A Unified CRT Framework for Zero-Divisor Graphs over General Composite Integers: Structural Reduction and Invariant Transfer

Abstract

The results on the zero-divisor graphs of modular rings are often given only for prime powers, semiprimes or products of two prime powers, and there is no single computational and theoretical model for an arbitrary composite integer. This paper builds one of these in the case of Γ(ℤₙ), where n=Πpᵢᵃⁱ is any product of distinct prime numbers and arbitrary positive exponents. Each nonzero zero-divisor is encoded by its signature, which is a set of signatures α=(min(vₚᵢ(x),aᵢ))ᵢ. Each nonzero zero-divisor is coded by its signature α=(min(vₚᵢ(x),aᵢ))ᵢ. Every class size has a multiplicative formula according to the Chinese remainder theorem and the zero product adjacency is the coordinate wise condition αᵢ+βᵢ≥aᵢ. This results in a reconstruction of Γ(ℤₙ) to be exactly a weighted quotient of a clique or independent set with τ(n)-2 classes. Adjacency and Laplacian spectra, pairwise distances and the Wiener index, domination, metric dimension bounds and demand-preserving coloring are then considered in general transfer results. The construction is not affected by the number or multiplicities of prime factors. No adjacency or spectral mismatch have been found after exhaustive deterministic verification of all 138 composite integers 4≤n≤180. The mean structural reduction of the quotient was 85.6% and the reductions of nontrivial cases (n≥6) varied from 33.3% to 97.8%. Ten moduli are shown which are representative examples of the effect on the transferred invariants of different factorization patterns. The main result is a family-independent reproducible CRT engine rather than a number of individual, family-specific derivations.