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ORIGINAL SCIENTIFIC PAPERS
Relating Sombor and Euler indices
Ivan Gutman
University of Kragujevac, Faculty of Science,
Kragujevac, Republic of Serbia, e-mail: [email protected],
ORCID Ю: https://orcid.org/0000-0001-9681-1550
DOI: https://doi.org/10.5937/vojtehg72-48818
FIELD: mathematics (mathematics subject classification: primary 05c07, secondary 05c09)
ARTICLE TYPE: original scientific paper Abstract:
Introduction/purpose: The Euler-Sombor index (EU) is a new vertex-degree-based graph invariant, obtained by geometric consideration. It is closely related to the Sombor index (SO). The actual form of this relation is established.
Methods: Combinatorial graph theory is applied.
Results: The inequalities between EU and SO are established.
Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.
Keywords: degree(of vertex), Sombor index, Euler-Sombor index.
Introduction
Vertex-degree-based (VDB) graph invariants are much studied in the current mathematical and applied-mathematical literature; see for instance the recent papers (Das et al, 2021; Hu et al, 2022; Liu, 2023a; Monsalve & Rada, 2021; Rada et al, 2022; Yuan, 2024). A few years ago, it was discovered that some of these graph invariants have a geometric interpretation (Gutman, 2021). Eventually, this triggered a whole series of geometry-based research studies on VDB invariants (Ali et al, 2024; Gutman, 2022; Gutman et al, 2024; Imran et al, 2022; Liu, 2023b; Tang et al, 2024). The first geometry-motivated VDB invariant is the Sombor index (Gutman, 2021), defined as
SO = SO(G) = £ 2 + d,2 . (1)
uveE (G)
Gutman, I., Relating Sombor and Euler indices, pp.1-12
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Although relatively new, the Sombor index has been a subject of numerous mathematical studies; see the review (Liu et al, 2022), the most recent papers (Attarzadeh & Behtoei, 2024; Chen & Zhu, 2024; Selenge & Horoldagva, 2024; Shetty & Bhat, 2024), and the references cited therein. The Sombor index found also noteworthy applications, especially in chemistry (Hayat et al, 2024; Rauf & Ahmad, 2024; Redzepovic, 2021).
In some recent studies, a similarly-looking quantity has been encountered (Ali et al, 2024, Gutman et al, 2024, Tang et al, 2024), namely
EU = EU(G) = X <Jdu2 + dv2 + dudv. (2)
uveE (G)
For the reasons explained below, it can be named the "Euler-Sombor index".
In this paper, we use the following notation and terminology. By G we denote a simple graph with n vertices and m edges. Let E(G) be its edge
sets, and then | E(G) |= m . The edge of the graph G, connecting the vertices u and v is denoted by uv. The degree du of a vertex u is the number of the first neighbors of this vertex.
For additional details of graph theory, see (Harary, 1969; Bondy & Murty, 1976).
A geometric approach to VDB invariants
The general form of a VDB graph invariant is X f (du, dv), where
uveE (G)
f is a pertinently chosen function with the property f(x,y)=f(y,x). In (Gutman,
2021), it was recognized that the vertex-degree pair (du,dv) can be
interpreted as a point in a 2-dimensional coordinate system, representing the edge uv, called the degree-point of the edge uv (point A in Figure 1). If
so, then (dv, du) would be the dual degree point, pertaining to the same edge uv (point B in Figure 1).
The (Eucldean) distance between the degree-point (du, dv) and the
origin O is •Я2 + dv2 , which then directly leads to the concept of the Sombor index, Eq. (1).
Figure 1 - A geometric representation of the edge uv of a graph G. Here du=a and dv=b. The distance between the origin O and either the degree-point A or the dual degree-point B leads to the Sombor index, Eq. (1). The distance between the points A and B pertains to the Albertson irregularity index, see (Gutman, 2021).
Recently, in (Gutman et al, 2024), a geometric model was proposed, in which the degree-point and the dual degree-point play equivalent roles: these are set to be the two foci of an ellipse passing through the origin (see Figure 2).
Figure 2 - Ellipse whose foci are the degree-point A and the dual degree-point B of the edge uv of a graph G. The point C is the center of the ellipse.
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In (Gutman et al, 2024), it was shown that the lengths of the semimajor and the semi-minor axes of the ellipse in Figure 2 are
r1 =y/ a2 + b2 ft du 2 + dv 2 and r2 = a + b = du + dv. (3)
Using formulas (3), the area of the ellipse, equal to ftr r2 , can
easily be calculated and related to a VDB graph invariant. On the other hand, the calculation of the perimeter of the ellipse is a difficult task and (because of its importance in astronomy) a large number of various approximations have been proposed; for details, see (Gutman et al, 2024). The approximate formula for the perimeter of an ellipse, proposed by
Leonhard Euler (Euler, 1773), is ft2(/j2 + r22) . When relations (3) are
substituted into this formula, and the multiplier abandoned (since it is irrelevant for the present considerations), we arrive at the expression
4dJ+dJ+dJv.
This expression directly leads to the VDB graph invariant (2). Because of its origin, the name Euler-Sombor index for it would be appropriate.
Evidently, there is a close algebraic analogy between the Sombor index, Eq. (1), and the Euler-Sombor index, Eq. (2). In what follows, we determine the actual form of the relation between these two VDB graph invariants.
Estimating SO by means of EU
From now on, in order to avoid trivialities, we restrict the consideration to connected graphs. The results obtained could then be directly extended to disconnected graphs, taking into account that, for a graph G consisting of disconnected components G1 and G2 ,
SO(G) = SO(G) + SO(G) and EU (G) = EU (G,) + EU (G2).
Theorem 1. Let G be a connected graph. Then
Я EU(G) < SO(G) < EU(G). (4)
The equality on the left-hand side is attained if and only if the graph G is regular.
Proof. It suffices to verify that the relations
2 Vx2 + У2 + xy <yjx2 + y2 <Jx2 + y2 + xy hold for all x > 0 and y > 0.
(5)
The right-hand side inequality in (5) is obvious. The equality in it occurs if and only if either x=0 or y=0 (or both), which in the case of vertex degrees of connected graphs cannot happen.
In order to obtain the left-hand side inequality in (5), we seek A satisfying
A^jx + y + xy <^x + y . (6)
From (6), we get
A2xy < (1 - A2)x2 + (1 - A2)y2
A2xy - 2(1 - A2)xy < (1 - A2)x2 + (1 - A2)y2 - 2(1 - A2)xy [A2 -2(1 -A2)]xy <(1 - A2)(x-y)2.
Assuming that 1-A2>0, we conclude that it must be A2 - 2(1 -A2) > 0 i.e., A>V273 .
Then the best choice for relation (5) is the smallest value of A, i.e.,
a=V273 .
From the above consideration, it is seen that the equality will hold if and only if x=y. Applying this to graphs, the equality will hold if the end-vertices of the edge uv have equal degrees. If this must be valid for all edges of the (connected) graph G, then all vertices of G must have equal degrees, i.e., then G must be regular.
This completes the proof of Theorem 1. ■
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Inequalities (4) provide lower and upper bounds for the Sombor index in terms of the Euler-Sombor index. Of course, these relations can be inverted, so that EU is estimated by means of SO:
SO(G) < EU(G) <J| SO(G).
Improving the inequality SO < EU
In this section, if the end-vertices of an edge uv have the degrees du=i and dv=j (or vice versa), we say that the edge uv is of the (i,j)-type.
The right-hand side inequality (4) in Theorem 1 is strict. Therefore, it would be of interest to modify it so as to get equality for some graphs. In order to achieve this goal, we first establish the following:
Lemma 1. For x, y > 0 , the function
F(x, y) = ylx2 + y2 + xy -yjx2 + y2 is monotonically increasing. More precisely, in the domain
D = {(x, y) e R2\x > 0, y > 0}, for any (x, yr),(x2, y2) e D, if x > x2
and y > y2, then F(x,y) > F(x2,y2) holds.
Proof. We need to show that the right-hand side of
SF(x, y) _ 2x + y 2x
оx 2^1 x2 + y2 + xy 2^J x2 + y2 ^
is positive-valued for all x, y > 0 . We start with an evident relation
x2y2 + y4 + 4xy3 > 0.
Using it, we have
4x4 + 4x2y2 + 4x3y + ^x2y2 + y4 + 4xy3 J > 4x4 + 4x2y2 + 4x3y
(4x4 + y2 + 4xy)(x2 + y2) > 4x2 (x2 + y2 + xy)
(2x + y)2 (x2 + y2) > (2x)2 (x2 + y2 + xy)
(2x + y)2 (2x)
>
2 2 2 2 x + y + xy x + y
2 x
2 x + y
sjx2 + y2 + xy
>
y[xF+y2
from which it immediately follows that the right-hand side expression in Eq. (7) is greater than zero.
Because of F(x,y)=F(y,x), we also have dF (x, y) / dy > 0. ■
In view of Lemma 1, we need to find the minumum value of F(x,y) when x and y are the degrees of adjacent vertices of some graph. Evidently, this would happen if x=1 and y=1, i.e., for an edge of the (1,1)-type, resulting in
F(1,1) <Vx2 + y2 + xy i.e.,
x2 + y2
4x^7 < Vx2 + y2 + xy - (Vs -y[2).
Recall that in the above inequalities, it is assumed that x and y pertain to the degrees of vertices of graphs.
Taking into account Eqs. (1) and (2), by summation over all edges of the undelying graph, we arrive at:
Theorem 2. Let G be a connected graph with n vertices and m edges. Then
SO(G) < EU(G) - (V3 -72)m.
The equality holds if and only if all edges of the graph G are of the
(1,1)-type, which at connected graphs can happen only if n=2, m=1, i.e., if G is the two-vertex path.
In a fully analogous manner, we obtain the following theorems, which hold not for all connected graphs, but for those satisfying some structural requirements.
Theorem 3. Let G be a connected graph with n>3 vertices and m edges. Then
SO(G) < EU(G) - (V7 - V5)m.
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The equality holds if and only if all the edges of the graph G are of the
(1.2) -type, which only can happen if n=3, m=2, i.e., if G is the three-vertex path.
Proof. If a connected graph has 3 or more vertices, then none of its edges can be of the (1,1)-type. Then the next-smallest value of F(x,y) is F(1,2), pertaining to an edge of the (1,2)-type. Graphs possessing (1,2)-edges exist for n>3, but only the 3-vertex path has all its edges of the (1,2)-type. ■
Theorem 4. Let G be a connected graph with n>3 vertices, m edges, and without vertices of degree 1. Then
SO(G) < EU(G) - (Vl2 - V8)m
The equality holds if and only if all edges of the graph G are of the
(2.2) -type, which is the case with the n-vertex cycles, n>3.
Proof. If pendent vertices (those of degree one) do not exit in the underlying graph, then the minimum possible value of F(x,y) is F(2,2), pertaining to an edge of the (2,2)-type. The connected graphs in which all edges are of the (2,2)-type are the cycles, and these exist for all n>3. Therefore, the claim of Theorem 4 is applicable to all graphs with 3 or more vertices. ■
Theorem 5. Let G be a connected graph with n vertices, m edges, and let 5 be its smallest vertex degree (5>1). Then
SO(G) < EU(G) - (yf3 -4l)8m
The equality holds if and only if all edges of the graph G are of the (5,5)-type, i.e., if G is regular of the degree 5. If 5 is even, then graphs of this kind exist for all n> 5+1. If 5 is odd, then graphs of this kind exist for all even-valued n, n> 5+1.
Proof. By the same argument as in the previous proofs, the minimum possible value of of F(x,y) is F(5,5). The equality requires that all edges be of the (5,5)-type. If so, then all vertices must be of the degree 5. Thus, the graphs for which the equality holds must be 5-regular. In the last part of the statement of Theorem 5, the well-known conditions for the number of vertices of 5-regular graphs are repeated (Harary, 1969; Bondy & Murty, 1976). ■
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Relacionando los indices de Sombor y Euler
Ivan Gutman
Universidad de Kragujevac, Facultad de Ciencias,
Kragujevac, Republica de Serbia
CAMPO: matematicas (clasificacion de materias de matematicas: primaria 05c07, secundaria 05c09)
TIPO DE ARTICULO: articulo cientifico original
Resumen:
Introduccion/objetivo: El indice de Euler-Sombor (EU) es un nuevo grafico invariante basado en grados de vertice, obtenido mediante consideracion geometrica. Esta estrechamente relacionado con el indice de Sombor (SO). Se establece la forma real de esta relacion.
Metodos: Se aplica la teoria combinatoria de grafos.
Resultados: Se establecen las desigualdades entre UE y SO.
Conclusion: El articulo contribuye a la teoria de las invariantes graficas similares al indice de Sombor.
Palabras claves: grado (de vertice), indice de Sombor, indice de Euler-Sombor.
Соотношение между индексами Сомбора и Эйлера 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:
Введение/цель: Сомборский индекс Эйлера является новым инвариантом графа, основанным на степени вершины, полученным путем геометрического анализа. И он соотносится с индексом Сомбора. В данной статье установлено математическое соотношение между этими двумя инвариантами графа.
Методы: В данной статье применяется комбинаторная теория графов.
Результаты: Верхняя и нижняя границы индекса Сомбора были определены в зависимости от индекса Эйлера-Сомбора и
Gutman, I., Relating Sombor and Euler indices, pp. 1-12
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наоборот. Затем эти границы были откорректированы с учетом структурных особенностей графов.
Выводы: Данное исследование вносит вклад в теорию инвариантов графа сомборского вида.
Ключевые слова: степень (вершины), индекс Сомбора, индекс Эйлера-Сомбора.
Веза измену Сомборског и О]леровог индекса Иван Гутман
Универзитет у Крагу]евцу, Природно-математички факултет,
Крагу]евац, Република Срби]а
ОБЛАСТ: математика
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:
Увод/циш: О]лер-сомборски индекс ]е нова, на степенима чворова заснована графовска инварианта, доби/'ена геометри]ским разматрашима. Сродан ]е Сомборском индексу. У раду су утвр^ене математичке везе измену ове две графовске инварианте.
Методе: Применена ]е комбинаторна теори]а графова.
Резултати: Одре^ене су горше и доше границе за Сомборски индекс у зависности од О]лер-сомборског индекса, и обратно. Ове границе су затим побошшане, узима}уЬи у обзир структурне карактеристике графова.
Закшучак: Рад доприности теори]и графовских инвари]анти сомборског типа.
Кшучне речи: степен (чвора), Сомборски индекс, О]лер-сомборски индекс.
Paper received on: 20.01.2024.
Manuscript corrections submitted on: 01.03.2024.
Paper accepted for publishing on: 03.03.2024.
© 2024 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/).
