Embeddings of the line graphs associated with the essential graphs of commutative rings

TL;DR

This paper studies the embeddings of line graphs derived from the essential graphs of finite commutative rings, classifying rings based on the planarity and surface genus of these graphs.

Contribution

It provides a complete classification of finite commutative rings whose essential graph line graphs are planar, outerplanar, or have low genus or crosscap number.

Findings

Classified rings with planar line graphs of essential graphs.

Identified rings with outerplanar and low-genus line graphs.

Characterized non-local rings with outerplanar zero-divisor graph line graphs.

Abstract

Let $A$ be a finite commutative ring with unity $1 \neq 0.$ An ideal of $A$ is said to be essential if it has a non-zero intersection with every non-zero ideal of $A.$ The essential graph of $A$ is a simple undirected graph whose vertex set consists of all non-zero zero-divisors of $A.$ Two different vertices $u$ and $v$ are connected by an edge precisely when the ideal formed by the annihilator of their product $uv$ is essential in $A.$ This paper examines the minimal embeddings of the line graph of the essential graph of $A$ into orientable surfaces as well as non-orientable surfaces. Our results include a complete classification of finite commutative rings for which the line graphs of their essential graphs is planar, outerplanar or have genus or crosscap number at most two. We also characterize all such non-local rings for which the line graph of their zero-divisor graph is outerplanar.