# Article

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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.
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