Analysis of Adjacency, Laplacian and Distance Matrices of Zero Divisor Graphs of 4-Radical Zero Completely Primary Finite Rings

Abstract

This study is an extension of our study on matrices of zero divisor graphs of classes of 3-radical zero completely primary finite rings. It focusses on Matrices of a class of finite rings R whose subset of the zero divisors Z(R) satisfies the condition (Z(R))⁴ = (0) and (Z(R))³ = (0) for all characteristics of R that is; p, p², p³ and p⁴. We have formulated the zero divisor graphs Γ(R) of R and associated them with three classes of matrices, namely, the Adjacency matrix [A], the Laplacian matrix [L] and the Distance matrix [d_ij]. The study has further characterized the properties of the graphs Γ(R) and the matrices mentioned.This study is an extension of our study on matrices of zero divisor graphs of classes of 3-radical zero completely primary finite rings. It focusses on Matrices of a class of finite rings R whose subset of the zero divisors Z(R) satisfies the condition (Z(R))⁴ = (0) and (Z(R))³ = (0) for all characteristics of R that is; p, p², p³ and p⁴. We have formulated the zero divisor graphs Γ(R) of R and associated them with three classes of matrices, namely, the Adjacency matrix [A], the Laplacian matrix [L] and the Distance matrix [d_ij]. The study has further characterized the properties of the graphs Γ(R) and the matrices mentioned.