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) = ∑ uv∈E(G) 2 d(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.