NOTE ON THE TEMPERATURE SOMBOR INDEX Текст научной статьи по специальности «Математика»

QPI/in/lHAflHI/1 HAyMHM PAflOBM OPI/in/lHAflbHblE HAyMHblE CTATbll ORIGINAL SCIENTIFIC PAPERS

Note on the temperature Sombor index

Ivan Gutman

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

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

DOI: 10.5937/vojtehg71 -44132; https://doi.org/10.5937/vojtehg71-44132

FIELD: mathematics (mathematics subject classification: primary 05c07,

secondary 05c09) ARTICLE TYPE: original scientific paper

Abstract:

Introduction/purpose: The temperature of a vertex of a graph of the order n is defined as d/(n-d), where d is the vertex degree. The temperature variant of the Sombor index is investigated and several of its properties established. Methods: Combinatorial graph theory is applied.

Results: Extremal values and bounds for the temperature Sombor index are obtained.

Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.

Keywords: temperature (of vertex), temperature vertex-degree-based graph invariant, Sombor index, temperature Sombor index.

Introduction

In this paper, we examine a class of vertex-degree-based (VDB) graph invariants. Let G be a simple graph with n vertices and m edges. Let

V(G) and E(G) be its vertex and edge sets, respectively. Then | V(G) |= n

and | E(G) |= m . The edge of the graph G, connecting the vertices u and v, will be denoted by uv. The degree du of the vertex u is the number of its first neighbors.

The graph in which any two vertices are adjacent is said to be complete and is denoted by Kn. It has m=n(n-1)/2 edges. Its complement,

denoted by Kn, is the edgeless graph, with m=0.

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

In the recent mathematical and chemical literature, a large number of graph invariants of the form

TI = TI(G) = X fd, dv) (1)

uveE(G)

are studied, where f is a pertinently chosen function with the property f(x,y)=f(y,x); for details, see (Gutman, 2023) and the references cited therein. The quantities defined via Eq. (1) are usually referred to as vertex-degree-based (VDB) graph invariants. Of these, one of the oldest is the first Zagreb index (Gutman & Trinajstic, 1972; Gutman & Das, 2004):

Ml = MX(G) = X (du + dv)

uveE (G)

whereas one of the most recent ones is the Sombor index (Gutman, 2021; Liu et al, 2022):

SO = SO(G) = X 4du u + dv

2

_ v

uveE(G)

According to Fajtlowicz (Fajtlowicz, 1988), the temperature of the vertex u of a graph with n vertices is defined as

d

K V (2)

n - du

where one should recall that in the case of n-vertex graphs, 0 < du < n — 1. Directly from this definition, it follows that

2m ^ ^ ^^

— < X tu < 2m.

n ueV(G)

The equality on the left-hand side holds if G = Kn, whereas the right-hand side equality holds if either G = Kn or G = Kn.

In Eq. (1), by replacing the vertex degrees with vertex temperatures, one obtains the respective temperature VDB graph invatiants, namely:

TTI = TTI (G) = X f (K, tv) .

uveE (G)

Such are the temperature first Zagreb index

TMX = TMX(G) = X (tu + tv) (3)

uveE (G)

and the temperature Sombor index

TSO = TSO(G) = X JtT+tJ.

uveE (G)

Several other temperature VDB graph invariants were studied in the literature (Narayankar et al, 2018; Kahsay et al, 2018; Kulli, 2019a; Kulli, 2019b; Kulli, 2021).

The temperature Sombor index was first considered by Kulli (Kulli, 2022). In this paper, we establish a few more of its properties.

Preparation: temperature first Zagreb index

Bearing in mind that for all vertices of any n-vertex graph, du < n -1, directly from Eq. (2), we obtain:

d

u

n

1 -

d

n

Xi ^

\ n J

Substituting this into Eq. (3) yields

1 TMi = X X nk (d-+ dvk )=XI n-

n uveE(G) k=1 n k=1 ^n

where we used the identity (Gutman, 2015)

X [g(u) + g(v)] = X dug(u)

uveE(G) ue V (G)

which holds for any quantity g determined by the vertex u. Note that TM1 had to be be divided by n3 because the maximum possible value of

X duk+1 is n(n - 1)k

ueV (G)

1

1

Xd

u

ueV (G)

k+1

(4)

vk+1 nk+2.

t

u

In connection with formula (4), one should note that for k=1 and k=2,

Zj k+1

du is equal to the well-known and much studied VDB

ueV (G)

invariants - the first Zagreb index Mi and the so-called forgotten index F (Furtula & Gutman, 2015), respectively. The same term for k=3 and k=4 coincides with the VBD invariants Y and S, recently introduced in (Alameri et al, 2020) and (Nagarajan et al, 2021), respectively.

Therefore, TM1 ~ — M1 + — F, which is an approximation that

n n

would satisfy all practical applications of the temperature first Zagreb index. A somewhat better, yet more perplexed approximation would be

TM— « — M— + -1 F + — Y + — S. n n n n

On the temperature Sombor index

It is evident from Eq. (2) that the temperature of a vertex is a monotonously increasing function of the respective vertex degree.

Therefore, by deleting an adge e e E(G) from the graph G, some of its vertex temperatures must decrease, and no vertex temperature will increase. This implies,

TSO(G - e) < TSO(G). (5)

From relation (5), we immediately conclude the following:

(1) The complete graph and its complement have the maximum and minimum temperature Sombor indices, i.e.,

0 = TSO(Kn) < TSO(G) < TSO(Kn) = -—= n(n -1)2.

-v/2

(2) The connected graph with the minimum value of TSO must be a tree.

(3) Based on a general result for VDB graph invariants (Cruz & Rada, 2019), the trees with the maximum and minimum temperature Sombor indices are the star and the path, respectively.

In what follows, we use the well-known inequality

a + b) < a + b

valid for a,b > 0, with the left-hand side equality if a=b, and the right-hand side inequality in the irrelevant case a=b=0. Applying it to TSO, we get

"i X (tu + tv) < TSO(G) < X (tu + tv)

V2 uveE(G) uveE(G)

i.e.,

-1 TMX (G) < TSO(G) < TM! (G) V2

With the left-hand side equality if and only if the graph G is regular, i.e., if all its vertices have mutually equal degrees.

Bearing in mind Eq. (4), we get

V2 k=i

1

1

X ^ X <TSO(G)<x ^ X d

k=1 Vn ueV(G) J n k=1 V n ueV(G)

1

k+1

nk+3 X d

Vn ueV (G) J

(6)

From (6), we immediately obtain the following lower bounds for TSO.

(7)

TSO(G) (— M— (G) + F(G) V 2 V n n

or, better, but more complicated,

TSO(G) >±= (—M— (G) + -i F (G) + \ Y (G) + -- S (G) V 2 ^ n n n n y

. (8)

O LO d. cp

x"

<1J T3

o

.Q

E o OT <1J

<1J

E <u

o

<1J

o

ro E

CD

The equality in (7) and (8) holds if G = Kn.

In order to get an upper bound for TSO, we modify the right-hand side of (6) as

1 1 ^ 1

TSO(G) < — M—(G) + - F(G) + X~(n - 1)k

\k+!

n

n

n

k=3

511

" from which it follows

<u

>-

f 1 „ 1 \

1_1 n

2

TSO(G) <1M (G) +1F(G) + (n -12

0 n n v n n~

> v y

CO

01

^^ References

a:

LU

ai Alameri, A., Al-Naggar, N., Al-Rumaima, M. & Alsharafi, M. 2020. Y-index of

o some graph operations. International Journal of Applied. Engineering Research,

0 15(2), pp. 173-179 [online]. Available at: <c https://www.ripublication.com/ijaer20/ijaerv15n2_11.pdf [Accessed: 19 April

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Cruz, R. & Rada, J. 2019. The path and the star as extremal values of vertex-degree-based topological indices among trees. MATCH Communications in Mathematical and in Computer Chemistry, 82, pp.715-732 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match82/n3/match82n3_715-732.pdf [Accessed: 19 April 2023].

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2 Furtula, B. & Gutman, I. 2015. A forgotten topological index. Journal of ^ Mathematical Chemistry, 53, pp.1184-1190. Available at:

https://doi.org/10.1007/s10910-015-0480-z.

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3 https://doi.org/10.22052/IJMC.2015.10107.

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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.

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O LO !±

<U T3

о

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https://doi.org/10.22061/JMNS.2018.3624.1032. I

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Narayankar, K.P., Kahsay, A.T. & Selvan, D. 2018. Harmonic temperature index of certain nanostructures. International Journal of Mathematics Trends and Technology (IJMTT), 56(3), pp.159-164.

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Заметка о температурном индексе города Сомбор Иван Гутман

Крагуевацкий университет, естественно-математический факультет, г. Крагуевац, Республика Сербия

РУБРИКА ГРНТИ: 27.29.19 Краевые задачи и задачи на собственные

значения для обыкновенных дифференциальных уравнений и систем уравнений

ВИД СТАТЬИ: оригинальная научная статья

« Резюме:

<и

Введение/цель: Температура вершины графа порядка п определяется как d/(n-d), в котором d представляет степень вершины. Исследован температурный вариант индекса Сомбора о и доказаны некоторые его свойства.

со см о см

Методы: В данной статье применяется комбинаторная теория графов.

of Результаты: В результате исследования были получены

предельные значения температурного индекса Сомбора и его 5 верхние и нижние пределы.

о Выводы: Данное исследования вносит вклад в теорию

< инвариантов графов сомборского типа. о

Ключевые слова: температура (вершины), температурный о инвариант графа, основанный на степени вершины, индекс

ш Сомбора, температурный индекс Сомбора.

>

^ Белешка о температурском сомборском индексу

Иван Гутман

Универзитет у Крагу]евцу, Природно-математички факултет, Крагу]евац, Република Срби]а

< ОБЛАСТ: математика

ёз КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад

о Сажетак:

Увод/цил>: Температура чвора у графу реда n дефинисана jе као d/(n-d), где jе d степен чвора. Истраживана jе температурска о варианта сомборског индекса и доказане су неке ъене особине.

о Методе: Применена jе комбинаторна теорба графова.

Резултати: Одре^ене су екстремне вредности за температурски сомборски индекс, и на^ене доъе и горне границе. Закъучак: Рад доприности теории графовских инвар^анти сомборског типа.

Къучне речи: температура (чвора), графовска инварианта зависна од степена чворова, сомборски индекс, температурски сомборски индекс.

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 о

ю

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.

х ф тз

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РЕДАКЦШСКИ КОМЕНТАР: Аутор овог чланка Иван Гутман je актуелни члан ^ Уре^ивачког одбора Во]нотехничког гласника. Због тога je уредништво спровело <u транспарентни]и и ригорозн^и двострукослепи процес рецензи]е. Уложило je додатни напор да одржи интегритет рецензи]е и необ]ективност сведе на на]ма^у ф могу^у меру тако што ]е други уредник сарадник водио процедуру рецензи]е ^ независно од уредника аутора, при чему ]е та] процес био апсолутно транспарентан. ф Уредништво ]е посебно водило рачуна да рецензент не препозна ко ]е написао рад и да не до^е до конфликта интереса.

о

Paper received on / Дата получения работы / Датум приема чланка: 23.04.2023. -g

Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 22.05.2023.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 24.05.2023. Е

u

© 2023 The Author. Published by Vojnotehnicki glasnik / Military Technical Courier CD

(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/).

© 2023 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/).

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