**Journal of Prime Research in Mathematics**

Vol. 1 (2017), Issue 1, pp. 16 – 29

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

**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

Copyright © 2017 Lotf Ali Mahdavi, Yahya Talebi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2017.

### 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.