Commuting graph of an aperiodic Brandt Semigroup

Document Type : Original paper

Authors

1 Department of Mathematics, Birla Institute of Technology and Science, Pilani

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

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 this paper, we study the commuting graph of an important class of inverse semigroups viz. Brandt semigroup $B_n$. In this connection, we obtain the automorphism group ${\rm Aut}(\Delta(B_n))$ and the endomorphism monoid End$(\Delta(B_n))$ of $\Delta(B_n)$. We show that ${\rm Aut}(\Delta(B_n)) \cong S_n \times \mathbb{Z}_2$, where $S_n$ is the symmetric group of degree $n$ and $\mathbb{Z}_2$ is the additive group of integers modulo $2$. Further, for $n \geq 4$, we prove that End$(\Delta(B_n)) = $Aut$(\Delta(B_n))$. Moreover,  we provide the vertex connectivity and edge connectivity of $\Delta(B_n)$. 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

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Available Online from 18 September 2023

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