The intersection graph of graded submodules of a graded module

Tariq Alraqad

doi:10.1515/math-2022-0005

Abstract

Abstract In this article, we introduce and study the intersection graph of graded submodules of a graded module. Let M M be a left G G -graded R R -module. We define the intersection graph of G G -graded R R -submodules of M M , denoted by Γ ( G , R , M ) \Gamma \left(G,R,M) , to be a simple undirected graph whose set of vertices consists of all nontrivial G G -graded R R -submodules of M M , where two vertices are adjacent if their intersection is nonzero. We study properties of these graphs, such as connectivity, diameter, and girth. We also investigate the intersection graph of graded submodules for certain types of gradings such as faithful and strong gradings.

Highlights

Studies of graphs associated with algebraic structures developed remarkably in recent years
Zero-divisors graph, total graphs, annihilating-ideal graph, and unit graphs are very interesting examples of graphs associated with rings, see [1–4]
In 2009, Chakrabarty et al [7] introduced and studied the intersection graph of ideals of a ring R, which is an undirected simple graph, denoted by G(R), whose vertices are the nontrivial left ideals of R and two vertices I and J are adjacent if their intersection is nonzero

Summary

Introduction

Studies of graphs associated with algebraic structures developed remarkably in recent years. For studies on graphs associated with graded rings and graded modules, in particular, see [5,6]. In 2009, Chakrabarty et al [7] introduced and studied the intersection graph of ideals of a ring R, which is an undirected simple graph, denoted by G(R), whose vertices are the nontrivial left ideals of R and two vertices I and J are adjacent if their intersection is nonzero. The intersection graph of submodules of M, denoted by G(M), is an undirected simple graph defined on S∗(M), where two non-trivial submodules are adjacent if they have a nonzero intersection Since they were introduced, intersection graphs of ideal and submodules have attracted many researchers to study their graph-theoretic properties and investigate their structures (see [9–17]). The intersection graph of G-graded submodules of M, denoted by Γ(G, R, M), is defined to be an undirected simple graph whose set of vertices is hS∗(M) and two vertices N and K are adjacent if N ∩ K ≠ {0}. The graph G(M) is disconnected if and only if it is null graph with at least two vertices

86 Tariq Alraqad

Intersection graph of types of gradings