On a conjecture of the harmonic index and the minimum degree of graphs

Abstract

The harmonic index of a graph G is dened as the sum of the weights 2/(d(u)+d(v)) of all edges uv of G, where d(u) denotes the degree of the vertex u in G. Cheng and Wang [9] proposed a conjecture:
For all connected graphs G with n≥4 vertices and minimum degree δ(G)≥k, where 1 ≤ k ≤[1/2]+1, then H(G) ≤H(K_{k,n-k}^{*}) with equality if and only if G =K_{k,n-k}^{*}. K_{k,n-k}^{*} is a complete split graph which has only two degrees, i.e. degree k and degree n-1, and the number of vertices of degree k is n-k, while the number of vertices of degree n-1 is k. In this work, we prove that this conjecture is true
when k ≤n/2, and give a counterexample to show that the conjecture is not correct when k = [1/2] + 1, n is even, that is k = n/2 + 1.