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.
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References
-
Alon, N., Friedland, S., Kalai, G.: Regular subgraphs of almost regular graphs. J. Comb. Theory Ser. B 37(1), 79–91 (1984)
-
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)
-
Baclawski, K., Garsia, A.M.: Combinatorial decompositions of a class of rings. Adv. Math. 39(2), 155–184 (1981)
-
Bruns, W., Conca, A., Römer, T.: Koszul homology and syzygies of Veronese subalgebras. Math. Ann. 351(4), 761–779 (2011)
-
Corso, A., Nagel, U.: Monomial and toric ideals associated to Ferrers graphs. Trans. Am. Math. Soc. 361(3), 1371–1395 (2009)
-
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)
-
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)
-
Fröberg, R.: Betti numbers of fat forests and their Alexander dual. J. Algebr. Comb. 56(4), 1023–1030 (2022)
-
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)
-
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)
-
Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2011)
-
Hoang, D.T.: On the Betti numbers of edge ideals of skew Ferrers graphs. Int. J. Algebra Comput. 30(1), 125–139 (2020)
-
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)
-
Jacques, S.: Betti numbers of graph ideals. PhD. thesis (2004). arXiv:math/0410107v1
-
Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Comb. Theory Ser. A 113(3), 435–454 (2006)
-
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)
-
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)
-
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
-
Peeva, I.: Graded Syzygies. Algebra and Applications, vol. 14. Springer, London (2011)
-
Rather, S.A., Singh, P.: On Betti numbers of edge ideals of crown graphs. Beitr. Algebra Geom. 60(1), 123–136 (2019)
-
Stanley, R.P.: Combinatorics and Commutative Algebra, vol. 41, 2nd edn. Birkhäuser Boston, Inc., Boston (1996)
-
Whieldon, G.: Jump sequences of edge ideals (2010). arXiv:1012.0108v1
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).
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Communicated by Siamak Yassemi.
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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
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DOI: https://doi.org/10.1007/s40840-024-01731-2