A Lower Bound for the Harmonic Index of a Graph With Minimum Degree at Least Two

Renfang Wu, Zikai Tang, Hanyuan Deng

The harmonic index $H(G)$ of a graph $G$ is defined as the sum of the weights $2\over d(u)+d(v)$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. We give a best possible lower bound for the harmonic index of a graph (a triangle-free graph, respectively) with minimum degree at least two and characterize the extremal graphs.