THE STRUCTURES OF MATRICES AND INDICES OF ZERO DIVISOR GRAPHS OF 3,4-RADICAL ZERO COMPLETELY PRIMARY FINITE RINGS

Abstract

A zero divisor graph of the ring R is a graph whose vertices are entirely from the set of zero divisors of the ring and two vertices of the graph are adjacent if and only if their product is zero. The study of zero divisor graphs is important for it provides a better way of relating graph geometry to matrix conformations and formulation of encryption algorithms in coding therefore fundamental in interpretation of patterns, maps and networks in computer programs and modelling. Reasonable research has been done concerning zero divisor graphs of commutative rings with identity 1 ̸ = 0, however the generalization of the structures of the matrices of zero divisor graphs is still not extensive in the existing literature. Much of the recent works on zero divisor graphs of finite commutative rings have been restricted to the algebraic properties of the graphs such as colouring, girth, spectral radii and classification in terms of their completeness up to isomorphism. This has left the characterization of finite commutative rings via the structures of the matrices and indices of their graphs fairly untouched. In particular, matrices and indices of the zero divisor graph Γ(R) of finite commutative rings of 3-radical zero and 4-radical zero have not been characterized. This research has determined and investigated the properties of the matrices and indices of Γ(R) of finite rings R with unique maximal ideal J(R) such that J(R)3 = (0) and J(R)2 ̸ = (0); J(R)4 = (0) and J(R)3 ̸ = (0). It has also established the singularity and the relationship that exist between the eigenvalue multiplicities in the spectrum to the nullity of the graphs. We have validated the construction of these classes of rings using idealization procedure and the zero divisor graphs drawn from the isolated zero divisors using the Tikz Software. The matrices have been formulated from the graphs using standard definitions and the Mathematica software applied in investigating some of their algebraic properties. The results of this study can find an application to networking such as Google PageRank algorithms, developing more improved codes for better graph interpretation in operation systems. It will also advance the ring classification problem by revealing the interplay between ring theory, graph theory and linear algebra therefore contributing fundamentally to the literature of advanced algebra