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)(v)ý, ýwhere d(u) denotes the degree of a vertex u in Gý. ýLet G(n,k) be the set of simple n-vertex graphs with minimum degree at least ký. ýIn this work we consider the problem of determining the minimum value of theý ýharmonic index and the corresponding extremal graphs among G(n,k)ý. ýWe 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−ký. ýThis 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 problemý.