On the co-intersection graph of subsemimodules of a semimodule

Document Type : Research Paper

Authors

Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

Abstract

Let $S$ be a semiring with identity and $U$ be a unitary left $S$-semimodule. The co-intersection graph of an $S$-semimodule $U$, denoted by $\Gamma(U)$, is defined to be the undirected simple graph whose vertices are in one-to-one correspondence with all non-trivial subsemimodules of $U$, and there is an edge between two distinct vertices $N$ and $L$ if and only if $N+L \neq U$. We study these graphs to relate the combinatorial properties of $\Gamma(U)$ to the algebraic properties of the $S$-semimodule $U$. We study the connectedness of $\Gamma(U)$. We investigate some properties of $\Gamma(U)$ for instance, we find the domination number and clique number of $\Gamma(U)$. Also, we study cycles in $\Gamma(U)$.

Keywords

[1] S. Akbari, A. Tavallaee and S. Khalashi Ghezelahmad, Intersection graph of submodule of a module, J. Algebra
Appl. 11 (2012), no. 1, 1250019 .
[2] A.H. Alwan and A.M. Alhossaini, On dense subsemimodules and prime semimodules, Iraqi J. Sci. 61 (2020), no.
6, 1446–1455.
[3] A.H. Alwan and A.M. Alhossaini, Dedekind multiplication semimodules, Iraqi J. Sci. 61 (2020), no. 6, 1488–1497.
[4] A.H. Alwan and A.M. Alhossaini, Endomorphism Semirings of Dedekind Semimodules, Int. J. Adv. Sci. Technol.
29 (2020), no. 4, 2361–2369.
[5] A.H. Alwan, Maximal ideal graph of commutative semirings, International Journal of Nonlinear Analysis and
Applications, 12(1) (2021) 913-926.
[6] A.H. Alwan, A graph associated to proper non-small subsemimodules of a semimodule, Int. J. Nonlinear Anal.
Appl. 12 (2021), no. 2, 499–509.
[7] A.H. Alwan, Maximal submodule graph of a module, J. Discrete Math. Sci. Crypto. 24 (2021), no. 7, 1941–1949.
[8] S.E. Atani and F.Saraei, On coatomic semimodules over commutative semirings, Cankaya Univ. J. Sci. Eng. 8
(2011), no. 2, 189–200.[9] J. Bosak, The graphs of semigroups, in theory of graphs and application, Academic Press, 1964.
[10] J.S. Golan, Semirings and their applications, Kluwer Academic Publishers, Dordrecht, 1999.
[11] Y. Katsov, T. Nam and N. Tuyen, On subtractive semisimple semirings, Algebra Colloq. 16 (2009), no. 3, 415–426.
[12] L.A. Mahdavi and Y. Talebi, Co-intersection graph of submodules of a module, J. Algebra Discrete Math. 21
(2016), no. 1, 128–143.
[13] Z.A. Nema and A.H. Alwan, Co-intersection graph of subsemimodules of a semimodule, J. Interdiscip. Math. to
appear.
[14] O. Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. 38, American Mathematical Society, Providence, RI, 1962.
[15] R. Wisbauer, Foundations of module and ring theory, Gordon and Breach Science Publishers, Philadelphia, 1991.
Volume 13, Issue 2
July 2022
Pages 2763-2770
  • Receive Date: 06 April 2022
  • Revise Date: 15 May 2022
  • Accept Date: 04 June 2022