Harmonic Index and Harmonic Polynomial on Graph Operations

Juan Hernández-Gómez,José Rodríguez,José M Sigarreta,J Méndez-Bermúdez

doi:10.3390/sym10100456

Abstract

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O ( n 2 ) .

Highlights

A single number representing a chemical structure, by means of the corresponding molecular graph, is known as topological descriptor
Where uv is the edge of G between vertices u and v, and du is the degree of vertex u. Both M1 and M2 have recently attracted much interest
We would like to stress that the symmetry property present in the operations on graphs studied here (Cartesian product, corona product, join, Cartesian sum and lexicographic product) was an essential tool in the study of the topological indexes, because it allowed us to obtain closed formulas for the harmonic polynomial and to deduce the optimal bounds for that index

Summary

Introduction

A single number representing a chemical structure, by means of the corresponding molecular graph, is known as topological descriptor. Both M1 and M2 have recently attracted much interest (see, e.g., [8,9,10,11]) (in particular, they are included in algorithms used to compute topological indices) Another remarkable topological descriptor is the harmonic index, defined in [12] as. We would like to stress that the symmetry property present in the operations on graphs studied here (Cartesian product, corona product, join, Cartesian sum and lexicographic product) was an essential tool in the study of the topological indexes, because it allowed us to obtain closed formulas for the harmonic polynomial and to deduce the optimal bounds for that index

Definitions and Background

Computation of the Harmonic Index of Graph Operations

Algorithm for the Computation of the Harmonic Polynomial