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 the present paper, we study various graph theoretic properties of the commuting graph ∆(B n ) of Brandt semigroup B n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(∆(B n )) and the endomorphism monoid End(∆(B n )) of ∆(B n ). We show that Aut(∆(B n )) ∼ = S n × Z 2 , where S n is the symmetric group of degree n and Z 2 is the additive group of integers modulo 2. Further, for n ≥ 4, we prove that End(∆(B n )) =Aut(∆(B n )). In order to provide an answer to the question posed in [2], we ascertained a class of inverse semigroups whose commuting graph is Hamiltonian.