Deriving Some Properties of Stanley-Reisner Rings from Their Squarefree Zero-Divisor Graphs

Abstract

Let ∆ be a simplicial complex, I∆ its Stanley-Reisner ideal and R = K[∆] its Stanley-Reisner ring over a field K. In 2018, the author introduced the squarefree zero-divisor graph of R, denoted by Γsf(R), and proved that if ∆ and ∆′ are two simplicial complexes, then the graphs Γsf(K[∆]) and Γsf(K[∆ ′]) are isomorphic if and only if the rings K[∆] and K[∆′] are isomorphic. Here we derive some algebraic properties of R using combinatorial properties of Γsf(R). In particular, we state combinatorial conditions on Γsf(R) which are necessary or sufficient for R to be Cohen-Macaulay. Moreover, we investigate when Γsf(R) is in some well-known classes of graphs and show that in these cases, I∆ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R) and their algebraic interpretations. Mathematics Subject Classification (2020): 13F55, 13C70, 05C25, 05E40