Independence Complexes of Well-Covered Circulant Graphs

We study the independence complexes of families of well-covered circulant graphs discovered by Boros–Gurvich–Milanič, Brown–Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g., vertex decomposable, shellable) or topological (e.g., Cohen–Macaulay, Buchsbaum) properties. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen–Macaulay, and/or Buchsbaum. A highlight of this search is an example of a graph whose independence complex is shellable but not vertex decomposable.