# Total graph of a $0$-distributive lattice

Document Type : Research Paper

Authors

Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

10.29252/cgasa.9.1.15

Abstract

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${\rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${\rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y \in £$, the vertices $x$ and $y$ are adjacent if and only if $x \vee y \in {\rm Z}(£)$. The basic properties of the graph ${\rm T}(G (£))$ and its subgraphs are studied. We investigate the properties of the total graph of $0$-distributive lattices as diameter, girth, clique number, radius, and the  independence number.

Keywords

#### References

[1] Anderson, D.D. and Naseer, M., Beck’s coloring of commutative rings, J. Algebra 159(2) (1993), 500-514.
[2] Anderson, D.F. and Badawi, A., The total graph of a commutative ring, J. Algebra 320(7) (2008), 2706-2719.
[3] Anderson, D.F. and Livingston, P.S., The zero-divisor graph of a commutative rings, J. Algebra 217(2) (1999), 434-447.
[4] Balasubramani, P. and Venkatanarasimhan, P.V., Characterizations of the 0-distributive lattice, Indian J. Pure Appl. Math. 32(3) (2001), 315-324.
[5] Beck, I., Coloring of commutative rings, J. Algebra 116(1) (1988), 208-226.
[6] Birkhoff, G., “Lattice Theory", Amer. Math. Soc., 1973.
[7] Bondy, A. and Murty, U.S.R., “Graph Theory with Applications", American Elsevier, 1976.
[8] Coykendall, J. Sather-Wagstaff, S., Sheppardson, L., and Spiroff, S., On zero-divisor graphs, Prog. Comm. Algebra 2 (2012), 241-299.
[9] Ebrahimi Atani, S., Dolati Pish Hesari, S., and Khoramdel, M., Total graph of a commutative semiring with respect to identity-summand elements, J. Korean Math. Soc. 51(3) (2014), 593-607.
[10] Ebrahimi Atani, S. and Sedghi Shanbeh Bazari, M., On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra and Appl. 36(2) (2016), 157-168.
[11] Estaji, E. and Khashyarmanesh, K., The zero-divisor graph of a lattice, Results Math. 61 (2012), 1-11.
[12] Grätzer, G., “General Lattice Theory", Birkhauser, 1978.
[13] Halaš, R. and Jukl, M., On Beck’s coloring of posets, Discrete Math. 309 (2009), 4584-4589.
[14] Joshi, V. and Khiste, A., On the zero-divisor graphs of pm-lattices, Discrete Math. 312 (2012), 2076-2082.
[15] Joshi, V. and Khiste, A., Complement of the zero-divisor graph of a lattice, Bull. Aust. Math. Soc. 89 (2014), 177-190.
[16] Nimbhorkar, S.K. Wasadikar, M.P., and Pawar, M.M., Coloring of lattices, Math. Slovaca 60 (2010), 419-434.
[17] Tamizh Chelvam, T. and Nithya, S., A note on the zero-divisor graph of a lattices, Trans. Comb. 3(3) (2014), 51-59.