Computing the size of zero divisor graphs

Abstract

A recent subject of study linking commutative ring theory with graph theory has been the concept of the zero divisor graph of a commutative ring. The zero divisor graph of a commutative ring exhibits a remarkable amount of graphical structure. Let G(R) be the zero divisor graph introduced by Beck [9], whose vertices are the elements of a ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Let Г(R) be the zero divisor graph introduced by Anderson, Livingston [5] whose vertices are the non-zero zero divisors of the ring R such that two distinct vertices x, y are adjacent provided that xy = 0. Here, the authors investigate the size of the graphs G(ℤ n ), Г(ℤ n ).