Geodesic
Betti numbers of some circulant graphs
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 593-607
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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 $S_2$.
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 $S_2$.
DOI : 10.21136/CMJ.2019.0606-16
Classification : 05C75, 13D02
Keywords: Betti number; Castelnuovo-Mumford regularity; projective dimension; circulant graph 
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Abdi Makvand, Mohsen; Mousivand, Amir. Betti numbers of some circulant graphs. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 593-607. doi: 10.21136/CMJ.2019.0606-16

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