# A note on the zero divisor graph of a lattice

Document Type: Research Paper

Authors

Manonmaniam Sundaranar University

Abstract

‎Let $L$ be a lattice with the least element $0$‎. ‎An element $x\in L$ is a zero divisor if $x\wedge y=0$ for some $y\in L^*=L\setminus \left\{0\right\}$‎. ‎The set of all zero divisors is denoted by $Z(L)$‎. ‎We associate a simple graph $\Gamma(L)$ to $L$ with vertex set $Z(L)^*=Z(L)\setminus \left\{0\right\}$‎, ‎the set of non-zero zero divisors of $L$ and distinct $x,y\in Z(L)^*$ are adjacent if and only if $x\wedge y=0$‎. ‎In this paper‎, ‎we obtain certain properties and diameter and girth of the zero divisor graph $\Gamma(L)$‎. ‎Also we find a dominating set and the domination number of the zero divisor graph $\Gamma(L)$‎.

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### References

D. D. Anderson and M. Naseer (1993). Beck’s coloring of a commutative ring. J. Algebra. 150, 500-514
D. F. Anderson and P. Livingston (1999). The zero divisor graph of a commutative ring. J. Algebra. 217, 434-447
D. F. Anderson, R. Levy and J. Shapiro, (2003). Zero divisor graphs, von Neumann regular rings and Boolean algebras. J. Pure Appl. Algebra. 180, 221-241
I. Beck (1988). Coloring of Commutative rings. J. Algebra. 116, 208-226
B. Bollobos and I. Rival (1979). The maximal size of the covering graph of a lattice. Algebra Universalis. 9, 371-373
G. Chartrand and P. Zhang (1986). Introduction to Graph theory. Wadsworth and Brooks/Cole, Monterey, CA.
B. A. Davey and H. A. Priestley (2002). Introduction to Lattices and Order. Cambridge University Press, New York.
F. R. Demeyer, T. Mckenzie and K. Schneider (2002). The zero divisor graph of a commutative semigroup. Semigroup Forum. 65, 206-214
D. Duffus and I. Rival (1977). Path length in the covering graph of a lattice. Discrete Math.. 19, 139-158
E. Estaji and K. Khashyarmanesh (2012). The zero divisor graph of a lattice,. Results Math., DOI 10.1007/s00025-010-0067-8.. 61, 1-11
E. Mendelson (2004). Boolean algebra and Switching Circuits. Tata McGraw-Hill, New Delhi.
N. D. Filipov (1980). Comparability graphs of partially ordered sets of different types. Colloq. Math. Soc. J´anos Bolyai. 33, 373-380
E. Gedeonov´a (1980). Lattices whose covering graphs are S-graphs. Colloq Math. Soc. J´anos Bolyai. 33, 407-435
S. K. Nimbhorkar, M. P. Wasadikar and M. M. Pawar (2010). Coloring of lattices. Math. Slovaca. 60, 419-434