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.