Zero Divisor Graphs Γ(ℤₙ) for Complex Systems Modelling: Structural Characterization, Graph Invariants and a CRT-Based Framework

Abstract

Zero divisor graphs provide a powerful combinatorial framework for studying the algebraic structure of commutative rings by representing
annihilation relations among elements. In this research, investigate the zero-divisor graph Γ(Zₙ) of the finite commutative ring Zₙ. Using the
prime factorisation of n, to establish new structural characterisations for adjacency, connectivity, clique number, chromatic number, and
dominating sets. Present new results describing the diameter of Γ(Zₙ) for arbitrary composite integers n and derive necessary and sufficient
conditions under which the graph is complete, bipartite, or multipartite. A formulation based on the Chinese Remainder Theorem (CRT) is
introduced to relate ring decomposition directly to key graph invariants. Illustrative examples and graphical representations are included to
demonstrate the behaviour of Γ(Zₙ) across different classes of integers. The results contribute new insights into the algebraic–combinatorial
nature of zero divisors in modular rings and extend existing classifications while providing a unified structural perspective.