In this article, for a lattice L, we define and investigate the annihilator graph ag(L) of L which contains the zero-divisor graph of L as a subgraph. Also, for a 0-distributive lattice L, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice L with non zero Z(L), we show that the annihilator graph ag(L) is the same as the zero-divisor graphof L if and only if L has exactly two minimal prime ideals. Among other things, we consider the annihilator graph ag(L) of the lattice L = (D(n); |) containing all positive divisors of a non-prime natural number and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which the annihilator graph ag(D(n)) and the zero-divisor graph of D(n) are planar, Eulerian or Hamiltonian.
