A note on the algebra of threshold graphs

Abstract

It is known that threshold graphs have edge rings with 2-linear resolutions. This was proved by Engström and Stamps, [4]. They used the fact that an edge ring of a graph G has a 2-linear resolution if and only if the complement graph is chordal. They also described a method to determine the Betti numbers. Our goal is to determine when edge rings of threshold graphs are Cohen-Macaulay. In order to do so, it is more convenient to use an alternative way to study edge rings of graphs, that is to interpret them as Stanley-Reisner rings. We also determine when the neighborhood complex of a threshold graph has a Cohen-Macaulay Stanley-Reisner ring.