On Harmonic Index and Diameter of Quasi-Tree Graphs

Abstract

The harmonic index of a graph G (HG) is defined as the sum of the weights 2/du+dv for all edges uv of G, where du is the degree of a vertex u in G. In this paper, we show that HG≥DG+5/3−n/2 and HG≥1/2+2/3n−2DG, where G is a quasi-tree graph of order n and diameter DG. Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.