2-Connected graphs with minimum general sum-connectivity index

The general sum-connectivity index of a graph G is χα(G) = ∑ uv∈E(G) (d(u)+d(v)), where d(u) denotes the degree of vertex u ∈ V (G), and α is a real number. In this paper, we show that in the class of graphs G of order n ≥ 3 and minimum degree δ(G) ≥ 2, the unique graph G having minimum χα(G) is K2 + Kn−2 if −1 ≤ α < α0 ≈ −0.867. Similarly, if we impose the supplementary condition for G to be triangle-free, the extremal graph is K2,n−2 for n ≥ 4 and −1 ≤ α < β0 ≈ −0.817. Since both extremal graphs are 2-connected, it follows that the properties are also true in the subclass of 2-connected graphs.