Order two element graph over a group.

The order two element graph of a group G is the simple undirected graph whose vertex set consists of all elements of G and two distinct vertices u , v are adjacent if and only if either u v o r v u ∈ { t 2 : t ∈ G } ∪ { a ∈ G : a 2 = e } \ { e } , where e is the identity element of G. We denote this graph by 2 (G). In this paper, we identify those commutative groups G for which the graph 2 (G) is connected. We also characterize the structures of the graph 2 (ℤ n) for any positive integer n. Moreover, we consider the symmetric group S n , the dihedral group D n for any positive integer n and for these groups, we look into the structure and other graph-theoretic properties of the corresponding graphs. Finally, we give a list of finite groups of order ≤ 1 6 with their corresponding order two element graph structures. [ABSTRACT FROM AUTHOR]