Harmonic Index of Graphs with More Than One Cut-Vertex

Abstract

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights of all edges \(uv\) of \(G\), where the weight of \(uv\) is \(\frac{2}{d(u) + d(v)}\), with \(d(u)\) denoting the degree of the vertex \(u\) in \(G\). In this work, we compute the harmonic index of a graph with a cut-vertex and with more than one cut-vertex. As an application, this topological index is computed for Bethe trees and dendrimer trees. Also, the harmonic indices of Fasciagraph and a special type of trees, namely, polytree, are computed.