Hilbert-Samuel Multiplicity of a Bipartite Graph

TL;DR

This paper calculates the Hilbert-Samuel multiplicity of bipartite graphs with specific toric ideals generated by quadratic binomials, under certain conditions, advancing algebraic understanding of these graph structures.

Contribution

It provides a formula for the Hilbert-Samuel multiplicity for bipartite graphs with quadratic binomial toric ideals satisfying particular conditions.

Findings

Derived explicit multiplicity formulas for specific bipartite graphs.

Extended algebraic understanding of toric ideals in graph theory.

Identified conditions under which the multiplicity can be computed.

Abstract

Let $G$ be a bipartite graph and $I(G)$ the toric ideal associated to the graph $G$. In this article we calculate Hilbert-Samuel multiplicity of the graph $G$ for which the toric ideal $I(G)$ is generated by a quadratic binomials and it satiesfies some conditions.