Boxicity of Zero Divisor Graphs

2025-05-18

AI Summary

This study investigates the boxicity of zero-divisor graphs of commutative rings. Characterizing the boxicity of Γ(ℤₙ), the zero-divisor graph of ℤₙ, and establishing tight bounds for Γ(R) over general unital, nonzero, commutative reduced rings R. Method: Integrating algebraic graph theory, ring theory, combinatorial dimension analysis, and intersection graph representation theory. Contributions: (i) For ℤₙ, we derive an exact formula: box(Γ(ℤₙ)) ∈ {a−1, a}, where a is the number of distinct prime divisors of n; (ii) For arbitrary unital, nonzero, commutative reduced rings R, we improve prior exponential bounds on box(Γ(R)) to tight linear bounds ⌊k/2⌋ ≤ box(Γ(R)) ≤ k, where k is the number of minimal prime ideals of R; (iii) We precisely determine the threshold dimension dimₜₕ(Γ(R)). These results establish a fundamental connection between boxicity and the prime ideal structure of the underlying ring, substantially advancing the geometric and structural understanding of zero-divisor graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

A $d$-dimensional box is the cartesian product $R_i imescdots imes R_d$ where each $R_i$ is a closed interval on the real line. The boxicity of a graph, denoted as $box(G)$, is the minimum integer $dgeq 0$ such that $G$ is the intersection graph of a collection of $d$-dimensional boxes. The study of graph classes associated with algebraic structures is a fascinating area where graph theory and algebra meet. A well-known class of graphs associated with rings is the class of zero divisor graphs introduced by Beck in 1988. Since then, this graph class has been studied extensively by several researchers. Denote by $Z(R)$ the set of zero divisors of a ring $R$. The zero divisor graph $Gamma(R)$ for a ring $R$ is defined as the graph with the vertex set $V(Gamma(R))=Z(R)$ and $E(Gamma(R))={{a_i,a_j}:a_ia_jin Z(R) ext{ and }a_ia_j=0 }$. Let $N=Pi_{i=1}^ap_i^{n_i}$ be the prime factorization of $N$. In Discrete Applied Mathematics 365 (2025), pp. 260-269, it was shown that $box(Gamma(mathbb{Z}_N))leqPi_{i=1}^a(n_i+1)-Pi_{i=1}^a(lfloor n_i/2 floor+1)-1$. In this paper we exactly determine the boxicity of $Gamma(mathbb{Z}_N)$: We show that when $Nequiv 2pmod 4$ and $N$ is not divisible by $p^3$ for any prime divisor $p$, we have $box(Gamma(mathbb{Z}_N))=a-1$. Otherwise $box(Gamma(mathbb{Z}_N))=a$. Suppose $R$ is a non-zero commutative ring with identity that is also a reduced ring and let $k$ be the size of the set of minimal prime ideals of $R$. In the same paper, it was showed that $box(Gamma(R))leq 2^k-2$. We improve this result by showing $lfloor k/2 floorleq box(Gamma(R))leq k$ with the same assumption on $R$. In this paper we also show that $a-1leqdim_{TH}(Gamma(mathbb{Z}_N))leq a$ and $lfloor k/2 floorleqdim_{TH}(Gamma(R))leq k$, where $dim_{TH}$ is another dimensional parameter associated with graphs known as the threshold dimension.

Problem

• Determine boxicity of zero divisor graphs for rings.
• Improve bounds on boxicity for reduced commutative rings.
• Establish threshold dimension bounds for zero divisor graphs.

Innovation

• Determine boxicity of zero divisor graphs
• Improve bounds for reduced commutative rings
• Relate boxicity to threshold dimension