Normality, factoriality and strong $F$-regularity of Lov\'asz-Saks-Schrijver rings

TL;DR

This paper explores algebraic properties of Lovász-Saks-Schrijver rings associated with graphs, linking these properties to graph invariants and establishing conditions for normality, factoriality, and strong F-regularity.

Contribution

It establishes new connections between algebraic properties of $R_G(d)$ and combinatorial invariants of graphs, providing conditions for various algebraic regularities based on graph parameters.

Findings

$R_G(d)$ is $F$-regular if $d \\geq \\text{pmd}(G)+k(G)$ in finite characteristic.

$R_G(d)$ has rational singularities in characteristic zero under the same conditions.

$R_G(d)$ is a UFD if $d \\geq \\text{pmd}(G)+k(G)+1$.

Abstract

Every simple finite graph $G$ has an associated Lov\'asz-Saks-Schrijver ring $R_G(d)$ that is related to the $d$-dimensional orthogonal representations of $G$. The study of $R_G(d)$ lies at the intersection between algebraic geometry, commutative algebra and combinatorics. We find a link between algebraic properties such as normality, factoriality and strong $F$-regularity of $R_G(d)$ and combinatorial invariants of the graph $G$. In particular we prove that if $d \geq \text{pmd}(G)+k(G)$ then $R_G(d)$ is $F$-regular in finite characteristic and rational singularity in characteristic $0$ and furthermore if $d \geq \text{pmd}(G)+k(G)+1$ then $R_G(d)$ is UFD. Here $\text{pmd}(G)$ is the positive matching decomposition number of $G$ and $k(G)$ is its degeneracy number.