EDGE IDEALS OF SQUARES OF TREES

[1] A. Adamaszek and M. Adamaszek: Uniqueness of graph square roots of girth six, Electron. J. Combin. 18 (2011), Paper 139, 5pp.

[2] D. Aingworth, R. Motwani and F. Harary: The difference between a graph and its square, Util. Math. 54 (1998), 223–228.

[3] M. Cheng and G.J. Chang: Families of graphs closed under taking powers, Graphs and Combin. 17 (2001), 207–212.

[4] H. Dao, C. Huneke and J. Schweig: Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), 37–55.

[5] G. Fan and R. Haggkvist: ¨ The square of a Hamiltonian cycle, SIAM J. Discrete Math. 7 (1994), 203–212.

[6] G. Fan and H.A. Kierstead: The square of paths and cycles, J. Combin Theorey Ser. B, 63 (1995), 55–64.

[7] R. Froberg: ¨ On Stanley-Reisner rings; in Topics in algebra, Part 2 (Warsaw, 1988), Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990, 57–70.

[8] H.T. Ha and A. Van Tuyl: ` Resolutions of square-free monomial ideals via facet ideal: a survey; in Algebra, geometry and their interactions, Contemporary Mathematics 488, Amer. Math. Soc. Providence, RI, 2007, 91–117.

[9] H.T. Ha and A. Van Tuyl: ` Monomial ideals, edge ideals of hypergraphs, and their Betti numbers, J. Algebraic Combin. 27 (2008), 215–245.

[10] F. Harary and I.C. Ross: The square of a tree, Bell System Tech. J. 39 (1960), 641–647.

[11] J. Herzog and T. Hibi: Monomial ideals, Graduate Texts in Mathematics, 260, Springer-Verlag, London, 2011.

[12] J. Herzog, T. Hibi and X. Zheng: Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), 23–32.

[13] J. Herzog and Y. Takayama: Resolutions by mapping cones; in The Roos Festschrift, vol. 2, Homology, Homotopy Appl. 4 (2002), 277–294.

[14] A. Hirano and K. Matsuda: Matching numbers and dimension edge ideals, Grahs and Combin. 37 (2021), 761–774.

[15] Z. Iqbal and M. Ishaq: Depth and Stanley depth of the edge ideals of the powers of paths and cycles, An. S¸tiint. Univ. “Ovidius” Constant¸a, Ser. Mat. 27 (2019), 113–135.

[16] G. Kalai and R. Meshulam: Intersections of Leray complexes and regularity of monomial ideals, J. Combin. Theory Ser. A 113 (2006), 1586–1592.

[17] M. Katzmann: Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), 435–454.

[18] K. Kimura: Non-vanishingness of Betti Numbers of Edge Ideals; in Harmony of Grobner Bases and in the ¨ Modern Industry Society, World Sci. Publ., Hackensack, NJ, 2012, 153–168.

[19] V.B. Le and N.N. Tuy: The square of a block graph, Discrete Math. 310 (2010), 734–741.

[20] V.B. Le and N.N. Tuy: A good characterization of squares of strongly chordal split graphs, Inform Process. Lett. 111 (2011), 120–123.

[21] A. Lubiw: Γ-Free Matrices. Master Thesis, University of Waterloo, 1982.

[22] S. Morey: Depths of powers of the edge ideal of a tree, Comm. Algebra 38 (2010), 4042–4055.

[23] S. Morey and R. Villarreal: Edge ideals: Algebraic and combinatorial property, in Progress in Commutative Algebra 1, de Gruyter, Berlin, 2012, 85–126.

[24] A. Raychaudhuri: On powers of interval and unit interval graphs, Congr. Numer. 59 (1987), 235–242.

[25] A. Raychaudhuri: On powers of strongly chordal graphs and circular arc graphs. Ars Combin. 34 (1992), 147–160.

[26] R. Scheidweiler and S. Wiederrecht: On chordal graph and line graph squares Discrete Appl. Math. 243 (2018), 239–247.

[27] R. Villareal: Monomial Algebras, 2nd Ed, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018.

[28] R. Woodroofe: Matchings, coverings, and Castelnuovo Mumford regularity, J. Commut. Algebra 6 (2014), 287–304.