A note on the zero divisor graph of a lattice

Abstract

‎Let $L$ be a lattice with the least element $0$‎. ‎An element $xin L$ is a zero divisor if $xwedge y=0$ for some $yin L^*=Lsetminus left{0right}$‎. ‎The set of all zero divisors is denoted by $Z(L)$‎. ‎We associate a simple graph $Gamma(L)$ to $L$ with vertex set $Z(L)^*=Z(L)setminus left{0right}$‎, ‎the set of non-zero zero divisors of $L$ and distinct $x,yin Z(L)^*$ are adjacent if and only if $xwedge y=0$‎. ‎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 $Gamma(L)$‎.