## Lotf Ali Mahdavi, Yahya Talebi

*Some results on the co-intersection graph of submodules of a module*

Comment.Math.Univ.Carolin. 59,1 (2018) 15-24.**Abstract:**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.

**Keywords:** co-intersection graph; core; clique number; planarity

**DOI:** DOI 10.14712/1213-7243.2015.230

**AMS Subject Classification:** 05C15 05C25 05C69 16D10

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