Power Graphs of Finite Groups Determined by Hosoya Properties.

Fawad Ali,Anwarud Din,Asad Ullah,Tareq Saeed,Bilal Ahmad Rather

doi:10.3390/e24020213

Abstract

Suppose is a finite group. The power graph represented by of is a graph, whose node set is , and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph is known as the Z-index or Hosoya index. The Z-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.

Highlights

Received: 6 January 2022A topological index is a numeric value that represents the symmetry of a molecular structure
It is a mathematical classification of a chemical graph that offers a mathematical function in a quantitative structure–property relationship (QSPR)
This work aimed to discuss the structural properties of the power graphs of finite non-abelian groups

Summary

Introduction

A topological index is a numeric value that represents the symmetry of a molecular structure It is a mathematical classification of a chemical graph that offers a mathematical function in a quantitative structure–property relationship (QSPR). Cameron et al [9] discussed the matching numbers and gave the upper, as well as the lower bounds for the perfect matching of power graphs of certain finite groups. The authors of [15] examined the maximum clique and found the largest number of edges of power graphs for all the finite cyclic groups. There are still several gaps in the current study about the determination of the Hosoya polynomials, the reciprocal Hosoya polynomials, and the Z-index or Hosoya index of the power graphs of a finite cyclic group Zn , the dihedral group D2m , and the generalized quaternion group Q4n. We look at one of these problems in this article

Basic Notions and Notations

Main Results

Hosoya Polynomial

Reciprocal Status Hosoya Polynomial

Hosoya Index

Concluding Remarks