Adjacency matrices of Ideal based-zero divisor graphs

In this article, we investigate the adjacency matrix and the eigenvalues of the Ideal-based-zerodivisor graph for a given ideal of the finite commutative ring . Additionally, we define the notion of projection graph of the Idealbased-zerodivisor graph as a graph with vertices for all and edge connecting with if every element in is connected with . We consider the ring for some special cases of . If where (where ) and is non-prime ideal of , then we determined the determinant of the adjacency matrices of the Idealbased-zerodivisor graph of and satisfies We showed that in the Ideal-based-zerodivisor graph , where , is not prime ideal of and not zero ideal of , then and whenever then .Finally, we investigate some basic properties of the edge ideal of the graph . We calculated the graph's depth, girth, diameter, Betti number, regularity, and projective dimension.