Atom divisor graph of a poset

Abstract

Let $P=(V,\le )$ be a poset with least element 0 and $A\subseteq V$. Then $A^{\ell }=\left(b\in V\mid b\le a\right)\left(\text{for every}a\in A\right)$ is a $lowercone$ of $A$. Denote $A^{{\ell }_0}=A^{\ell }\setminus \left\{0\right\}$. In this paper, we introduce the atom-divisor graph of $P$, denoted by $ADG\left(P\right)$, it is a graph with the vertex set $P\setminus \{0,1\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\{x,y\}}^{{\ell }_0}\subseteq At\left(P\right)$. Since $ADG\left(P\right)$ is generally not connected, the condition for ADG(P) to be connected is determined. If $ADG\left(P\right)$ contains a cycle, then we have shown that its girth is $3\text{or even}$. Moreover, we obtain a class of posets for which the girth of $ADG\left(P\right)$ is $3,4\text{or}2n,n\ge 3$. We obtain necessary and sufficient conditions for $ADG\left(P\right)$ to be in the class of cycles, paths, stars, complete bipartite, complete, complete multipartite, or sun graphs. The bounds on domination and clique number of $ADG\left(P\right)$ are determined. Further, we give the class of posets for which $ADG\left(P\right)$ is not weakly perfect.