Weakly Zero Divisor Graph of a Lattice

Abstract

For a lattice L, we associate a graph W ZG(L) called a weakly zero divisor graph of L. The vertex set of W ZG(L) is Z * (L), where Z * (L) = {r ∈ L | r ̸ = 0, ∃ s ̸ = 0 such that r ∧ s = 0} and for any distinct u and v in Z * (L), u − v is an edge in W ZG(L) if and only if there exists p ∈ Ann(u) \ {0} and q ∈ Ann(v) \ {0} such that p ∧ q = 0. In this paper, we determined the diameter, girth, independence number and domination number of W ZG(L). We characterized all lattices whose W ZG(L) is complete bipartite or planar. Also, we find a condition so that W ZG(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.