L(2, 1)-labeling of some zero-divisor graphs associated with commutative rings

TL;DR

This paper studies the $L(2,1)$-labeling of zero-divisor graphs of finite commutative rings, introducing the partite truncation operation to relate complex graphs to simpler Boolean ring graphs.

Contribution

It introduces the concept of partite truncation for zero-divisor graphs and establishes relationships between their $ ext{lambda}$-numbers, simplifying the analysis of these graphs.

Findings

Partite truncation reduces the size of zero-divisor graphs.

$ ext{lambda}$-numbers are related between original and truncated graphs.

Zero-divisor graphs of reduced rings can be contracted to Boolean rings.

Abstract

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The $\lambda$-number of $\mathcal{G}$, denoted by $\lambda(\mathcal{G})$, is the smallest positive integer $\ell$ such that $\mathcal{G}$ has an $L(2,1)$-labeling with all labels as members of the set $\{0,1, \dots,\ell\}$. The zero-divisor graph of a finite commutative ring $R$ with unity, denoted by $\Gamma(R)$, is the simple graph whose vertices are all zero divisors of $R$ in which two vertices $u$ and $v$ are adjacent if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labeling of some zero-divisor graphs. We study the partite truncation, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between $\lambda$-numbers of the graph and its partite truncated one. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.