Structural Properties of Zero-Divisor Graphs of Multilocal Finite Rings

Abstract

Let N =Yti=1pnii , t ≥ 2, where the primes p1, . . . , pt are distinct and ni ≥ 1, and let R = Z/NZ. 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 Γ(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 diam Γ(R) = 3 whenever t ≥ 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 Aut(Γ(R)); no assertion is made that this subgroup is always the full automorphism group. A complete calculation for Z/12Z illustrates the layer sizes, adjacency pattern, diameter, clique number, and canonical automorphism subgroup.