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 \(Z^{* }(L)= \{ r\in L\mid r eq 0 , \ \exists \ s eq 0\) such that \(r\wedge s=0 \}\) and for any distinct \(u\) and \(v\) in \(Z^{* }(L)\), \(u-v\) is an edge in \(WZG(L)\) if and only if there exists \(p \in \Ann(u)\setminus\{0\}\) and \(q \in \Ann(v) \setminus \{0\}\) such that \(p\wedge q=0\). 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 annihilator-ideal graph of lattices.