The intersection graph of graded submodules of a graded module

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.