betti-numbers-of-edge-ideals-of-grimaldi-graphs-and-their-complements-|-bulletin-of-the-malaysian-mathematical-…-–-springer

Betti Numbers of Edge Ideals of Grimaldi Graphs and Their Complements | Bulletin of the Malaysian Mathematical … – Springer

Abstract

Let (nge 2) be an integer. The Grimaldi graph G(n) is defined by taking the elements of the set ({ 0, ldots , n-1 }) as vertices. Two distinct vertices x and y are adjacent in G(n) if and only if (gcd (x+y, n) =1). In this paper, we examine the Betti numbers of the edge ideals of these graphs and their complements.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Availability of Data, Code and Materials

Data sharing not applicable to this work as no data sets were generated or analysed during the current study.

References

  1. Alon, N., Friedland, S., Kalai, G.: Regular subgraphs of almost regular graphs. J. Comb. Theory Ser. B 37(1), 79–91 (1984)

    Article  MathSciNet  Google Scholar 

  2. Ashitha, T., Asir, T., Hoang, D.T., Pournaki, M.R.: Cohen–Macaulayness of a class of graphs versus the class of their complements. Discrete Math. 344(10), 112525 (2021)

    Article  MathSciNet  Google Scholar 

  3. Baclawski, K., Garsia, A.M.: Combinatorial decompositions of a class of rings. Adv. Math. 39(2), 155–184 (1981)

    Article  MathSciNet  Google Scholar 

  4. Bruns, W., Conca, A., Römer, T.: Koszul homology and syzygies of Veronese subalgebras. Math. Ann. 351(4), 761–779 (2011)

    Article  MathSciNet  Google Scholar 

  5. Corso, A., Nagel, U.: Monomial and toric ideals associated to Ferrers graphs. Trans. Am. Math. Soc. 361(3), 1371–1395 (2009)

    Article  MathSciNet  Google Scholar 

  6. Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebr. Comb. 38(1), 37–55 (2013)

    Article  MathSciNet  Google Scholar 

  7. Engström, A., Go, C., Stamps, M.T.: Betti numbers and anti-lecture hall compositions of random threshold graphs. Pac. J. Math. 319(1), 75–98 (2022)

    Article  MathSciNet  Google Scholar 

  8. Fröberg, R.: Betti numbers of fat forests and their Alexander dual. J. Algebr. Comb. 56(4), 1023–1030 (2022)

    Article  MathSciNet  Google Scholar 

  9. Grimaldi, R.P.: Graphs from rings. In: Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). Congressus Numerantium, vol. 71, pp. 95–103 (1990)

  10. Hà, H.T., Van Tuyl, A.: Resolutions of square-free monomial ideals via facet ideals: a survey. In: Algebra, Geometry and Their Interactions. Contemporary Mathematics, vol. 448. American Mathematical Society, Providence, pp. 91–117 (2007)

  11. Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2011)

    Book  Google Scholar 

  12. Hoang, D.T.: On the Betti numbers of edge ideals of skew Ferrers graphs. Int. J. Algebra Comput. 30(1), 125–139 (2020)

    Article  MathSciNet  Google Scholar 

  13. Hoang, D.T., Maimani, H.R., Mousivand, A., Pournaki, M.R.: Cohen–Macaulayness of two classes of circulant graphs. J. Algebr. Comb. 53(3), 805–827 (2021)

    Article  MathSciNet  Google Scholar 

  14. Jacques, S.: Betti numbers of graph ideals. PhD. thesis (2004). arXiv:math/0410107v1

  15. Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Comb. Theory Ser. A 113(3), 435–454 (2006)

    Article  MathSciNet  Google Scholar 

  16. Kimura, K.: Non-vanishingness of Betti numbers of edge ideals. In: Harmony of Gröbner Bases and the Modern Industrial Society. World Scientific Publishing, Hackensack, pp. 153–168 (2012)

  17. Martínez-Bernal, J., Pizá-Morales, O.A., Valencia-Bucio, M.A.: Nonvanishing Betti numbers of edge ideals of weakly chordal graphs. J. Algebr. Comb. 58(1), 279–290 (2023)

    Article  MathSciNet  Google Scholar 

  18. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

    Google Scholar 

  19. Peeva, I.: Graded Syzygies. Algebra and Applications, vol. 14. Springer, London (2011)

    Google Scholar 

  20. Rather, S.A., Singh, P.: On Betti numbers of edge ideals of crown graphs. Beitr. Algebra Geom. 60(1), 123–136 (2019)

    Article  MathSciNet  Google Scholar 

  21. Stanley, R.P.: Combinatorics and Commutative Algebra, vol. 41, 2nd edn. Birkhäuser Boston, Inc., Boston (1996)

    Google Scholar 

  22. Whieldon, G.: Jump sequences of edge ideals (2010). arXiv:1012.0108v1

Download references

Acknowledgements

A part of this work was completed during the third author’s visit to the Vietnam Institute for Advanced Study in Mathematics (VIASM). He wishes to express his gratitude toward VIASM for its support and hospitality. The research of D. T. Hoang was in part supported by a grant from Hanoi University of Science and Technology, Vietnam (Ref. T2023-PC-082). The research of T. Asir was in part supported by a grant from The Council of Scientific and Industrial Research, India (CSIR Project—Ref. 25/0323/23/EMR-II). The research of M. R. Pournaki was in part supported by a grant from The World Academy of Sciences, Italy (TWAS–UNESCO Associateship—Ref. 3240295905).

Author information

Authors and Affiliations

  1. Department of Mathematics, Deva Matha College, Kuravilangad, Kerala, 686 633, India

    T. Ashitha

  2. Department of Mathematics, Pondicherry University, Puducherry, 605 014, India

    T. Asir

  3. Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

    D. T. Hoang

  4. Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran

    M. R. Pournaki

Contributions

All authors have contributed equally to this work.

Corresponding author

Correspondence to D. T. Hoang.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest

Ethical Approval and Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Additional information

Communicated by Siamak Yassemi.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ashitha, T., Asir, T., Hoang, D.T. et al. Betti Numbers of Edge Ideals of Grimaldi Graphs and Their Complements. Bull. Malays. Math. Sci. Soc. 47, 136 (2024). https://doi.org/10.1007/s40840-024-01731-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-024-01731-2

Keywords

Mathematics Subject Classification