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Keywords:
co-intersection graph; core; clique number; planarity
Summary:
Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega(M)$, is an undirected simple graph whose vertex set $V(\Omega)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\neq M$. We study the connectivity, the core and the clique number of $\Omega(M)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega(M)$ is infinite and $\Omega(M)$ is a planar graph. Moreover, we give several examples for which $n$ the graph $\Omega(\mathbb{Z}_{n})$ is connected, bipartite and planar.
References:
[1] Akbari S., Nikandish R., Nikmehr M. J.: Some results on the intersection graphs of ideals of rings. J. Algebra Appl. 12 (2013), no. 4, 1250200, 13 pp. DOI 10.1142/S0219498812502003 | Zbl 1264.05056
[2] Akbari S., Tavallaee A., Khalashi Ghezelahmad S.: Intersection graph of submodule of a module. J. Algebra Appl. 11 (2012), no. 1, 1250019, 8 pp. DOI 10.1142/S0219498811005452
[3] Akbari S., Tavallaee A., Khalashi Ghezelahmad S.: On the complement of the intersection graph of submodules of a module. J. Algebra Appl. 14 (2015), 1550116, 11 pp. DOI 10.1142/S0219498815501169
[4] Akbari S., Tavallaee A., Khalashi Ghezelahmad S.: Some results on the intersection graph of submodules of a module. Math. Slovaca 67 (2017), no. 2, 297–304. DOI 10.1515/ms-2016-0267
[5] Anderson F. W., Fuller K. R.: Rings and Categories of Modules. Springer, New York, 1992. Zbl 0765.16001
[6] Bondy J. A., Murty U. S. R.: Graph Theory. Graduate Texts in Mathematics, 244, Springer, New York, 2008. Zbl 1134.05001
[7] Bosak J.: The graphs of semigroups. in Theory of Graphs and Its Application, Academic Press, New York, 1964, pp. 119–125. Zbl 0161.20901
[8] Chakrabarty I., Gosh S., Mukherjee T. K., Sen M. K.: Intersection graphs of ideals of rings. Discrete Math. 309 (2009), 5381–5392. DOI 10.1016/j.disc.2008.11.034
[9] Clark J., Lomp C., Vanaja N., Wisbauer R.: Lifting Modules. Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkhäuser, Basel, 2006.
[10] Cohn P. M.: Introduction to Ring Theory. Springer Undergraduate Mathematics Series, Springer, London, 2000.
[11] Csakany B., Pollak G.: The graph of subgroups of a finite group. Czechoslovak Math. J. 19 (1969), 241–247.
[12] Jafari S., Jafari Rad N.: Planarity of intersection graphs of ideals of rings. Int. Electron. J. Algebra 8 (2010), 161–166.
[13] Kayacan S., Yaraneri E.: Finite groups whose intersection graphs are planar. J. Korean Math. Soc. 52 (2015), no. 1, 81–96. DOI 10.4134/JKMS.2015.52.1.081
[14] Laison J. D., Qing Y.: Subspace intersection graphs. Discrete Math. 310 (2010), 3413–3416. DOI 10.1016/j.disc.2010.06.042
[15] Mahdavi L. A., Talebi Y.: Co-intersection graph of submodules of a module. Algebra Discrete Math. 21 (2016), no. 1, 128–143.
[16] Northcott D. G.: Lessons on Rings, Modules and Multiplicaties. Cambridge University Press, Cambridge, 1968.
[17] Shen R.: Intersection graphs of subgroups of finite groups. Czechoslovak Math. J. 60(4) (2010), 945–950. DOI 10.1007/s10587-010-0085-4
[18] Talebi A. A.: A kind of intersection graphs on ideals of rings. J. Mathematics Statistics 8 (2012), no. 1, 82–84. DOI 10.3844/jmssp.2012.82.84
[19] Yaraneri E.: Intersection graph of a module. J. Algebra Appl. 12 (2013), no. 5, 1250218, 30 pp. DOI 10.1142/S0219498812502180
[20] Zelinka B.: Intersection graphs of finite abelian groups. Czechoslovak Math. J. 25(2) (1975), 171–174.
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