Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs

Abstract

Let ΔΔ be a simplicial complex, IΔIΔ its Stanley-Reisner ideal and R=K[Δ]R=K[Δ] its Stanley-Reisner ring over a field KK. In 2018, the author introduced the squarefree zero-divisor graph of RR, denoted by Γsf(R)Γsf(R), and proved that if ΔΔ and Δ′Δ′ are two simplicial complexes, then the graphs Γsf(K[Δ])Γsf(K[Δ]) and Γsf(K[Δ′])Γsf(K[Δ′]) are isomorphic if and only if the rings K[Δ]K[Δ] and K[Δ′]K[Δ′] are isomorphic. Here we derive some algebraic properties of RR using combinatorial properties of Γsf(R)Γsf(R). In particular, we state combinatorial conditions on Γsf(R)Γsf(R) which are necessary or sufficient for RR to be Cohen-Macaulay. Moreover, we investigate when Γsf(R)Γsf(R) is in some well-known classes of graphs and show that in these cases, IΔIΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R)Γsf(R) and their algebraic interpretations.