Graphs and toric varieties

The toric ideal associated to a finite graph is obtained by taking the kernel of the monomial map that is defined by the edges of the graph. Equivalently one obtains a toric variety by defining edge cones (or edge polytopes) where the extremal rays (or vertices) are the columns of the incidence matrix of the graph. In this talk, we explain the interplay between graphs and their associated toric varieties appearing in different areas such as Fano, (matrix) Schubert and Kazhdan-Lusztig varieties.