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# δ-Small Intersection Graphs of Modules

## Abstract

Let R be a commutative ring with unit and M be a unitary left R-module. The δ-small intersection graph of non-trivial submodules of , denoted by , is an undirected simple graph whose vertices are the non-trivial submodules of , and two vertices are adjacent if and only if their intersection is a -small submodule of . In this article, we study the interplay between the algebraic properties of , and the graph properties of such as connectivity, completeness and planarity. Moreover, we determine the exact values of the diameter and girth of , as well as give a formula to compute the clique and domination numbers of

## References

1. Akbari S, Tavallaee HA, Khalashi Ghezelahmad S. Intersection graph of submodules of a module. J Algebra Appl 2012;11(1):1250019. https://doi.org/10.1142/S0219498812500198
2. Alwan AH. Maximal ideal graph of commutative semirings. Int J Nonlinear Anal Appl 2021;12(1):913e26. https://doi.org/10.22075/ijnaa.2020.22074.2049
3. Alwan AH. A graph associated to proper non-small subsemimodules of a semimodule. Int J Nonlinear Anal Appl 2021;12(2):499e509. https://doi.org/10.22075/ijnaa.2021.23532.2326
4. Alwan AH. Maximal submodule graph of a module. J Discrete Math Sci Cryptogr 2021;24(7):1941e9. https://doi.org/10.1080/09720529.2021.1962766
5. Alwan AH, Nema ZA. On the co-intersection graph of subsemimodules of a semimodule. Int J Nonlinear Anal Appl 2022;13(2):2763e70. https://doi.org/10.22075/ijnaa.2022.5344
6. Alwan AH. Small intersection graph of subsemimodules of a semimodule. Commun Combin Cryptogr Comput Sci 2022;1:15e22.
7. Beck I. Coloring of commutative rings. J Algebra 1988;116:208e26.
8. Bondy JA, Murty USR. Graph theory, Graduate texts in mathematics 244. New York: Springer; 2008.
9. Bosak J. The graphs of semigroups. In: Theory of graphs and its applications. New York: Academic Press; 1964. p. 119e25.
10. Chakrabarty I, Ghosh S, Mukherjee TK, Sen MK. Intersection graphs of ideals of rings. Discrete Math 2009;309:5381e92.
11. Haynes TW, Hedetniemi ST, Slater PJ, editors. Fundamentals of domination in graphs. New York, NY: Marcel Dekker, Inc.; 1998.
12. Inankil H, Halicioglu S, Harmanci A. A generalization of supplemented modules. Algebra Discrete Math 2011;11(1):59e74.
13. Mahdavi LA, Talebi Y. On the small intersection graph of submodules of a module. Algebraic Structures and Their Applications 2021;8(1):117e30.
14. Byukasik BN, Lomp C. When d-semiperfect rings are semiperfect. Turk J Math 2010;34:317e24.
15. Turkmen BN, Turkmen E. dss-supplemented modules and rings. An St Univ Ovidius Constanta 2020;28(3):193e216.
16. Wisbauer R. Foundations of module and ring theory. Gordon & Breach; 1991.
17. Zhou Y. Generalizations of perfect, semiperfect, and semiregular rings. Algebra Colloq 2000;7(3):305e18.

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