Minimum Variable Connectivity Index of Trees of a Fixed Order

Shamaila Yousaf,Akhlaq Ahmad Bhatti,Akbar Ali

doi:10.1155/2020/3976274

Abstract

The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a nonnegative real number, EG is the edge set of G, and dt denotes the degree of an arbitrary vertex t in G. Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for γ>0, have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order n, where n≥4.

Highlights

All the graphs that we discuss in the present study are simple, connected, undirected, and finite
Denote by Pn and Sn the n-vertex path graph and the n-vertex star graph, respectively. e class of all n-vertex trees is denoted by Tn
QSPR/QSAR studies are progressive fields of chemical research that focus on the behavior of this dependency. e quantitative relationships are mathematical models that either enable the prediction of a continuous variable or the classification of a discrete variable from structural parameters

Summary

Introduction

All the graphs that we discuss in the present study are simple, connected, undirected, and finite. E number |N(v)| is called the degree of a vertex, v ∈ G, and it is denoted by d(v). A graph of order n is called an n-vertex graph.

Results

Conclusion