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

Every simple finite graph G has an associated Lov\'asz-Saks-Schrijver ring RG(d) that is related to the d-dimensional orthogonal representations of G. The study of RG(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 RG(d) and combinatorial invariants of the graph G. In particular we prove that if d ≥ pmd(G)+k(G) then RG(d) is F-regular in finite characteristic and rational singularity in characteristic $0$ and furthermore if d ≥ pmd(G)+k(G)+1 then RG(d) is UFD. Here pmd(G) is the positive matching decomposition number of G and $k(G)$ is its degeneracy number.