Abstract
The general sum-connectivity index of a graph G, denoted by chi(alpha) (G), 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(w) denotes the degree of a vertex w is an element of V(G), and ff is a non-zero real number. For alpha D= -1/2 and n >= 11, Wang et al. [On the sumconnectivity index, Filomat 25 (2011) 29-42] proved that K-2 + (K) over bar (n-2) is the unique graph with minimum chi(alpha) value among all the n-vertex graphs having minimum degree at least 2, where K-2 + (K) over bar (n-2) is the join of the 2-vertex complete graph K-2 and the edgeless graph (K) over bar (n-2) on n 2 vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135-141] proved that the result of Wang et al. holds also for n >= 3 and -1 <= alpha < -0.867. In this paper, it is shown that the aforementioned result of Wang et al. remains valid if the graphs under consideration are connected, n >= 6 and -1 <= alpha < alpha(0), where alpha(0) approximate to -0.68119 is the unique real root of the equation chi(alpha) (K-2 + (K) over bar (4)) - chi(alpha) (C-6) = 0, and C-6 is the cycle on 6 vertices.