The embedding of line graphs associated to the annihilator graph of commutative rings

TL;DR

This paper investigates the topological properties of the line graph of the annihilator graph of finite commutative rings, including genus, crosscap, and book thickness, providing complete classifications for specific cases.

Contribution

It characterizes all finite rings where the line graph of the annihilator graph has genus or crosscap at most two and classifies rings with line graph book thickness at most four.

Findings

Complete classification of rings with line graph genus or crosscap ≤ 2

Determination of the inner vertex number of the line graph

Classification of rings with line graph book thickness ≤ 4

Abstract

The annihilator graph $AG(R)$ of the commutative ring $R$ is an undirected graph with vertex set as the set of all non-zero zero divisors of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy) \neq ann(x) \cup ann(y)$. In this paper, we study the embedding of the line graph of $AG(R)$ into orientable or non-orientable surfaces. We completely characterize all the finite commutative rings such that the line graph of $AG(R)$ is of genus or crosscap at most two. We also obtain the inner vertex number of $L(AG(R))$. Finally, we classify all the finite rings such that the book thickness of $L(AG(R))$ is at most four.