Betti Numbers of Some Circulant Graphs

Let o ( n ) be the greatest odd integer less than or equal to n . In this paper we provide explicit formulae to compute ℕ-graded Betti numbers of the circulant graphs C 2 n (1, 2, 3, 5, …, o ( n )). We do this by showing that this graph is the product (or join) of the cycle C n by itself, and computing Betti numbers of C n * C n . We also discuss whether such a graph (more generally, G * H ) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or S 2 .