Distributive lattices and some related topologies in comparison with zero-divisor graphs

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 $0$-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 $mathcal L$.