Weakly Zero Divisor Graph of a Lattice
DOI:
https://doi.org/10.26713/cma.v14i3.2455Keywords:
Zero divisor graph, Base of the element, Atom, PlanarAbstract
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 \(Z^{\ast }(L)\), where \(Z^{\ast }(L)= \{ r\in L\mid r \neq 0 , \ \exists \ s\neq 0\) such that \(r\wedge s=0 \}\) and for any distinct \(u\) and \(v\) in \(Z^{\ast }(L)\), \(u-v\) 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.
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