Equal-Square Graphs Associated with Finite Groups

Shafiq Ur Rehman,Tayaba Tariq,Ghulam Farid,Ebenezer Bonyah,Gohar Ali

doi:10.1155/2022/9244325

Abstract

The graphical representation of finite groups is studied in this paper. For each finite group, a simple graph is associated for which the vertex set contains elements of group such that two distinct vertices x and y are adjacent iff x 2 = y 2 . We call this graph an equal-square graph of the finite group G , symbolized by E S G . Some interesting properties of E S G are studied. Moreover, examples of equal-square graphs of finite cyclic groups, groups of plane symmetries of regular polygons, group of units U n , and the finite abelian groups are constructed.

Highlights

Graphs are studied very extensively by taking into account some characteristics and properties, which are used to prepare set of vertices and constructing edges
Combinatorics, and number theory play a very significant role in this study; concepts of abstract algebraic structures are studied with the help of graphs
Visualizing groups using graphs is a rapidly growing trend in algebraic graph theory [1,2,3]. e automorphism group of a graph and the Cayley graph of a group motivates to study the interplay between graphs and groups, see [4,5,6,7,8]

Summary

Introduction

Graphs are studied very extensively by taking into account some characteristics and properties, which are used to prepare set of vertices and constructing edges. We construct a graph whose vertex set is a finite group G and has two distinct vertices a and b adjacent iff a2 b2. We call this graph an equal-square graph of G and will be represented by ES(G). Corresponding to each finite group G, we will denote by ES(G), the equal-square graph of G, whose vertex set consists of all the elements of G such that two distinct vertices a and b are adjacent iff a2 b2. If G is an abelian group of finite order |G| 2αλ, provided λ is odd and α ≥ 1, G can have α distinct (nonisomorphic) equalsquare graphs (Corollary 3). Each vertex in G is adjacent to e and so ES(G) is connected

Equal-Square Graph of Cyclic Groups

Equal-Square Graph of Dihedral Groups

Equal-Square Graph of Finite Abelian Groups

Concluding Remarks