The small intersection graph of filters of a bounded distributive lattice

Let L be a lattice with 1 and 0. The small intersection graph of filters of L, denoted by Γ(L), is defined to be a graph whose vertices are in one to one correspondence with all non-trivial filters of L and two distinct vertices are adjacent if and only if the intersection of correspond-ing filters of L is a small filter of L. In this paper, the basic properties and possible structures of the graph Γ(L) are investigated. Moreover, the complemented property, the domination number and the planar property of Γ(L) are considered.