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 comparatively less explored. In this paper, we investigate the zero-divisor graph associated with the finite semilocal ring \( R=\mathbb Z_{p^{n}q^{m}}, \) where \(p\) and \(q\) are distinct primes and \(n,m\ge2\). Using a valuation-theoretic approach, we establish a natural decomposition of the vertex set into valuation layers. This decomposition enables us to determine the connectivity, shortest-path structure, eccentricities of valuation layers, diameter, radius, centre, and peripheral vertices of the graph. The results demonstrate that the valuation-layer decomposition completely governs the structural, metric, and symmetry properties of the zero-divisor graph, providing a unified framework for the study of finite semilocal rings and extending several known results on zero-divisor graphs.
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 quo...
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
Let R be a finite commutative ring with identity, and let Γ′ᵣ(R) denote its reduced cozero-divisor graph. We organize Γ′ᵣ(R) through the poset Prin*(R) of nonzero proper principal ideals and prove that adjacency is exactly incomparability in this poset. Consequently, cliques correspond to antichains, independent sets c...
Dong-Ze Du· Theoretical and Natural Scie...· 0 citations
This study investigates the spectral properties of the total zero-divisor graph associated with the ring $\mathbb{Z}_{p^2q^2}$, where $p$ and $q$ are distinct primes. Using an explicit equitable partition of the graph, we obtain a complete description of the adjacency spectrum. We show that the spectrum consists of two...
Meet J. Mashru, P. Lalchandani· Journal of Algebra Combinato...· 0 citations
Consider a finite ring R, and let U( R ) denote the set of all its unit elements. The unit graph G_R of R is defined as follows: the vertex set V( G_R ) consists of all elements of R. Two distinct elements a,\ b∈ \ G_R are adjacent if and only if a + b∈ U( R ). Recent research has provided necessary and sufficient cond...
P. W. Prasetyo, Muhammad Ardiyansyah, Joe Repka· Baghdad Science Journal· 0 citations
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...
Mohd Shariq, Jitender Kumar· 0 citations
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