The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains. Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint. In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).
The weakly zero-divisor graph $W\Gamma(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$, and two distinct vertices $x$, $y$ are adjacent if and only if there exists $w\in {\rm ann}(x)$ and $ z\in {\rm ann}(y)$ such that $wz =0$. In this paper, first we prove t...
Let R be a finite commutative ring with identity and let I be a proper ideal of R. The ideal-based zero-divisor graph \Gamma_I(R) has vertices outside I that annihilate some element outside I modulo I, with x adjacent to y whenever xy\in I. This paper studies the adjacency energy of \Gamma_I(R). General trace bounds ar...
Rosalio G. Artes, R. Malalay, M. Mbah et al.· International Journal of Mat...· 0 citations
The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain co...
Presley Kiplagat· Earthline Journal of Mathema...· 0 citations
The boxicity of a graph $G$, denoted by $box(G)$, is the minimum integer $d\geq 0$ such that $G$ is the intersection graph of axis-parallel boxes in $\mathbb{R}^d$. The class of zero divisor graphs introduced by Beck (1988) is a popular class of graphs and has been studied extensively by several researchers. Suppose $Z...
Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $\Gamma_E(R)$ and the associate class graph $\Gamma_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. Thi...
UDC 519.17; 512.55
Let \(R\) be a commutative ring with identity \(1 \neq 0 ,\) and let \(Z(R)\) be the set of zero divisors of \(R.\) The zero divisor graph \(\Gamma(R)\) is defined as a simple graph with the set of vertices formed by nonzero divisors \(Z^{+}(R) = Z(R) \setminus \{0\}\) of the zero element of $R$ such...
B. Rather, P. Ali, Muhammed Imran et al.· Ukrains'kyi Matematychnyi Zh...· 0 citations
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