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On the co-intersection graph of subsemimodules of a semimodule | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 222، دوره 13، شماره 2، مهر 2022، صفحه 2763-2770 اصل مقاله (387.85 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.27521.3637 | ||
| نویسندگان | ||
| Ahmed H. Alwan؛ Zahraa A. Nema* | ||
| Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq | ||
| تاریخ دریافت: 17 فروردین 1401، تاریخ بازنگری: 25 اردیبهشت 1401، تاریخ پذیرش: 14 خرداد 1401 | ||
| چکیده | ||
| 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 $\Gamma(U)$. | ||
| کلیدواژه‌ها | ||
| Semimodule؛ Co-intersection graph؛ Connectivity؛ Domination number؛ Clique number | ||
| مراجع | ||
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[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. | ||
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