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

The harmonic index of a graph G is defined as H(G)=\sum\limits_{uv\in E(G)}\frac{۲}{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 (۱\le k\le n/۲) 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/۲ \le k\le n-۱) give a complete solution of the problem‎.

Keywords:

‎harmonic index‎ , ‎minimum degree‎ , ‎extremal graphs