Commuting graph of an aperiodic Brandt Semigroup

Document Type : Original paper

Authors

1 Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani-333031, India

2 School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, Odisha 752050, India

Abstract

The commuting graph of a finite non-commutative semigroup S, denoted by Δ(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(Δ(Bn)) and the endomorphism monoid End(Δ(Bn)) of Δ(Bn). We show that Aut(Δ(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(Δ(Bn))=Aut(Δ(Bn)). Moreover,  we provide the vertex connectivity and edge connectivity of Δ(Bn). This paper provides a partial answer to a question posed in \cite{a.Araujo2011} and so we ascertained  a class of inverse semigroups whose commuting graph is Hamiltonian.

Keywords

Main Subjects

References

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Communications in Combinatorics and Optimization
Volume 10, Issue 1
March 2025
Pages 127-150

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