Equal‐Square Graphs Associated with Finite Groups

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.

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

“…are widely used in spectral graph theory. Various structural properties and matrices associated with graphs of groups are studied in [1][2][3][4][5] . In [6][7][8] various structural properties of the square power graph of the finite Abelian group and its complement graph are studied.…”

Topological Indices and Structural Properties of Cubic Power Graph of Dihedral Group

“…A co-prime order graph of a finite group G is a graph with a vertex set G in which two distinct vertices a, b have an edge iff gcd(o(a), o(b)) = 1 or a prime number. Various properties of equal-square graphs associated with finite groups are studied in [10]. An equal-square graph of a finite group is a simple graph with two distinct vertices x, y adjacent iff x 2 = y 2 .…”

A New Graphical Representation of the Old Algebraic Structure

“…Upper bounds for the clique number and maximum degree associated with square graphs are given in [24,25], where authors also computed the result regarding the number of edges of this graph. Some authors calculated some results of equal-square graphs, a subclass of simple graphs where all the elements of the fnite group are the vertices and any two distinct vertices x 1 , x 2 are adjacent if x 2 1 � x 2 2 see [26]. With the aid of loop structures and a fnite Boolean commutative ring, the authors of [27,28] discovered properties of balanced bipartite graphs and zero-divisor graphs, while [29] gave a concept of directed inverse graphs of antiautomorphic inverse property loops and star graphs of substructures of these loops through edge labelings.…”