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 $\Delta(S)$, 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

[1] A. Alilou and J. Amjadi, The sum-annihilating essential ideal graph of a commutative ring, Commun. Comb. Optim. 1 (2016), no. 2, 117–135.  https://doi.org/10.22049/cco.2016.13555
[2] J. Araújo, W. Bentz, and K. Janusz, The commuting graph of the symmetric inverse semigroup, Israel J. Math. 207 (2015), no. 1, 103–149.  https://doi.org/10.1007/s11856-015-1173-9
[3] J. Araújo, M. Kinyon, and J. Konieczny, Minimal paths in the commuting graphs of semigroups, European J. Combin. 32 (2011), no. 2, 178–197.  https://doi.org/10.1016/j.ejc.2010.09.004
[4] C. Bates, D. Bundy, S. Perkins, and P. Rowley, Commuting involution graphs for finite Coxeter groups, J. Group Theory 6 (2003), no. 4, 461–476.  https://doi.org/10.1515/jgth.2003.032
[5] C. Bates, D. Bundy, S. Perkins, and P. Rowley, Commuting involution graphs for symmetric groups, J. Algebra 266
(2003), no. 1, 133–153.  https://doi.org/10.1016/S0021-8693(03)00302-8
[6] T. Bauer and B. Greenfeld, Commuting graphs of boundedly generated semigroups, European J. Combin. 56 (2016), 40–45.  https://doi.org/10.1016/j.ejc.2016.02.009
[7] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Elsevier Publishing, New York, 1976.
[8] D. Bundy, The connectivity of commuting graphs, J. Combin. Theory Ser. A 113 (2006), no. 6, 995–1007.  https://doi.org/10.1016/j.jcta.2005.09.003
[9] P.J. Cameron, Graph homomorphisms, Combinatorics Study Group Notes (2006), Manuscript.
[10] I. Chakrabarty and J.V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. Comb. Optim. 7 (2022), no. 2, 121–167.  https://doi.org/10.22049/cco.2021.26990.1176
[11] M. Ciric and S. Bogdanovic, The five-element Brandt semigroup as a forbidden divisor, Semigroup Forum 61 (2000), no. 3, 363–372.  https://doi.org/10.1007/PL00006035
[12] S. Dalal, Graphs associated with groups and semigroups, Ph.D. thesis, BITS Pilani, Pilani, 2021.
[13] S. Dalal and J. Kumar, Chromatic number of the cyclic graph of infinite semigroup, Graphs Combin. 36 (2020), no. 1, 109–113.  https://doi.org/10.1007/s00373-019-02120-4
[14] N.D. Gilbert and M. Samman, Endomorphism seminear-rings of Brandt semigroups, Comm. Algebra 38 (2010), no. 11, 4028–4041.  https://doi.org/10.1080/00927870903286892
[15] Y. Hao, X. Gao, and Y. Luo, On the Cayley graphs of Brandt semigroups, Comm. Algebra 39 (2011), no. 8, 2874–2883. https://doi.org/10.1080/00927872.2011.568028
[16] P. Hell and J. Nešetřil, The core of a graph, Discrete Math. 109 (1992), no. 1-3, 117–126.  https://doi.org/10.1016/0012-365X(92)90282-K
[17] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1995.
[18] J.M. Howie and M.I.M. Ribeiro, Rank properties in finite semigroups, Comm. Algebra 27 (1999), no. 11, 5333–5347. https://doi.org/10.1080/00927879908826758
[19] J.M. Howie and M.I.M. Ribeiro, Rank properties in finite semigroups II: The small rank and the large rank, Southeast Asian Bull. Math. 24 (2000), no. 2, 231–237. https://doi.org/10.1007/s10012-000-0231-2
[20] A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, J. Algebra .Appl. 7 (2008), no. 1, 129–146. https://doi.org/10.1142/S0219498808002710
[21] M. Jackson and M. Volkov, Undecidable problems for completely 0-simple semigroups, J. Pure Appl. Algebra 213 (2009), no. 10, 1961–1978.  https://doi.org/10.1016/j.jpaa.2009.02.011
[22] K. Kátai-Urbán and C. Szabó, Free spectrum of the variety generated by the five element combinatorial Brandt semigroup, 73 (2006), no. 2, 253–260.  https://doi.org/10.1007/s00233-006-0615-4
[23] B. Khosravi and B. Khosravi, A characterization of Cayley graphs of Brandt semigroups, Bull. Malays. Math. Sci. Soc. 35 (2012), no. 2, 399–410.
[24] A. Kumar, L. Selvaganesh, P.J. Cameron, and T.T. Chelvam, Recent developments on the power graph of finite groups–a survey, AKCE Int. J. Graphs Comb. 18 (2021), no. 2, 65–94.  https://doi.org/10.1080/09728600.2021.1953359
[25] J. Kumar, Affine near-semirings over Brandt semigroups, Ph.D. thesis, IIT Guwahati, Guwahati, 2014.
[26] J. Kumar, S. Dalal, and P. Pandey, On the structure of the commuting graph of Brandt semigroups, International Conference on Semigroups and Applications, Springer Proceedings in Mathematics & Statistics, 2019, pp. 95–105.
[27] S. Margolis, J. Rhodes, and P.V. Silva, On the subsemigroup complex of an aperiodic Brandt semigroup, 97 (2018), no. 1, 7–31.  https://doi.org/10.1007/s00233-018-9927-4
[28] J.D. Mitchell, Turán's graph theorem and maximum independent sets in Brandt semigroups, Semigroups and languages, World Sci. Publ., River Edge, NJ, 2004, pp. 151–162.
[29] F. Movahedi, The energy and edge energy of some Cayley graphs on the abelian group $Z^4_n$, Commun. Comb. Optim.9 (2024), no. 1, 119–130. https://dx.doi.org/10.22049/cco.2023.28642.1647.
[30] R.P. Panda, S. Dalal, and J. Kumar, On the enhanced power graph of a finite group, Comm. Algebra 49 (2021), no. 4, 1697–1716.  https://doi.org/10.1080/00927872.2020.1847289
[31] M.M. Sadar, Pseudo-amenability of Brandt semigroup algebras, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 413–419.  http://eudml.org/doc/33324
[32] M.M. Sadr, Morita equivalence of Brandt semigroup algebras, Int. J. Math. Math. Sci. 2012, Article ID 280636.  https://doi.org/10.1155/2012/280636
[33] Y. Segev, On finite homomorphic images of the multiplicative group of a division algebra, Annals Math. 149 (1999), no. 1, 219–251.  https://doi.org/10.2307/121024
[34] Y. Segev, The commuting graph of minimal nonsolvable groups, Geometriae Dedicata 88 (2001), no. 1, 55–66. http://dx.doi.org/10.1023/A:1013180005982
[35] Y. Segev and G.M. Seitz, Anisotropic groups of type an and the commuting graph of finite simple groups, Pacific J. Math. 202 (2002), no. 1, 125–225.  http://dx.doi.org/10.2140/pjm.2002.202.125
[36] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001.
Communications in Combinatorics and Optimization
Volume 10, Issue 1
March 2025
Pages 127-150

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