Some properties of Square element graphs over semigroups. Issue 1 (2nd January 2020) - Biswas, Bijon; Sen Gupta, Raibatak; Sen, M.K.; Kar, S.
Abstract: The Square element graph over a semigroup S is a simple undirected graph S q ( S ) whose vertex set consists precisely of all the non-zero elements of S, and two vertices a, b are adjacent if and only if either a b or b a belongs to the set t 2 : t ∈ S ∖ 1 , where 1 is the identity of the semigroup (if it exists). In this paper, we study the various properties of S q ( S ) . In particular, we concentrate on square element graphs over three important classes of semigroups. First, we consider the semigroup Ω n formed by the ideals of Z n . Afterwards, we consider the symmetric groups S n and the dihedral groups D n . For each type of semigroups mentioned, we look into the structural and other graph-theoretic properties of the corresponding square element graphs.