On the harmonic index and the girth of a graph

Abstract

The harmonic index H(G) of a graph G is defined as the sum of the weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we prove that H(G) + g(G) ≥ 11/2-6/n + 4/n+1 and H(G). g(G) ≥ 15/2-18/n + 12/n+1. for any connected graph G on n with girth g(G), with equalities if and only if G = S⁺_n is obtained from the star K_1,n-1 by adding a new edge between two pendant vertices.