On the harmonic index and the average eccentricity of graphs

Abstract

The harmonic index H(G) of a graph G is defined as the sum of the weights 2/d(u)ld(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. The eccentricity e(v) of a vertex v in G is the maximum distance from v to any other vertex, and the average eccentricity e(G) of G is the mean value of the eccentricities of all vertices in G. In this note, we present the sharp lower and upper bounds of H(G) + e(G) and H{G) ▪ e(G) for connected graphs G and characterize the corresponding extremal graphs.