The Harmonic Index on Unicyclic Graphs

Lingping Zhong1
1Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016, P.R. China

Abstract

The Harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of weights \(\frac{2}{d(u)+d(v)}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we consider the Harmonic index of unicyclic graphs with a given order. We give the lower and upper bounds for Harmonic index of unicyclic graphs and characterize the corresponding extremal graphs.