Saeid Bagheri - Department of Mathematics, Malayer University, P.O.Box: ۶۵۷۱۹-۹۵۸۶۳, Malayer, Iran.
mahtab Koohi Kerahroodi - Department of Mathematics, Malayer University, P.O.Box: ۶۵۷۱۹-۹۵۸۶۳, Malayer, Iran.

In this paper,for a distributive lattice \mathcal L, we study and compare some lattice theoretic features of \mathcal L and topological properties of the Stone spaces {\rm Spec}(\mathcal L) and {\rm Max}(\mathcal L) with the corresponding graph theoretical aspects of the zero-divisor graph \Gamma(\mathcal L).Among other things,we show that the Goldie dimension of \mathcal L is equal to the cellularity of the topological space {\rm Spec}(\mathcal L) which is also equal to the clique number of the zero-divisor graph \Gamma(\mathcal L). Moreover, the domination number of \Gamma(\mathcal L) will be compared with the density and the weight of the topological space {\rm Spec}(\mathcal L). For a ۰-distributive lattice \mathcal L, we investigate the compressed subgraph \Gamma_E(\mathcal L) of the zero-divisor graph \Gamma(\mathcal L) and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of \mathcal L.

Distributive lattice, Goldie dimension, compressed zero-divisor graph, domination number