Structural Properties of Zero-Divisor Graphs of Multilocal Finite Rings
Presley Kiplagat *
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Lao Hussein Mude
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Zachary Kaunda Kayiita
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Let
\[
N=\prod_{i=1}^t p_i^{n_i}, \quad t \geq 2
\]
where the primes \(p_1, \ldots, p_t\) are distinct and \(n_i \geq 1\), and let \(R=\mathbb{Z} / N \mathbb{Z}\). The nonzero zero-divisors of \(R\) are partitioned by their truncated prime-adic valuation vectors. This paper develops the resulting valuation-layer description of the zero-divisor graph \(\Gamma(R)\). A complete formula is obtained for the size of every valuation layer, including layers containing elements that vanish in one or more Chinese-remainder components. Adjacency is shown to depend only on coordinatewise sums of valuation vectors, and the graph is therefore a blow-up of a finite weighted layer graph. This representation yields a direct proof that \(\operatorname{diam} \Gamma(R)=3\) whenever \(t \geq 2\), together with a criterion distinguishing vertex pairs at distances one, two, and three. The clique number is expressed exactly as a weighted clique optimization problem on the layer graph. In addition, independent permutations within each valuation layer are shown to form a canonical direct-product subgroup of \(\operatorname{Aut}(\Gamma(R))\); no assertion is made that this subgroup is always the full automorphism group. A complete calculation for \(\mathbb{Z} / 12 \mathbb{Z}\) illustrates the layer sizes, adjacency pattern, diameter, clique number, and canonical automorphism subgroup.
Keywords: Zero-divisor graph, finite semilocal ring, valuation layer, clique number, automorphism group