Automorphism group of a graph constructed from a lattice

Let L be a lattice and S be a ∧-closed subset of L. The graph ΓS(L) is a simple graph with all elements of L as vertex set and two distinct vertex x, y are adjacent if and only if x ∨ y ∈ S. In this paper, we verify the automorphism group of ΓS(L) and the relation by automorphism group of the lattice L. Also we study some properties of the graph ΓS(L), where S is a prime filter or an ideal such as the perfect maching