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 $\Gamma(L)$, 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 corresponding filters of $L$ is a small filter of $L$. In this paper, the basic  properties and possible structures of the graph $\Gamma(L)$ are investigated. Moreover, the complemented property, the domination number and the planar property of $\Gamma(L)$ are considered.