Regularity of the vanishing ideal over a parallel composition of paths

TL;DR

This paper calculates the Castelnuovo--Mumford regularity of the vanishing ideal over a specific class of graphs formed by parallel paths, extending known results to new graph structures.

Contribution

It introduces a formula for the regularity of the vanishing ideal over graphs formed by parallel paths, a case previously unresolved, and establishes new inequalities linking regularity to graph structure.

Findings

Computed regularity for the parallel path graph class.

Extended understanding of regularity beyond trees and cycles.

Established inequalities relating regularity to graph properties.

Abstract

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of the regularity in the bipartite case to the computation of the regularity of the blocks of G. We also prove new inequalities relating the Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.