A note on the Rees algebra of a bipartite graph

Let R=K[x1,…,xn] be a polynomial ring over a field K and let I be an ideal of R generated by a set xα1,…,xαq of square-free monomials of degree two such that the graph G defined by those monomials is bipartite. We study the Rees algebra R(I) of I, by studying both the Rees cone R+A′ generated by the set A′={e1,…,en,(α1,1),…,(αq,1)} and the matrix C whose columns are the vectors in A′. It is shown that C is totally unimodular. We determine the irreducible representation of the Rees cone in terms of the minimal vertex covers of G. Then we compute the a-invariant of R(I).