On the Adjacency and Incidence matrices of the Zero divisor graphs of A class of the Square Radical Zero finite commutative Rings

Abstract

The characterization of the zero divisor graphs of commutative finite rings has attracted a lot of research for quite sometime, however not so much has been done concerning their Adjacency and Incidence matrices. In computer modelling, matrices are better under stood than graphs and therefore the representation of graphs by matrices is worth studying. Given an arbitrary square matrix Mn, it is not known in general the classes of finite rings for which it represents the zero divisors. In spite of that , there exist some expositions on the adjacency and incidence matrices of the zero divisor graphs of commutative finite rings( reference can be made to [3, 7, 9] among others). Let R be a square radical zero finite commu tative ring. This paper characterizes the adjacency and incidence matrices of the zero divisor graphs Γ(R) of such rings of characteristic p and p 2 . We have drawn a zero divisor graph of the classes of rings studied using TikZ software and studied its properties, then generalized the properties of such graphs in the same category. By using the standard algebraic concepts, we have formulated the Adjacency and Incidence matrices of the graphs. A cursory study of these matrices has been undertaken on some of their algebraic properties. We also extend our findings on the adjacency matrices [Aij ] as transformations. The results provide an extension on the classification problem of rings in which the product of any two zero divisors is zero.