On the extremal graphs for general sum-connectivity index ([formula omitted]) with given cyclomatic number when [formula omitted

Abstract

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 χα(G) and is defined as ∑uv∈E(G)(du+dv)α, where uv is the edge connecting the vertices u,v∈V(G), du is the degree of the vertex u and α is a 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 ν. In this paper, it is proved that the unique graph obtained from the star Sn by adding ν edge(s) between a fixed pendant vertex u and ν other pendant vertices, has the maximum χα value in the collection of all n-vertex connected graphs having cyclomatic number ν with the constraints ν=5, n≥6, α>1 or 6≤ν≤n−2, α≥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 χα value among all n-vertex connected graphs (and also among all n-vertex connected molecular graphs) having cyclomatic number ν with the conditions ν≥3, n≥5(ν−1) and α>1.