The Harmonic Index on Bicyclic Graphs

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 establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.