Line Graphs with a Cohen–Macaulay or Gorenstein Clique Complex

Let H be a simple undirected graph and G = L(H) be its line graph. Assume that Δ (G) denotes the clique complex of G. We present a complete characterization of those H for which Δ (G) is Cohen–Macaulay or Gorenstein. In addition, we show that if Δ (G) is pure, then it is Cohen–Macaulay if and only if it is shellable if and only if it is vertex decomposable.