Vanishing ideals over graphs and even cycles
  Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components. 
Jorge Neves, Maria Vaz Pinto, and Rafael H. Villarreal
math.AC cs.IT math.AG math.CO math.IT