Weakly Zero Divisor Graph of a Lattice - Kulal, Vikas; Khairnar, Anil; Masalkar, Krishnat; Kadam, Lata (Communications in Mathematics and Applications, 2023)
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 \(Z^* (L)= { rLr eq 0 , s eq 0\) such that \(rs=0 }\) and for any distinct \(u\) and \(v\) in \(Z^* (L)\), \(u-v\) is an edge in \(WZG(L)\) if and only if there exists \(p (u){0}\) and \(q (v) {0}\) such that \(pq=0\). 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 annihilator-ideal graph of lattices.