ABSTRACT
In this chapter we study monomial subrings associated to graphs and their toric ideals. We relate the even closed walks and circuits of the vector matroid of a graph with Gro¨bner bases. A description of the integral closure of the edge subring of a multigraph will be presented along with a description of the circuits of its toric ideal. The Smith normal form and the invariant factors of the incidence matrix of a graph are fully determined. As an application we compute the multiplicity of edge subrings in terms of relative volumes. The family of ring graphs is studied here. These graphs are characterized in algebraic and combinatorial terms.
