Diameter of the zero divisor graph of semiring of matrices over Boolean semiring

L John,Harita Vijay

doi:10.12988/imf.2014.47131

Abstract

Let S be a semiring and let Z(S) be its set of nonzero zero divisors. We denote the zero divisor graph of S by ( S) whose vertex set is Z(S) and there is an edge between the vertices x and y (x6 y) in ( S) if and only if either xy = 0 or yx = 0. In this paper we study the zero divisor graph of the semiring of matrices Mn(B), (n > 1) over the Boolean semiring B. We investigate the properties of the right zero divisors and the left zero divisors of Mn(B) and then use these results to prove that the diameter of ( Mn(B)) is 3.

Highlights

The concept of a zero divisor graph was first introduced by Beck [5] in the study of commutative rings and later redefined by Anderson and Livingston [3]
Redmond [11] extended this concept to the non commutative case
We consider the graph Γ(S) whose vertices are the elements of Z(S)∗ and whose edges are those pairs of distinct non zero zero divisors x, y such that either xy = 0 or yx = 0

Summary

Introduction

The concept of a zero divisor graph was first introduced by Beck [5] in the study of commutative rings and later redefined by Anderson and Livingston [3]. We consider the graph Γ(S) whose vertices are the elements of Z(S)∗ and whose edges are those pairs of distinct non zero zero divisors x, y such that either xy = 0 or yx = 0. Theorem 2.3 Any matrix of Mn(B) having atleast one zero row is a right zero divisor. Any matrix of Mn(B) having atleast one zero row is a right zero divisor.

Results

Conclusion