Vanishing ideals over graphs and even cycles

TL;DR

This paper investigates the algebraic properties of vanishing ideals over graphs, providing explicit generators, degree bounds, and formulas for regularity and code length related to even cycles and bipartite graphs.

Contribution

It offers explicit descriptions and bounds for vanishing ideals over graphs, including formulas for regularity and code length, extending understanding of algebraic toric sets associated with graphs.

Findings

Explicit generators for I(X) when X is based on even cycles or bipartite graphs

Degree bounds and a formula for the regularity of I(X)

A formula for the length of the associated linear code

Abstract

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.