The annihilator graph of a 0-distributive lattice

Abstract

‎‎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 $Z(mathcal L)neqlbrace 0rbrace$‎, ‎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.