index of graphs. Advances and Applications in Mathematical Sciences, 20(8), pp. 1559-1569 [online]. Available at: ^ https://www.mililink.com/upload/article/1367760163aams_vol_208_june_2021_a -§ 20_p1559-1569_a._saleh_and_i._n._canguli.pdf [Accessed: 10 August 2023]. о Todeschini, R. & Consonni, V. 2000. Handbook of molecular descriptors. Wein- ф

heim: Wiley-VCH. ISBN: 3-52-29913-0. J

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ВИД СТАТЬИ: оригинальная научная статья

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Иван Гутман

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Резултати: На^ена jе нова општа формула за Т1еЫ. За графове без троуглова и четвороуглова ова формула се значаjно поjедносmавл>уjе.

^ Закъучак: Рад доприноси теории графовских инваршанти

щи зависе од степена чворова и степена грана.

1 Къучне речи: потпуни тополошки индекси, графовске инварианте ще зависе од степена чворова и грана, степен (чвора), степен (гране).

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EDITORIAL NOTE: The author of this article, Ivan Gutman, is a current member of the Editorial Board of the Military Technical Courier. Therefore, the Editorial Team has ensured that the double blind reviewing process was even more transparent and more rigorous. The Team made additional effort to maintain the integrity of the review and to minimize any bias by having another associate editor handle the review procedure independently of the editor - author in a completely transparent process. The Editorial Team has taken special care that the referee did not recognize the author's identity, thus avoiding the conflict of interest. КОММЕНТАРИЙ РЕДКОЛЛЕГИИ: Автор данной статьи Иван Гутман является действующим членом редколлегии журнала «Военно-технический вестник». Поэтому редколлегия провела более открытое и более строгое двойное слепое рецензирование. Редколлегия приложила дополнительные усилия для того чтобы сохранить целостность рецензирования и свести к минимуму предвзятость, вследствие чего второй редактор-сотрудник управлял процессом рецензирования независимо от редактора-автора, таким образом процесс рецензирования был абсолютно прозрачным. Редколлегия во избежание конфликта интересов позаботилась о том, чтобы рецензент не узнал кто является автором статьи.

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РЕДАКЦШСКИ КОМЕНТАР: Аутор овог чланка Иван Гутман je актуелни члан Уре^ивачког одбора Во^отехничког гласника. Због тога je уредништво спровело i транспарентни|и и ригорозни|и двоструко слепи процес рецензи]е. Уложило je додатни напор да одржи интегритет рецензи]е и необ]ективност сведе на на]ма^у могу^у меру g. тако штсф други уредниксарадник водио процедуру рецензи]е независно од уредника аутора, при чему ]е та] процес био апсолутно транспарентан. Уредништво ]е посебно о водило рачуна да рецензент не препозна ко]е написао рад и да недоле до конфликта интереса.

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Paper received on / Дата получения работы / Датум приема чланка: 14.08.2023. о Manuscript corrections submitted on / Дата получения исправленной версии работы / о Датум достав^а^а исправки рукописа: 22.11.2023. -а

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Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 23.11.2023.

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© 2023 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier ^

(http://vtg.mod.gov.rs, http://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/).

© 2023 Авторы. Опубликовано в "Военно-технический вестник / VojnotehniCki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, httpV/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/).

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