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Betti Numbers of Some Circulant Graphs

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Abstract

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 C2n(1, 2, 3, 5, …, o(n)). We do this by showing that this graph is the product (or join) of the cycle Cn by itself, and computing Betti numbers of Cn * Cn. We also discuss whether such a graph (more generally, G * H) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or S2.

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Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Mohsen Abdi Makvand

  2. Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Tehran, Iran

    Amir Mousivand

Authors
  1. Mohsen Abdi Makvand
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  2. Amir Mousivand
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Corresponding author

Correspondence to Amir Mousivand.

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Acknowledgements

We gratefully acknowledge the use of computer algebra systems CoCoA [1] and Macaulay2 [7] which was valuable for our work. We also appreciate the referee’s careful reading and useful comments.

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Makvand, M.A., Mousivand, A. Betti Numbers of Some Circulant Graphs. Czech Math J 69, 593–607 (2019). https://doi.org/10.21136/CMJ.2019.0606-16

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  • DOI: https://doi.org/10.21136/CMJ.2019.0606-16

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