Maximum Matching of Zero-Divisor Graph Over a Commutative Ring

The zero-divisor graph Γ(Z_n ) is constructed by taking the nonzero zero-divisor of a commutative ring Z_n as vertices, with the edges connecting two vertices if their product is zero in Z_n. In this paper, we investigate α'(Γ(Z_n )) and α(Γ(Z_n )), the matching number and independence number of the zero-divisor graph Γ(Z_n ), respectively, for several values of n, when n=kq, with integer k∈{2,3,5,7}, and when n=pq, where p and q are distinct prime numbers and p<q. We prove that in these cases, the graph Γ(Z_n ) is isomorphic to complete bipartite graphs, which allows for the exact determination of α'(Γ(Z_n )) and α(Γ(Z_n )). This study demonstrates the relationship between the algebraic structure of Z_n and the graph-theoretic properties of its zero-divisor graph.