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 x_{i} x_{j} where x_{i} and x_{j} 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)}=I^{n} 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.