Binomial Ideals from Graphs

This paper contains an examination of polynomial ideals associated to graphs and conjectures about them. We will describe how to obtain an ideal IG from a graph G. This ideal encodes the relations among all spanning trees of G: There are ve conjectures we are concerned with. The rst claims that IG is generated by quadratic binomials. We will discuss the relationships uncovered between dierent classes of graphs and their ideals. For instance, for any tree or one-circuit graph, IG = 0: Also, we can contract edges that are not part of any circuit. The number of circuits in a graph has a direct correlation to its ideal.