Journal of Prime Research in Mathematics

# Properties of co-intersection graph of submodules of a module

Lotf Ali Mahdavi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.
Yahya Talebi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

$$^{1}$$Corresponding Author: l.a.mahdavi154@gmail.com

### Abstract

Let $$R$$ be a ring with identity and $$M$$ be a unitary left Rmodule. The co-intersection graph of proper submodules of $$M, Ω(M)$$ is an undirected simple graph whose vertices are non-trivial submodule of $$M$$ in which two vertices N and K are joined by an edge, if and only if $$N + K \neq M$$. In this paper, we study several properties of $$Ω(M)$$. We prove that, if $$Ω(M)$$ is a path, then $$Ω(M) \cong P_2$$or $$Ω(M) \cong P_3$$. We show that, if $$Ω(M)$$ is a forest, then each component of $$Ω(M)$$ is complete or star graph. We determine the conditions under which $$Ω(M)$$ is weakly perfect. Moreover, we introduce the universal vertices and the dominating sets of $$Ω(M)$$ and their relationship with the non-trivial small submodules of $$M$$.

#### Keywords:

Co-intersection graph, forest, weakly perfect graph, universal vertex, dominating set.