Circulant \(S_2\) graphs

Recently, Earl, Vander Meulen, and Van Tuyl characterized some families of Cohen-Macaulay or Buchsbaum circulant graphs discovered by Boros-Gurvich-Milani\(\check{\text{c}}\), Brown-Hoshino, and Moussi. In this paper, we will characterize those families of circulant graphs which satisfy Serre's condition \(S_2\). More precisely, we show that for some families of circulant graphs, \(S_2\) property is equivalent to well-coveredness or Buchsbaumness, and for some other families it is equivalent to Cohen-Macaulayness. We also give examples of infinite families of circulant graphs which are Buchsbaum but not \(S_2\), and vice versa.