On the extremal graphs for general sum-connectivity index (chi(alpha)) with given cyclomatic number when alpha > 1

摘要

Let V(G) and E(G) be, respectively, the vertex set and edge set of a graph G. The general sum-connectivity index of a graph G is denoted by chi(alpha) (G) and is defined as Sigma(uv is an element of E(G))(d(u) + d(v))(alpha), where uv is the edge connecting the vertices u, v is an element of V(G), d(u) is the degree of the vertex u and a is alpha non-zero real number. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by v. In this paper, it is proved that the unique graph obtained from the star S-n by adding v edge(s) between a fixed pendant vertex u and v other pendant vertices, has the maximum chi(alpha) value in the collection of all n-vertex connected graphs having cyclomatic number v with the constraints v = 5, n >= 6, alpha > 1 or 6 <= v <= n - 2, alpha >= 2. It is also proved that only those graphs which consist of (only) vertices of degrees 2, 3, such that no two vertices of degree 3 are adjacent, have the minimum chi(alpha) value among all n-vertex connected graphs (and also among all n-vertex connected molecular graphs) having cyclomatic number v with the conditions v >= 3, n >= 5(v - 1) and alpha > 1. (C) 2018 Elsevier B.V. All rights reserved.