Harmonic Index of Dense Graphs

Abstract

The harmonic weight of an edge is defined as reciprocal of the average degree of its end-vertices. The harmonic index of a graph \(G\) is defined as the sum of all harmonic weights of its edges. In this work, we give the minimum value of the harmonic index for any \(n\)-vertex connected graphs with minimum degree \(\delta\) at least \(k(\geq n/2)\) and show the corresponding extremal graphs have only two degrees,i.e., degree \(k\)and degree \(n – 1\), and the number of vertices of degree \(k\) is as close to \(n/2\) as possible.