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.
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