On the zero-divisor graph of a ring

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Gamma(R)=Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy=0. In this article, we study Gamma(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} not equal Nil(R) subset of zR for all z is an element of Z(R)\Nil(R).