2-Connected graphs with minimum general sum-connectivity index

Abstract

The general sum-connectivity index of a graph G is χα(G)=∑uv∈E(G)(d(u)+d(v))α, 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.