Skip to main content
SpringerLink
Log in

Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

In this paper, we introduce the unbalanced crown graphs \({\mathcal {C}}_{m,n}\), and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained.

This is a preview of subscription content, log in via an institution to check access.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Fernández-Ramos, O., Gimenez, P.: Regularity 3 in edge ideals associated to bipartite graphs. J. Algebr. Combin. 39(4), 919–937 (2014)

    Article  MathSciNet  Google Scholar 

  2. Fröberg, R.: On Stanley–Reisner rings. Topics in algebra, Part 2 (Warsaw, 1988), pp. 57–70, Banach Center Publ., vol. 26, Part 2, PWN, Warsaw (1990)

  3. Grayson, D., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/

  4. Hochster, M.: Cohen–Macaulay rings, combinatorics, and simplicial complexes. Ring theory II. Proc. Second Conf., Univ. Oklahoma, Norman, pp. 171–223 (1975)

  5. Jacques, S.: Betti numbers of graph ideals, Ph.D. thesis, University of Sheffield, Great Britain

  6. Lyubeznik, G.: The minimal non-Cohen–Macaulay monomial ideals. J. Pure Appl. Algebra 51, 261–266 (1988)

    Article  MathSciNet  Google Scholar 

  7. Rather, S.A., Singh, P.: Graded Betti numbers of crown edge ideals. Commun. Algebra 47, 1690–1698 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Singh, P., Rather, S.A.: On minimal free resolution of edge ideals of multipartite crown graphs. Commun. Algebra 48, 1314–1326 (2020)

    Article  MathSciNet  Google Scholar 

  10. Villarreal, R.H.: Monomial Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 2nd edn. CRC Press, Boca Raton (2015)

    Google Scholar 

  11. Wegner, G.: \(d\)-collapsing and nerves of families of convex sets. Arch. Math. (Basel) 26, 317–321 (1975)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous referee(s) for his valuable comments and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Central University of Kashmir, Ganderbal, 191201, J&K, India

    Shahnawaz Ahmad Rather

Authors
  1. Shahnawaz Ahmad Rather
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shahnawaz Ahmad Rather.

Additional information

Communicated by Rashid Zaare Nahandi.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rather, S.A. Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs. Bull. Iran. Math. Soc. 48, 1023–1035 (2022). https://doi.org/10.1007/s41980-021-00560-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-021-00560-4

Keywords

Mathematics Subject Classification

Search

Navigation