Irreducible divisor graphs and numerical monoids

The factorization of an element x from a numerical monoid, unique or non-unique, can be represented visually by an irreducible divisor graph G(x). The graph G(x) consists of a vertex set corresponding to the irreducible elements that appear in some factorization of x. We connect two vertices a and b if and only if both a and b appear in the same factorization of x. In this thesis, we bring together aspects of algebra and graph theory to study these structures. In particular, we determine precisely when certain graph theoretic properties hold for irreducible divisor graphs of elements in monoids of the form N = ⟨n, n + 1, ..., 2n - 1⟩ and N = ⟨n, n + 1, ..., n + t⟩ where 0 ≤ t < n - 1. Finally, we give examples of irreducible divisor graphs that are isomorphic to each of the 31 distinct graphs on at most five vertices.