Geodesic
Some results on the co-intersection graph of submodules of a module
Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 1, pp. 15-24.

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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.
DOI : 10.14712/1213-7243.2015.230
Classification : 05C15, 05C25, 05C69, 16D10
Keywords: co-intersection graph; core; clique number; planarity 
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Mahdavi, Lotf Ali; Talebi, Yahya. Some results on the co-intersection graph of submodules of a module. Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 1, pp. 15-24. doi : 10.14712/1213-7243.2015.230. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.230/

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