Irreducible divisor graphs for numerical monoids

Abstract

The factorization of an element $x$ from a numerical monoid can be represented visually as an irreducible divisor graph $G\left(x\right)$. The vertices of $G\left(x\right)$ are the monoid generators that appear in some representation of $x$, with two vertices adjacent if they both appear in the same representation. In this paper, we determine precisely when irreducible divisor graphs of elements in monoids of the form $N=\langle n,n+1,\dots ,n+t\rangle$ where $0\le t<n$ are complete, connected, or have a maximum number of vertices. Finally, we give examples of irreducible divisor graphs that are isomorphic to each of the $31$ mutually nonisomorphic connected graphs on at most five vertices.