A lower bound for the harmonic index of a graph with minimum degree at least two

Abstract

The harmonic index H(G) of a graph G is defined as the sum of the weights 2 d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. We give a best possible lower bound for the harmonic index of a graph (a triangle-free graph, respectively) with minimum degree at least two and characterize the extremal graphs.

Degree-Based Topological Indices

“…His works were followed by the recent paper. 153 No chemical applications of the harmonic index were reported so far, but, knowing the present situation in mathematical chemistry, such researches are very much to be expected.…”

The harmonic index of unicyclic graphs with given matching number

“…Li and Shiu [10] studied how the harmonic index behaves when the graph is under perturbations and provided a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, lower bounds for harmonic index are also obtained in [10] and [14], respectively. Recently, Deng et al [4] studied the relationship between the harmonic index and the chromatic number of a graph G, and obtained the lower bound for H(G) in terms of its chromatic number.…”

Some Properties on the Harmonic Index of Molecular Trees

“…Xu [6] and Deng et al [7,8] established some relationship between the harmonic index of a graph and its topological indices, such as Randić index, atom-bond connectivity index, chromatic number, and radius, respectively. Wu et al [9] determined the graph with minimum harmonic index among all the graphs (or all triangle-free graphs) with minimum degree at least two. More information on the harmonic index of a graph can be found in [10].…”