In this paper we discuss a connection between graph theory and ring theory. Given a graph G there exists a corresponding edge ideal I generated by xi xj where xi and xj are vertices in G connected by an edge. Simis, Vasconcelos, and Villarreal show that a graph G is bipartite (contains only even cycles) if and only if its corresponding edge ideal I satisfies I(n)=In for all n greater than or equal to 1. We explore what happens when G is not bipartite - in particular, when G is an odd sided polygon.