Weakly Zero Divisor Graph of a Lattice





Zero divisor graph, Base of the element, Atom, Planar


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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D. F. Anderson, T. Asir, A. Badawi and T. T. Chelvam, Graphs from Rings, 1st edition, Springer International Publishing (2021).

I. Beck, Coloring of commutative rings, Journal of Algebra 116(1) (1988), 208 – 226, DOI: 10.1016/0021-8693(88)90202-5.

G. Birkhoff, Lattice Theory, Colloquium Publications, Vol. 25, American Mathematical Society, (1940), DOI: 10.2307/2268183.

T. T. Chelvam and S. Nithya, A note on the zero divisor graph of a lattice, Transactions on Combinatorics 3(3) (2014), 51 – 59, DOI: 10.22108/TOC.2014.5626.

T. T. Chelvam and S. Nithya, Zero-divisor graph of an ideal of a near-ring, Discrete Mathematics, Algorithms and Applications 5(1) (2013), 1350007, DOI: 10.1142/S1793830913500079

E. Estaji and K. Khashyarmanesh, The zero divisor graph of a lattice, Results in Mathematics 61(1) (2012), 1 – 11, DOI: 10.1007/s00025-010-0067-8.

G. Grätzer, Lattice Theory: Foundation, 1st edition, Birkhäuser, Basel, xxx + 614 pages (2011), DOI: 10.1007/978-3-0348-0018-1.

A. Khairnar and B. N. Waphare, Zero-divisor graphs of laurent polynomials and laurent power series, in: Algebra and its Applications, S. Rizvi, A. Ali and V. Filippis (editors), Springer Proceedings in Mathematics & Statistics, Vol. 174, Springer, Singapore (2016), DOI: 10.1007/978-981-10-1651-6_21.

V. Kulal, A. Khairnar and K. Masalkar, Annihilator ideal graph of a lattice, Palestine Journal of Mathematics 11(iv) (2022), 195 – 204, URL: https://pjm.ppu.edu/paper/1214-annihilator-ideal-graph-lattice.

M. J. Nikmehr, A. Azadi and R. Nikandish, The weakly zero-divisor graph of a commutative ring, Revista de la Unión Matemática Argentina 62(1) (2021), 105 – 116, DOI: 10.33044/revuma.1677.

D. B. West, Introduction to Graph Theory, 2nd edition, Pearson (2000).




How to Cite

Kulal, V., Khairnar, A., Masalkar, K., & Kadam, L. (2023). Weakly Zero Divisor Graph of a Lattice. Communications in Mathematics and Applications, 14(3), 1167–1180. https://doi.org/10.26713/cma.v14i3.2455



Research Article