On sufficient condition for t-toughness of a graph in terms of eccentricity-based indices

IF 1.2 4区 综合性期刊 Q3 MULTIDISCIPLINARY SCIENCES
Rajkaran Kori, Abhyendra Prasad, Ashish K. Upadhyay
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引用次数: 0

Abstract

Let \(\omega (G)\) be the number of components of graph G. For \(t\geqslant 0\) we call G t-tough if \(t\cdot \omega (G-X)\leqslant |X|\), for every \(X\subseteq V(G)\) with \(\omega (G-X)\geqslant 2\). \(1-\)tough graphs are also called Hamiltonian graphs. The eccentric connectivity index of a connected graph G denoted by \(\xi ^c(G)\), is defined as \(\xi ^c(G) = \sum _{v \in V(G)} \epsilon ({v}) d(v)\). The eccentric distance sum of a connected graph G is denoted by \(\xi ^d(G)\), is defined as \(\xi ^d(G) = \sum _{v \in V(G)} \epsilon (v) D(v)\). The connective eccentricity index of a connected graph G denoted as \(\xi ^{ce}(G)\), is defined as \(\xi ^{ce}(G) = \sum _{v \in V(G)} \frac{d(v)}{\epsilon (v)}\), where \(\epsilon (v)\) is the eccentricity of the vertex v, D(v) is the sum of the distance from to all other vertices, and d(v) is the degree of vertex v. Finding sufficient conditions for a graph to possess certain properties is a meaningful and important problem. In this article, we give sufficient conditions for t-toughness graphs in terms of the eccentric connectivity index, eccentric distance sum, and connective eccentricity index.

基于偏心指数的图的 t-韧度的充分条件
让\(\omega (G)\) 是图 G 的分量数。对于\(t\geqslant 0\) 如果\(t\cdot \omega (G-X)\leqslant|X||),对于每一个\(X\subseteq V(G)\)都有\(\omega (G-X)\geqslant 2\) ,我们称 G 为 t-tough。\韧图也被称为哈密顿图。连通图 G 的偏心连通性指数用 \(\xi ^c(G)\)表示,定义为 \(\xi ^c(G) = \sum _{v \in V(G)} \epsilon ({v}) d(v)\)。连通图 G 的偏心距和用 \(\xi ^d(G)\)表示,定义为 \(\xi ^d(G) = \sum _{v \in V(G)} \epsilon (v) D(v)\)。连通图 G 的连通偏心指数用 \(\xi ^{ce}(G)\) 表示,定义为 \(\xi ^{ce}(G) = \sum _{v\in V(G)} \frac{d(v)}{\epsilon (v)}\)、其中 \(\epsilon (v)\) 是顶点 v 的偏心率,D(v) 是到所有其他顶点的距离之和,d(v) 是顶点 v 的度数。寻找图形具备某些属性的充分条件是一个有意义的重要问题。本文从偏心连通指数、偏心距离总和和连通偏心指数三个方面给出了 t-韧度图的充分条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
National Academy Science Letters
National Academy Science Letters 综合性期刊-综合性期刊
CiteScore
2.20
自引率
0.00%
发文量
86
审稿时长
12 months
期刊介绍: The National Academy Science Letters is published by the National Academy of Sciences, India, since 1978. The publication of this unique journal was started with a view to give quick and wide publicity to the innovations in all fields of science
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