Homological properties of the edge ideals for trees with small diameter

[1] T. Ashitha, T. Asir, D.T. Hoang, and M.R. Pournaki, Betti numbers of edge ideals of Grimaldi graphs and their complements, Bull. Malays. Math. Sci. Soc. 47 (2024), no. 4, 136. https://doi.org/10.1007/s40840-024-01731-2

[2] D. Bayer, H. Charalambous, and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra 221 (1999), no. 2, 497–512. https://doi.org/10.1006/jabr.1999.7970

[3] M. Bigdeli and J. Herzog, Betti diagrams with special shape, Homological and Computational Methods in Commutative Algebra: Dedicated to Winfried Bruns on the Occasion of his 70th Birthday, Springer, 2017, pp. 33–52.
https://doi.org/10.1007/978-3-319-61943-9_2

[4] W. Bruns and H.J. Herzog, Cohen-Macaulay Rings, no. 39, Cambridge university press, 1998.

[5] A. Corso and U. Nagel, Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1371–1395.

[6] A. Engström, C. Go, and M.T. Stamps, Betti numbers and anti-lecture hall compositions of random threshold graphs, Pacific J. Math. 319 (2022), no. 1, 75–98. https://doi.org/10.2140/pjm.2022.319.75

[7] R. Fröberg, On Stanley–Reisner rings, Topics in Algebra, Part 2 (Warsaw, 1988) 26 (1990), no. 2, 57–70.

[8] D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry.

[9] N.T. Hang and T. Vu, Projective dimension and regularity of 3-path ideals of unicyclic graphs, Graphs Combin. 41 (2025), no. 1, 18. https://doi.org/10.1007/s00373-024-02877-3

[10] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.

[11] J. Herzog and G. Rinaldo, On the extremal Betti numbers of binomial edge ideals of block graphs, Electron. J. Combin. 25 (2018), no. 1, #P1.63. https://doi.org/10.37236/7689

[12] T. Hibi, K. Kimura, and K. Matsuda, Extremal Betti numbers of edge ideals, Arch. Math. 113 (2019), no. 2, 149–155.
https://doi.org/10.1007/s00013-019-01322-9

[13] T. Hibi, K. Kimura, and S. Murai, Betti numbers of chordal graphs and $f$-vectors of simplicial complexes, J. Algebra 323 (2010), no. 6, 1678–1689. https://doi.org/10.1016/j.jalgebra.2009.12.029

[14] M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in” ring theory II”, Lect. Notes in Pure Appl. Math. (1977), no. 26, 171–223.

[15] S. Jacques, Betti numbers of graph ideals, Ph.D. thesis, The University of Sheffield, Western Bank, Sheffield S10 2TN, United Kingdom, 2004.

[16] M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), no. 3, 435–454. https://doi.org/10.48550/arXiv.math/0408016

[17] K. Kimura, Non-vanishingness of Betti numbers of edge ideals, Harmony of Gröbner bases and the modern industrial society, World Scientific, 2012, pp. 153–168. https://doi.org/10.1142/9789814383462_0009

[18] D.N. Kozlov, Combinatorial Algebraic Topology, vol. 21, Springer, Berlin, 2008.

[19] J. Martínez-Bernal, O.A. Pizá-Morales, and M.A. Valencia-Bucio, Nonvanishing Betti numbers of edge ideals of weakly chordal graphs, J. Algebraic Combin. 58 (2023), no. 1, 279–290. https://doi.org/10.1007/s10801-023-01248-0

[20] F. Mohammadi and S. Moradi, Resolution of unmixed bipartite graphs, Bull. Malays. Math. Sci. Soc. 52 (2015), no. 3, 977–986. https://doi.org/10.4134/BKMS.2015.52.3.977

[21] H.D. Nguyen and T. Vu, Linearity defect of edge ideals and Fröberg’s theorem, J. Algebraic Combin. 44 (2016), no. 1, 165–199. https://doi.org/10.1007/s10801-015-0662-6

[22] B.A. Rather, Betti numbers of edge ideals of some graphs with application to graphs assigned to groups, Filomat 38 (2024), no. 6, 2185–2204. https://doi.org/10.2298/FIL2406185R

[23] B.A. Rather, M Brunetti, and J.F. Wang, Betti numbers of chain edge ideals and some cozero divisor graphs, submitted.

[24] B.A. Rather, A. Diene, and M. Imran, Linear strand of edge ideals of comaximal graphs of commutative rings, Comm. Algebra 52 (2024), no. 4, 1486–1500. https://doi.org/10.1080/00927872.2023.2263559

[25] M. Roth and A.V. Tuyl, On the linear strand of an edge ideal, Comm. Algebra 35 (2007), no. 3, 821–832.
https://doi.org/10.1080/00927870601115732

[26] P. Singh and R. Verma, Betti numbers of edge ideals of some split graphs, Comm. Algebra 48 (2020), no. 12, 5026–5037. https://doi.org/10.1080/00927872.2020.1777559

[27] R. Verma, On some algebraic properties of edge ideals of ladder graphs, Comm. Algebra 50 (2022), no. 6, 2296–2311.
https://doi.org/10.1080/00927872.2021.2006206

[28] R.H. Villarreal, Monomial Algebras, Chapman and Hall/CRC, Boca Raton, FL, 2015.

[29] R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra 6 (2014), no. 2, 287–304. https://doi.org/10.1216/JCA-2014-6-2-287