Weakly Zero Divisor Graph of a Lattice

For a lattice L, we associate a graph WZG(L) called a weakly zero divisor graph of L. The vertex set of WZG(L) is Z *(L), where Z*(L) = {r is an element of L | r not equal 0, there exists s not equal 0 such that r <^> s = 0} and for any distinct u and v in Z*(L), u- v is an edge in WZG(L) if and only if there exists p is an element of Ann(u)\{0} and q is an element of Ann(v)\{0} such that p <^> q = 0. In this paper, we determined the diameter, girth, independence number and domination number of WZG(L). We characterized all lattices whose WZG(L) is complete bipartite or planar. Also, we find a condition so that WZG(L) is Eulerian or Hamiltonian. Finally, we study the affinity between the weakly zero divisor graph, the zero divisor graph and the annihilatorideal graph of lattices.