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.