Commuting graph of an aperiodic Brandt Semigroup

The commuting graph of a finite non-commutative semigroup S, denoted by ∆(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. In this paper, we study the commuting graph of an important class of inverse semigroups viz. Brandt semigroup Bn. In this connection, we obtain the automorphism group Aut(∆(Bn)) and the endomorphism monoid End(∆(Bn)) of ∆(Bn). We show that Aut(∆(Bn)) ≅ Sn × Z2, where Sn is the symmetric group of degree n and Z2 is the additive group of integers modulo 2. Further, for n ≥ 4, we prove that End(∆(Bn)) =Aut(∆(Bn)). Moreover, we provide the vertex connectivity and edge connectivity of ∆(Bn). This paper provides a partial answer to a question posed in [3] and so we ascertained a class of inverse semigroups whose commuting graph is Hamiltonian.