On the commuting graphs of Brandt semigroups
Abstract
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 (Bn) of Brandt semigroup Bn including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, 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)). In order to provide an answer to the question posed in [2], we ascertained a class of inverse semigroups whose commuting graph is Hamiltonian.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.