On the harmonic index and the average eccentricity of a graph

Abstract

The harmonic index H(G) of a graph G is defined as the sum of the weights of 2/d(u) + d(v) all edges uv of G, where d(u) denotes the degree of the vertex u in G. The eccentricity g(v) of a vertex v in G is the maximum distance from it to any other vertex, and the average eccentricity ϵ(G) in G is the mean value of the eccentricities of all vertices of G. In this work, a sharp upper bound and a lower bound of H(G) +ϵ(G) (H(G) ϵ(G), respectively) are presented, and the corresponding extremal graphs are characterized.