On the commuting graphs of Brandt semigroups

TL;DR

This paper investigates the graph-theoretic properties of the commuting graph of Brandt semigroups, including automorphisms and endomorphisms, and characterizes when these graphs are Hamiltonian.

Contribution

It provides a detailed analysis of the structure, automorphisms, and endomorphisms of the commuting graph of Brandt semigroups, including new results on Hamiltonian properties.

Findings

Automorphism group of elta(B_n) is S_n d7 Z_2.

For n b7 4, endomorphism monoid equals automorphism group.

Identified classes of inverse semigroups with Hamiltonian commuting graphs.

Abstract

The commuting graph of a finite non-commutative semigroup S, denoted by \Delta(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 \Delta(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(\Delta(Bn)) and the endomorphism monoid End(\Delta(Bn)) of \Delta(Bn). We show that Aut(\Delta(Bn)) = S_n \times 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 \geq 4, we prove that End(\Delta(Bn)) =Aut(\Delta(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.