| Title:
|
Betti numbers of some circulant graphs (English) |
| Author:
|
Abdi Makvand, Mohsen |
| Author:
|
Mousivand, Amir |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
593-607 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $S_2$. (English) |
| Keyword:
|
Betti number |
| Keyword:
|
Castelnuovo-Mumford regularity |
| Keyword:
|
projective dimension |
| Keyword:
|
circulant graph |
| MSC:
|
05C75 |
| MSC:
|
13D02 |
| idZBL:
|
Zbl 07088806 |
| idMR:
|
MR3989268 |
| DOI:
|
10.21136/CMJ.2019.0606-16 |
| . |
| Date available:
|
2019-07-24T11:14:08Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147778 |
| . |
| Reference:
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[1] Abbott, J., Bigatti, A. M., Robbiano, L.: CoCoA: a system for doing Computations in Commutative Algebra.Available at http://cocoa.dima.unige.it. SW: https://swmath.org/software/00143 |
| Reference:
|
[2] Boros, E., Gurvich, V., Milanič, M.: On CIS circulants.Discrete Math. 318 (2014), 78-95. Zbl 1281.05073, MR 3141630, 10.1016/j.disc.2013.11.015 |
| Reference:
|
[3] Brown, J., Hoshino, R.: Independence polynomials of circulants with an application to music.Discrete Math. 309 (2009), 2292-2304. Zbl 1228.05178, MR 2510357, 10.1016/j.disc.2008.05.003 |
| Reference:
|
[4] Brown, J., Hoshino, R.: Well-covered circulant graphs.Discrete Math. 311 (2011), 244-251. Zbl 1222.05208, MR 2739910, 10.1016/j.disc.2010.11.00 |
| Reference:
|
[5] Bruns, W., Herzog, J.: Cohen-Macaulay Rings.Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1993). Zbl 0788.13005, MR 1251956, 10.1017/CBO9780511608681 |
| Reference:
|
[6] Earl, J., Meulen, K. N. Vander, Tuyl, A. Van: Independence complexes of well-covered circulant graphs.Exp. Math. 25 (2016), 441-451. Zbl 1339.05333, MR 3499708, 10.1080/10586458.2015.1091753 |
| Reference:
|
[7] Grayson, D. R., Stillman, M. E., Eisenbud, D.: Macaulay2, a software system for research in algebraic geometry.Available at http://www.math.uiuc.edu/Macaulay2/. SW: https://swmath.org/software/00537 |
| Reference:
|
[8] Hà, H. T., Tuyl, A. Van: Splittable ideals and the resolutions of monomial ideals.J. Algebra 309 (2007), 405-425. Zbl 1151.13017, MR 2301246, 10.1016/j.jalgebra.2006.08.022 |
| Reference:
|
[9] Hà, H. T., Tuyl, A. Van: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers.J. Algebr. Comb. 27 (2008), 215-245. Zbl 1147.05051, MR 2375493, 10.1007/s10801-007-0079-y |
| Reference:
|
[10] Hoshino, R.: Independence polynomials of circulant graphs.Ph.D. Thesis, Dalhousie University, Halifax (2008). MR 2710951 |
| Reference:
|
[11] Jacques, S.: Betti numbers of graph ideals.Ph.D. Thesis, University of Sheffield, Sheffield. Available at https://arxiv.org/abs/math/0410107 (2004). |
| Reference:
|
[12] Jacques, S., Katzman, M.: The Betti numbers of forests.Available at https://arxiv.org/abs/math/0501226, 2005, 12 pages. |
| Reference:
|
[13] Katzman, M.: Characteristic-independence of Betti numbers of graph ideals.J. Comb. Theory, Ser. A 113 (2006), 435-454. Zbl 1102.13024, MR 2209703, 10.1016/j.jcta.2005.04.005 |
| Reference:
|
[14] Mahmoudi, M., Mousivand, A., Tehranian, A.: Castelnuovo-Mumford regularity of graph ideals.Ars Comb. 125 (2016), 75-83. Zbl 06644150, MR 3468037 |
| Reference:
|
[15] Mousivand, A.: Algebraic properties of product of graphs.Commun. Algebra 40 (2012), 4177-4194. Zbl 1263.13021, MR 2982931, 10.1080/00927872.2011.605408 |
| Reference:
|
[16] Mousivand, A.: Circulant $S_2$ graphs.Available at https://arxiv.org/abs/1512.08141, 2015, 11 pages. |
| Reference:
|
[17] Moussi, R.: A characterization of certain families of well-covered circulant graphs.M.Sc. Thesis, St. Mary's University, Halifax. Available at http://library2.smu.ca/handle/01/24725, 2012. |
| Reference:
|
[18] Roth, M., Tuyl, A. Van: On the linear strand of an edge ideal.Commun. Algebra 35 (2007), 821-832. Zbl 1123.13012, MR 2305234, 10.1080/00927870601115732 |
| Reference:
|
[19] Terai, N.: Alexander duality in Stanley-Reisner rings.Affine Algebraic Geometry Osaka University Press, Osaka T. Hibi (2007), 449-462. Zbl 1128.13014, MR 2330484 |
| Reference:
|
[20] Meulen, K. N. Vander, Tuyl, A. Van: Shellability, vertex decomposability, and lexicographical products of graphs.Contrib. Discrete Math. 12 (2017), 63-68. Zbl 1376.05171, MR 3739053, 10.11575/cdm.v12i2.62777 |
| Reference:
|
[21] Meulen, K. N. Vander, Tuyl, A. Van, Watt, C.: Cohen-Macaulay circulant graphs.Commun. Algebra 42 (2014), 1896-1910. Zbl 1327.13077, MR 3169680, 10.1080/00927872.2012.749886 |
| Reference:
|
[22] Villarreal, R. H.: Monomial Algebras.Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). Zbl 1002.13010, MR 1800904 |
| Reference:
|
[23] Whieldon, G.: Jump sequences of edge ideals.Available at https://arxiv.org/abs/1012.0108, 2010, 27 pages. |
| Reference:
|
[24] Zheng, X.: Resolutions of facet ideals.Commun. Algebra 32 (2004), 2301-2324. Zbl 1089.13014, MR 2100472, 10.1081/AGB-120037222 |
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