# Sum-Essential Graphs of Modules

## TL;DR

This paper introduces and studies the sum-essential graph of modules, exploring how module properties influence the graph's structure and properties, and analyzing subgraphs related to essential submodules.

## Contribution

It defines the sum-essential graph of modules and investigates its properties and the relationship between module characteristics and graph structure.

## Key findings

- Characterization of the sum-essential graph's properties
- Analysis of the subgraph induced by non-essential submodules
- Insights into the relationship between module properties and graph features

## Abstract

The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the graph $\mathcal{S}_R(M)$ and its subgraph $\mathcal{P}_R(M)$ induced by vertices which are not essential as submodules of $M$ are investigated. The interplay between module properties of $M$ and properties of those graphs is studied.