The minimum harmonic index for bicyclic graphs with given diameter

Abstract

The harmonic index of a graph G, is defined as the sum of weights 2d(u)+d(v) of all edges uv of G, where d(u) is the degree of the vertex u in G. In this paper we find the minimum harmonic index of bicyclic graph of order n and diameter d. We also characterized all bicyclic graphs reaching the minimum bound.