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

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 χ α ( G ) and is defined as ∑ u v ∈ E ( G ) ( d u + d v ) α , where u v is the edge connecting the vertices u , v ∈ V ( G ) , d u 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 S n by adding ν edge(s) between a fixed pendant vertex u and ν other pendant vertices, has the maximum χ α 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 χ α 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 .