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Power graphs of a class of completely 0-simple semigroups

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Abstract

We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we obtain a characterization that two \(G^{0}\)-normal completely 0-simple orthodox semigroups S and T with abelian group \(\mathcal {H}\)-classes are isomorphic based on their power graphs. We also present an algorithm to determine that S and T are isomorphic or not.

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References

  1. Abdollahi, A., Shahverdi, H.: Characterization of the alternating group by its non-commuting graph. J. Algebra 357, 203–207 (2012)

    Article  MathSciNet  Google Scholar 

  2. Ashegh Bonabi, Y., Khosravi, B.: On characterization of a completely simple semigroup by its power graph and Green Relations. J. Algebraic Combin. 55, 1123–1137 (2022)

    Article  MathSciNet  Google Scholar 

  3. Cameron, P.J.: The power graph of a finite group II. J. Group Theory. 13, 779–783 (2010)

    Article  MathSciNet  Google Scholar 

  4. Cameron, P.J., Ghosh, S.: The power graph of a finite group. Discrete Math. 311, 1220–1222 (2011)

    Article  MathSciNet  Google Scholar 

  5. Chakrabarty, I., Ghosh, S., Sen, M.K.: Undirected power graphs of semigroups. Semigroup Forum. 78, 410–426 (2009)

    Article  MathSciNet  Google Scholar 

  6. Ćirić, M., Bogdanović, S.: The five-element Brandt semigroup as a forbidden divisor. Semigroup Forum. 61(3), 363–372 (2000)

    Article  MathSciNet  Google Scholar 

  7. Dalal, S., Kumar, J., Singh, S.: On the connectivity and equality of some graphs on finite semigroups. Bull. Malays. Math. Sci. Soc. 46(1), 1–20 (2023)

    Article  MathSciNet  Google Scholar 

  8. Gharibkhajeh, A., Doostie, H.: On the graphs related to Green relations of finite semigroups. Iran. J. Math. Sci. Inf. 9, 43–51 (2014)

    MathSciNet  Google Scholar 

  9. Godsil, C., Royle, G.F.: Algebraic Graph Theory. Springer, New York (2001)

    Book  Google Scholar 

  10. Hao, Y., Gao, X., Luo, Y.: On the Cayley graphs of Brandt semigroups. Commun. Algebra 39(8), 2874–2883 (2011)

    Article  MathSciNet  Google Scholar 

  11. Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, London (1976)

    Google Scholar 

  12. Kelarev, A.V.: On Cayley graphs of inverse semigroups. Semigroup Forum. 72, 411–418 (2006)

    Article  MathSciNet  Google Scholar 

  13. Kelarev, A.V., Quinn, S.J.: Directed graph and combinatorial properties of semigroups. J. Algebra 251, 16–26 (2002)

    Article  MathSciNet  Google Scholar 

  14. Khosravi, B., Khosravi, B.: A characterization of Cayley graphs of Brandt semigroups. Bull. Malays. Math. Sci. Soc. 35(2), 399–410 (2012)

    MathSciNet  Google Scholar 

  15. Khosravi, B., Khosravi, B., Khosravi, B.: A characterization of the finite simple group \(L_{16}(2)\) by its prime graph. Manuscr. Math. 126, 49–58 (2008)

    Article  Google Scholar 

  16. Kumar, J., Dalal, S., Pandey, P.: On the commuting graphs of Brandt semigroups. (2020). https://doi.org/10.48550/arXiv.2010.05000

  17. Lewis, M.L.: An overview of graphs associated with character degrees. Rocky Mt. J. Math. 38(1), 175–211 (2008)

    Article  Google Scholar 

  18. Panma, S., Knauer, U., Arworn, S.: On transitive Cayley graphs of strong semilattices of right (left) groups. Discrete Math. 309, 5393–5403 (2009)

    Article  MathSciNet  Google Scholar 

  19. Petrich, M., Reilly, N.R.: Completely Regular Semigroups. Wiley, New York (1990)

    Google Scholar 

  20. Solomon, R.M., Woldar, A.J.: Simple groups are characterized by their non-commuting graphs. Int. J. Group Theory. 16, 793–824 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Yanliang Cheng, Yong Shao & Lingli Zeng

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  1. Yanliang Cheng
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  2. Yong Shao
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  3. Lingli Zeng
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Correspondence to Yong Shao.

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The paper is supported by National Natural Science Foundation of China (11971383) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY023).

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Cheng, Y., Shao, Y. & Zeng, L. Power graphs of a class of completely 0-simple semigroups. J Algebr Comb 59, 697–710 (2024). https://doi.org/10.1007/s10801-024-01306-1

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  • DOI: https://doi.org/10.1007/s10801-024-01306-1

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