‎ In this paper we continue our study of perpendicular graph of modules‎,‎ that was introduced
in\cite {Hokkaido}‎.‎ Let $ R $ be a ring and $ M $ be an $ R $-module‎.‎ Two modules $ A …

Let $ R $ be a ring and $ M $ be an $ R $-module. Two modules $ A $ and $ B $ are called
orthogonal, written $ A\perp B $, if they do not have non-zero isomorphic submodules. We …

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-
module M, denoted by G (M), is defined to be the undirected simple graph whose vertices …

A non-parallel submodules graph of M, denoted by G∦(M), is an undirected simple graph
whose vertices are in one-to-one correspondence with all non-zero proper submodules of M …

Let R be a ring with identity and M be a unitary left R-module. The complement of the
intersection graph of submodules of M, denoted by Γ (M), is defined to be a graph whose …

Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The
intersection graph of proper R-submodules of V is an undirected graph without loops and …

Let M be a unitary left R-module, where R is a (not necessarily commutative) ring with
identity. The small intersection graph of nontrivial submodules of M, denoted by Γ (M), is an …

Let R be a commutative ring with identity and M be an unitary R-module. The intersection
graph of an R-module M, denoted by Γ (M), is a simple graph whose vertices are all non …

Abstract 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 …

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 …