On the Hilbert series of vertex cover algebras of unmixed bipartite graphs

TL;DR

This paper investigates the algebraic structure of vertex cover algebras associated with unmixed bipartite graphs, focusing on their Hilbert series and Gröbner bases, revealing new algebraic insights into these combinatorial objects.

Contribution

It provides a computation of the Gröbner basis for the toric ideal and analyzes the Hilbert series of vertex cover algebras for a specific class of bipartite graphs.

Findings

Computed the reduced Gröbner basis of the toric ideal.

Studied the Hilbert series of the vertex cover algebra.

Characterized algebraic properties of unmixed bipartite graphs.

Abstract

We compute the reduced Gr\"{o}bner basis of the toric ideal with respect to a suitable monomial order and we study the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is an unmixed bipartite graph without isolated vertices.