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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