Intersection Graph of a Module

TL;DR

This paper explores the intersection graph of submodules of a module over a ring, analyzing its properties like connectivity, domination, coloring, and planarity to connect graph theory with module algebra.

Contribution

It introduces the study of intersection graphs of modules, establishing relationships between graph properties and algebraic structure, including results on domination number, chromatic number, and planarity.

Findings

Determined the domination number of the intersection graph.

Calculated the chromatic number in specific cases.

Characterized modules whose intersection graphs are planar.

Abstract

Let $V$ be a left $R$-module where $R$ is a (not necessarily commutative) ring with unit. The intersection graph $\cG(V)$ of proper $R$-submodules of $V$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper $R$-submodules of $V,$ and there is an edge between two distinct vertices $U$ and $W$ if and only if $U\cap W\neq 0.$ We study these graphs to relate the combinatorial properties of $\cG(V)$ to the algebraic properties of the $R$-module $V.$ We study connectedness, domination, finiteness, coloring, and planarity for $\cG (V).$ For instance, we find the domination number of $\cG (V).$ We also find the chromatic number of $\cG(V)$ in some cases. Furthermore, we study cycles in $\cG(V),$ and complete subgraphs in $\cG (V)$ determining the structure of $V$ for which $\cG(V)$ is planar.