Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 593-607, 2019
Betti numbers of some circulant graphs
Mohsen Abdi Makvand, Amir Mousivand
Received November 22, 2016. Published online July 9, 2019.
Abstract: Let $o(n)$ be the greatest odd integer less than or equal to $n$. In this paper we provide explicit formulae to compute $\mathbb{N}$-graded Betti numbers of the circulant graphs $C_{2n}(1,2,3,5,\ldots,o(n))$. 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 $S_2$.
Affiliations: Mohsen Abdi Makvand, Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran, e-mail: mohsenabdimakvand@yahoo.com; Amir Mousivand (corresponding author), Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Tehran, Iran, e-mail: amirmousivand@gmail.com, amir.mousivand@iaufb.ac.ir