One can associate to a bipartite graph a so-called edge ring and its spectrum is an affine normal toric variety. We first characterize the faces of the (edge) cone associated to this toric variety in terms of certain independent sets of the bipartite graph. Then, we give first examples of rigid toric varieties associated to bipartite graphs. We show their rigidity combinatorially, to wit, purely in terms of graphs. In the next chapters, we determine the two and three-dimensional faces of the edge cone. With this information, we show that these toric varieties are smooth in codimension two and the non-simplicial three-dimensional faces are generated by exactly four extremal rays. In the latter case, we get non rigid toric varieties. Lastly, we study torus actions on matrix Schubert varieties. In the toric case, we present a classification for their rigidity.
