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

The harmonic index of a graph G is defined as H(G)=∑uv∈E(G)2d(u)+d(v)‎, ‎where d(u) denotes the degree of a vertex u in G‎. ‎Let 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 G(n,k)‎. ‎We solve the problem for each integer k(1≤k≤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≤k≤n−1) give a complete solution of the problem‎.