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 $S_2$.
Keywords:  Betti number; Castelnuovo-Mumford regularity; projective dimension; circulant graph
Classification MSC:  13D02, 05C75
DOI:  10.21136/CMJ.2019.0606-16


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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


 
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