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 $k(\ge n/2)$ 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.