General sum-connectivity index of unicyclic graphs with given diameter

Abstract

For α∈R, the general sum-connectivity index of a graph G is defined as χα(G)=∑uv∈E(G)[degG(u)+degG(v)]α, where E(G) is the edge set of G and degG(v) is the degree of a vertex v in G. Let Un,d be the set of unicyclic graphs with n vertices and diameter d.We present the graph with the smallest general sum-connectivity index among the graphs in Un,d for −1≤α<0 and the graph with the largest general sum-connectivity index among the graphs in Un,d for 0<α<1. Sharp lower bounds on the classical sum-connectivity index and the harmonic index for graphs in Un,d follow from the lower bound on the general sum-connectivity index.