Geodesic
Co-intersection graph of submodules of~a~module
Algebra and discrete mathematics, Tome 21 (2016) no. 1, pp. 128-143.

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Let $M$ be a unitary left $R$-module where $R$ is a ring with identity. The co-intersection graph of proper submodules of $M$, denoted by $\Omega(M)$, is an undirected simple graph whose the vertex set $V(\Omega)$ is a set of all non-trivial submodules of $M$ and there is an edge between two distinct vertices $N$ and $K$ if and only if $N+K\neq M$. In this paper we investigate connections between the graph-theoretic properties of $\Omega(M)$ and some algebraic properties of modules . We characterize all of modules for which the co-intersection graph of submodules is connected. Also the diameter and the girth of $\Omega(M)$ are determined. We study the clique number and the chromatic number of $\Omega(M)$.
Keywords: co-intersection graph, clique number, chromatic number. 
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Lotf Ali Mahdavi; Yahya Talebi. Co-intersection graph of submodules of~a~module. Algebra and discrete mathematics, Tome 21 (2016) no. 1, pp. 128-143. http://geodesic.mathdoc.fr/item/ADM_2016_21_1_a8/

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