On Zero-Divisor Graphs of Zn When n Is Square-Free

In this article, some properties of the zero-divisor graph Γ(Zn) are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph Γ(Zn) of ring Zn is a (2k−2)-partite graph when the prime decomposition of n contains k distinct square-free primes using the method of congruence relation. We present some examples, accompanied by graphic representations, to achieve the desired results. It is also obtained that the zero-divisor graph Γ(Zn) is Eulerian if n is a square-free odd integer. Since Zn is a semisimple ring when n is square-free, the results can be generalized to characterize semisimple rings and modules, as well as rings satisfying Artinian and Noetherian conditions through the properties of their zero-divisor graphs. We endeavored to show that Γ(R) is a partite graph with a certain condition on n and also that Γ(R) is a complete graph when n=p2 for a prime p as part of a corollary. To prove these results, we employed the assistance of several theoretic congruence relations that grabbed our attention, making the investigation more interesting.