University of Isfahan Transactions on Combinatorics 2251-8657 7 3 2018 09 01 The annihilator graph of a 0-distributive lattice 1 18 22285 10.22108/toc.2017.104919.1507 EN Saeid Bagheri Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran Mahtab Koohi Kerahroodi Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran. Journal Article 2017 06 29 ‎‎In this article‎, ‎for a lattice $\mathcal L$‎, ‎we define and investigate‎ ‎the annihilator graph $\mathfrak {ag} (\mathcal L)$ of $\mathcal L$ which contains the zero-divisor graph of $\mathcal L$ as a subgraph‎. ‎Also‎, ‎for a 0-distributive lattice $\mathcal L$‎, ‎we study some properties of this graph such as regularity‎, ‎connectedness‎, ‎the diameter‎, ‎the girth and its domination number‎. ‎Moreover‎, ‎for a distributive lattice $\mathcal L$ with $Z(\mathcal L)\neq\lbrace 0\rbrace$‎, ‎we show that $\mathfrak {ag} (\mathcal L) = \Gamma(\mathcal L)$ if and only if $\mathcal L$ has exactly two minimal prime ideals‎. ‎Among other things‎, ‎we consider the annihilator graph $\mathfrak {ag} (\mathcal L)$ of the lattice $\mathcal L=(\mathcal D(n),|)$ containing all positive divisors of a non-prime natural number $n$ and we compute some invariants such as the domination number‎, ‎the clique number and the chromatic number of this graph‎. ‎Also‎, ‎for this lattice we investigate some special cases in which $\mathfrak {ag} (\mathcal D(n))$ or $\Gamma(\mathcal D(n))$ are planar‎, ‎Eulerian or Hamiltonian. https://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdf