A specific intersection graph of a module applying semisimple property

Abstract

The exploration of graphs linked to algebraic structures has emerged as a dynamic field, bridging graph theory and module theory to reveal deeper insights into module properties. In this study, we introduce and analyze the semisimple intersection graph, denoted as \(GSS_R(M)\) associated with a right \(R \)-module \(M\). The vertices of \(GSS_R(M)\) are the nonzero submodules of \(M\), with two vertices \(N\) and \(K\) connected if their intersection forms a nonzero semisimple module. We examine fundamental properties of \(GSS_R(M)\) including its connectivity, diameter, and domination number, in relation to the module-theoretic characteristics of \(M\). Additionally, the girth of \(GSS_R(M)\) is determined.