Solution to the minimum harmonic index of graphs with given minimum degree

Abstract

The harmonic index of a graph $G$ is defined as $ H(G)=\sum\limits_{uv\in E(G)}\frac{2}{d(u)+d(v)}$‎, ‎where $d(u)$ denotes the degree of a vertex $u$ in $G$‎. ‎Let $\mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$‎. ‎In this work we consider the problem of determining the minimum value of the‎ ‎harmonic index and the corresponding extremal graphs among $\mathcal{G}(n,k)$‎. ‎We solve the problem for each integer $k (1\le k\le n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$‎. ‎This result together with our previous result which solve the problem for each integer $k (n/2 \le k\le n-1)$ give a complete solution of the problem‎.

Keywords

• ‎harmonic index‎
• ‎minimum degree‎
• ‎extremal graphs