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The strongly regular graph with parameters \((100,22,0,6)\): hidden history and beyond. (English) Zbl 1430.05137
Written for the community of algebraic graph theorists, this elaborate and endearingly indulgent tribute to Dale Marsh Mesner (1923–2009) recovers Mesner’s construction in his 1956 thesis and a 1964 set of mimeographed notes of the titular strongly regular graph with parameters \((100,22,0,6)\) and explores a rich collection of associated ideas. Over a decade after Mesner’s initial research, this graph was independently constructed by D. G. Higman and C. C. Sims [Math. Z. 105, 110–113 (1968; Zbl 0186.04002)] alongside a sporadic simple group that marked a milestone in the program to classify all finite simple groups (see [A. Steingart, “A group theory of group theory”, Soc. Stud. Sci. 42, No. 2, 185–213 (2012); doi:10.1177/0306312712436547)]. Examples and illustrations join an extensive biobliography supporting the authors’ effusive and wide-ranging appreciation of Mesner’s work, which the authors regret was not better recognized and appreciated in its time. Those with the requisite familiarity with the relevant branches of combinatorics will find a stimulating perspective on the field inspired by the authors’ association with Mesner. The paper may be read in segments, and begins with a succinct introduction, preliminary conceptual survey, and overview to orient the reader.
MSC:
05E30 Association schemes, strongly regular graphs
05C51 Graph designs and isomorphic decomposition
05B05 Combinatorial aspects of block designs
05B07 Triple systems
05-03 History of combinatorics
01A60 History of mathematics in the 20th century
Software:
GAP; GRAPE; nauty
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References:
[1] E. Artin, “Geometric Algebra”, Interscience Publishers, Inc., New York, 1957. · Zbl 0077.02101
[2] M. Aschbacher, “Sporadic Groups”, Cambridge Univ. Press, Cambridge, 1994. · Zbl 0804.20011
[3] E. F. Assmus, Jr. and H. F. Mattson, Jr., Perfect codes and the Mathieu groups, Arch. Math. 17 (1966), 122–135.
[4] E. F. Assmus, Jr. and C. J. Salwach, The (16, 6, 2)-designs, Internat. J. Math. & Math. Sci. 2 (1979), 261–281.
[5] A. Baartmans and M. S. Shrikhande, Designs with no three mutually disjoint blocks, Discrete Math. 40 (2-3) (1982), 129–139. · Zbl 0488.05009
[6] B. Bagchi, No extendable biplane of order nine, J. Combin. Theory Ser. A 49 (1988), 1–12. · Zbl 0662.05010
[7] B. Bagchi, Corrigendum, J. Combin. Theory Ser. A 57 (1991), p.162.
[8] B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly and A. Schürmann, Experimental study of energy-minimizing point confgurations on spheres, Experiment. Math. 18 (3) (2009), 257–283.
[9] E. Bannai and T. Ito, “Algebraic Combinatorics I. Association Schemes”, Benjamin/Cummings, Menlo Park, 1984. · Zbl 0555.05019
[10] E. Bannai, R. L. Griess, Jr., C. E. Praeger and L. Scott, The mathematics of Donald Gordon Higman, Michigan Math. J, 58 (1) (2009), 3–30. · Zbl 1165.01309
[11] R. A. Bailey, “Association Schemes: Designed Experiments, Algebra and Combinatorics”, Cambridge University Press, Cambridge, 2004. · Zbl 1051.05001
[12] D. J. Bergstrand, New uniqueness proofs for the (5, 8, 24), (5, 6, 12) and related Steiner systems, J. Combin. Theory Ser. A 33 (3) (1982), 247–272.
[13] T. Beth, D. Jungnickel and H. Lenz, “Design Theory”, Cambridge Univ. Press, New York, London, 1993.
[14] Bhagwandas and S. S. Shrikhande, Seidel-equivalence of strongly regular graphs, Sankhy¯a: The Indian Journal of Statistics Series A (1961-2002) Vol. 30, No. 4 (1968), 359-368.
[15] N. L. Biggs, “Algebraic Graph Theory”, Second edition (First edition 1974), Cambridge University Press, Cambridge, 1993. · Zbl 0797.05032
[16] N. L. Biggs, Strongly regular graphs with no triangles. Research Report, September 2009, arXiv:0911.2160v1.
[17] N. L. Biggs and A. T. White, “Permutation Groups and Combinatorial Structures”, London Math. Soc. Lec. Note Series 33, Cambridge Univ. Press, London, 1979. · Zbl 0415.05002
[18] R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (2) (1963), 389–419. · Zbl 0118.33903
[19] R. C. Bose and D. M. Mesner, On linear associative algebras corresponding to association schemes of partially balanced designs, Ann. Math. Statist. 30 (1) (1959), 21–38. · Zbl 0089.15002
[20] R. C. Bose and K. R. Nair, Partially balanced incomplete block designs, Sankhya 4 (1939), 337–372. 54Mikhail H. Klin, Andrew J. Woldar
[21] R. C. Bose and T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Statist. Assoc. 47 (1952), 151–184. · Zbl 0048.11603
[22] A. E. Brouwer, The uniqueness of the strongly regular graph on 77 points, Math. Centr. Report ZW147, Amsterdam (Nov. 1980), J. Graph Th. 7 (1983), 455–461. · Zbl 0523.05021
[23] A. E. Brouwer, A slowly growing collection of graph descriptions, accessed July-2016, http://www.win.tue.nl/ aeb/graphs/index.html.
[24] A. E. Brouwer and A. M. Cohen, A. Neumaier, “Distance-Regular Graphs”, SpringerVerlag, New York, 1989. · Zbl 0747.05073
[25] A. E. Brouwer, W. H. Haemers, Structure and uniqueness of the (81,20,1,6) strongly regular graph, Discrete Math.106/107 (1992), 77–82. · Zbl 0764.05098
[26] A. E. Brouwer, W. H. Haemers, “Spectra of Graphs”, Springer, Berlin, 2012. · Zbl 1231.05001
[27] A. E. Brouwer and J. H. van Lint, Strongly regular graphs and partial geometries, in: D. M. Jackson, S. A. Vanstone (eds.), “Enumeration and Design”, Academic Press, London, 1984, 85–122.
[28] R. A. Brualdi and H. J. Ryser, “Combinatorial Matrix Theory”, Encyclopedia of Mathematics, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0746.05002
[29] R. H. Bruck, Finite nets. I. Numerical invariants, Canad. J. Math. 3 (1951), 94–107. · Zbl 0042.38802
[30] R. H. Bruck, Finite nets. II. Uniqueness and imbedding. Pacific J. Math. 13 (1963), 421– 457. · Zbl 0124.00903
[31] A. A. Bruen and J. W. P. Hirschfeld, Applications of line geometry over finite fields, II. The Hermitian surface, Geom. Dedicata 7 (3) (1978), 333–353. · Zbl 0394.51006
[32] J. M. J. Buczak, “Some topics in group theory”, Ph.D. thesis, Oxford University, 1980.
[33] F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory Ser. A 27 (1979), 121–151. · Zbl 0419.51003
[34] F. C. Bussemaker, W. H. Haemers, R. Mathon and H. A. A. Wilbrink, (49, 16, 3, 6) strongly regular graph does not exist, European J. Combin. 10 (5) (1989), 413–418.
[35] A. R. Calderbank and P. Morton, Quasi-symmetric 3-designs and elliptic curves, SIAM J. Discrete Math. 3 (2) (1990), 178–196. · Zbl 0742.05013
[36] P. J. Cameron, Extending symmetric designs, J. Combin. Theory Ser. A 14 (2) (1973), 215–220.
[37] P. J. Cameron, Biplanes, Math. Zeitschrift 131 (1973), 85–101.
[38] P. J. Cameron, Locally symmetric designs, Geom. Dedicata 3 (1974), 65–76. · Zbl 0284.05014
[39] P. J. Cameron, Partial quadrangles, Quart. J. Math. Oxford 26 (1975), 61–73. · Zbl 0301.05009
[40] P. J. Cameron, 6-transitive graphs, J. Combin. Theory Ser. B 28 (1980), 168–179. · Zbl 0451.05038
[41] P. J. Cameron, “Permutation Groups”, London Mathematical Society Student Texts, 45, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0922.20003
[42] P. J. Cameron and J. H. van Lint, “Designs, Graphs, Codes and their Links”, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0743.05004
[43] R. D. Carmichael, Tactical configurations of rank two, Amer. J. Math. 53 (1931), 217–240. · JFM 57.0110.04
[44] R. D. Carmichael, “Introduction to the Theory of Groups of Finite Order”, Ginn and Co., 1937. · Zbl 0019.19702
[45] W. H. Clatworthy, Partially balanced incomplete block designs with two associate classes and two treatments per block, J. Res. Nat. Bureau Stds. 54 (1955), 177–190. · Zbl 0068.13403
[46] W. H. Clatworthy, Tables of two-associate partially balanced designs, National Bureau of Standards, Washington D.C., 1973. Acta Univ. M. Belii, ser. Math. 25 (2017), 5–6255
[47] A. Clebsch, Über die Flachen vierter Ordung, welche eine Doppelcurve zweiten Grades besitzen, J. für Math. 69 (1868), 42–84.
[48] H. Cohn and A. Kumar, The densest lattice in twenty-four dimensions, Electron. Res. Announc. Amer. Math. Soc. 10, S 1079-6762(04)00130-110, 2004.
[49] C. J. Colbourn and J. H. Dinitz, “Handbook of Combinatorial Designs”, 2nd edition, Chapman & Hall/CRC, 2007. · Zbl 1101.05001
[50] W. S. Connor, The uniqueness of the triangular association scheme, Ann. Math. Stat. 29 (1958), 262–266. · Zbl 0085.35601
[51] W. S. Connor, W. H. Clatworthy, Some theorems for partially balanced designs, Ann. Math. Stat. 25 (1954), 100–112. · Zbl 0055.13303
[52] J. H. Conway, A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398–400. · Zbl 0186.32401
[53] J. H. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 79 (1) (1969), 79–88. · Zbl 0186.32304
[54] J. H. Conway and N. J. A. Sloane, “Sphere Packings, Lattices and Groups”, 3rd edition, Springer-Verlag, New York, Berlin, 1999. · Zbl 0915.52003
[55] G. M. Conwell, The 3-space P G(3, 2) and its group, Ann. Math. 11 (1910), 60–76.
[56] H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413–455. · Zbl 0040.22803
[57] R. T. Curtis, A new combinatorial approach to M24, Math. Proc. Cambridge Philos. Soc. 79 (1979), 25–42.
[58] R. T. Curtis, Symmetric generation of the Higman-Sims group, J. Algebra 171 (1995), 567–586. · Zbl 0824.20014
[59] D. M. Cvetković, M. Doob and H. Sachs, “Spectra of Graphs”, 3rd edition, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995.
[60] E. R. van Dam and M. Muzychuk, Some implications on amorphic association schemes, J. Combin. Theory Ser. A 117 (2) (2010), 111–127. · Zbl 1286.05183
[61] A. Delandtsheer, Finite geometric lattices with highly ransitive automorphism groups, Arch. Math. 42 (1984), 376–383. · Zbl 0519.06006
[62] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. No. 10, 1973.
[63] P. Dembowski, “Finite Geometries”, Springer-Verlag, New York, 1968. · Zbl 0159.50001
[64] J. D. Dixon and B. Mortimer, “Permutation Groups”, Graduate Texts in Mathematics, vol. 163, Springer, Berlin, Heidelberg, New York, 1996. · Zbl 0951.20001
[65] W. L. Edge, The geometrical construction of Maschke’s quartic surfaces, Proc. Edinburgh Math. Soc. Series 2, Vol. 7, Part II (1945), 93–103. · Zbl 0063.01214
[66] W. L. Edge, Conics on a Maschke surface, Proc. Edinburgh Math. Soc. Series 2, Vol. 7 Part III (1946), 153–161. · Zbl 0063.01215
[67] W. L. Edge, The Kummer quartic and the tetrahedroid based on the Maschke forms, Proc. Cambridge Philos. Soc. London A 189 (1947), 326–358.
[68] W. L. Edge, The geometry of the linear fractional group LF (4, 2), Proc. London Math. Soc. 3-4 (1954), 317–342.
[69] W. L. Edge, Some implications of the geometry of the 21-point plane, Math. Zeitschrift 87 (1965), 348–362. · Zbl 0136.15303
[70] W. L. Edge, A new look at the Kummer surface, Canad. J. Math. 19 (1967), 952–967. 56Mikhail H. Klin, Andrew J. Woldar · Zbl 0156.41304
[71] A. Emch, Triple and multiple systems, their geometric configurations and groups, Trans. Amer. Math. Soc. 31 (1929), 25–42. · JFM 55.0062.05
[72] I. A. Faradžev and M. H. Klin, Computer package for computations with coherent configu- rations, in: S. M. Watt (ed.), “Proceedings ISSAC-91”, Bonn, ACM Press, 1991, 219–223.
[73] I. A. Faradžev, M. H. Klin and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, in: I. A. Faradžev, A. A. Ivanov, M. H. Klin, A. J. Woldar (eds.), “Investigations in Algebraic Theory of Combinatorial Objects”, Kluwer Publ., Dordrecht, 1994.
[74] M. A. Fiol and E. Garriga, On outindependent subgraphs of strongly regular graphs, Linear and Multilinear Alg. 54 (2) (2006), 123–140. · Zbl 1114.05100
[75] J. S. Frame, Computation of characters of the Higman-Sims group and its automorphism group, J. Algebra 20 (1972), 320–349. · Zbl 0226.20009
[76] D. Garbe and J. Mennicke, Some remarks on the Mathieu groups, Canad. Math. Bull. 7 (2) (1964), 201–212. · Zbl 0129.01802
[77] A. Gardiner, Antipodal covering graphs, J. Combin. Theory Ser. B 16 (1974), 255–273. · Zbl 0267.05111
[78] A. D. Gardiner, C. D. Godsil, A. D. Hensel and G. F. Royle, Second neighbourhoods of strongly regular graphs, Discrete Math. 103 (2) (1992), 161–170. · Zbl 0764.05100
[79] A. L. Gavrilyuk and A. Makhnev, On Krein graphs without triangles, Doklady Math. 72 (1) (2005), 591–594. · Zbl 1126.05102
[80] A. Gewirtz, The uniqueness of g(2, 2, 10, 56), Trans. New York Acad. Sci. 31 (1969), 656– 675.
[81] A. Gewirtz, Graphs with maximal even girth, Canad. J. Math. 21 (1969), 915–934. · Zbl 0181.51801
[82] G. Glauberman, Normalizers of p-subgroups in finite groups, Pacific J. Math. 29 (1) (1969), 137–144.
[83] C. Godsil and G. Royle, “Algebraic Graph Theory”, Springer-Verlag, New York, 2001. · Zbl 0968.05002
[84] J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math. 19 (1967), 1001–1010. · Zbl 0155.35601
[85] J. M. Goethals and J. J. Seidel, Quasisymmetric block designs, in: R. Guy (ed.), “Combinatorial Structures and Their Applications”, 111–116, Gordon-Breach, New York, 1970.
[86] J. M. Goethals and J. J. Seidel, Strongly regular graphs derived from combinatorial designs, Canad. J. Math. 22 (1970), 597–614. · Zbl 0198.29301
[87] M. J. E. Golay, Notes on digital coding, Proc. IEEE 37 (1949), p.657.
[88] R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955), 1–7. · Zbl 0064.17901
[89] R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1–102. · Zbl 0498.20013
[90] L. C. Grove, “Classical Groups and Geometric Algebra”, Graduate Studies in Mathematics Vol. 39, Amer. Math. Soc., Providence, R.I., 2002.
[91] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (eds.),“Algebraic Combinatorics and Applications”, Springer, Berlin, 2001, 196–211. · Zbl 0974.05054
[92] W. H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra & Applic., Vol. 226228 (1995), 593–616. · Zbl 0831.05044
[93] W. H. Haemers, Strongly regular graphs with maximal energy, Linear Algebra & Applic. 429 (11-12) (2008), 2719–2723.
[94] P. R. Hafner, On the graphs of Hoffman-Singleton and Higman-Sims, Electron. J. Com- bin.11 (1), Research Paper 77, 2004. Acta Univ. M. Belii, ser. Math. 25 (2017), 5–6257 · Zbl 1060.05073
[95] P. de la Harpe, Spin models for link polynomials, strongly regular graphs and Jaeger’s Higman-Sims model, Pacific J. Math. 162 (1994), 57–96. · Zbl 0795.57002
[96] A. Heinze and M. Klin, Loops, Latin squares and strongly regular graphs: An algorithmic approach via algebraic combinatorics, in (M. Klin et al eds.) “Algorithmic Algebraic Combinatorics and Gröbner Bases”, Springer-Verlag Berlin, 2009, pp.3-65. · Zbl 1186.05036
[97] M. D. Hestenes and D. G. Higman, Rank 3 groups and strongly regular graphs, SIAM Amer. Math. Soc. Proc. 4 (1971), 141–160.
[98] D. G. Higman, Finite permutation groups of rank 3, Math. Zeitschrift 86 (1964), 145–156. · Zbl 0122.03205
[99] D. G. Higman, Intersection matrices for finite permutation groups, J. Algebra 6 (1967), 22–42. · Zbl 0183.02704
[100] D. G, Higman, “Lectures on Permutation Representations”, notes taken by Wolfgang Hauptmann Vorlesungen aus dem Mathematischen Institut Giessen, Heft 4. Mathematisches Institut Giessen, Giessen, 1977, 106 pp. · Zbl 0365.20001
[101] D. G. Higman, C. C. Sims, A simple group of order 44, 352, 000, Math./ Zeitschrift 105 (1968), 110–113.
[102] G. Higman, On the simple group of D. G. Higman and C. C. Sims, Illinois J. Math. 13 (1969), 74–80. · Zbl 0165.04001
[103] J. W. P. Hirschfeld, “Projective Geometries over Finite Fields”, 2nd edition, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1998. · Zbl 0899.51002
[104] G. Hiss, Die sporadischen Gruppen, Jahresber. Deutsch. Math.-Verein. 105 (2003), 169– 193. · Zbl 1042.20007
[105] A. J. Hoffman and R. R. Singleton, On Moore graphs of diameter two and three, IBM J. Res. Develop. 4 (1960), 497–504, · Zbl 0096.38102
[106] R. W. H. T. Hudson, “Kummer’s Quartic Surface”, Cambridge 1905, revised reprint: Cambridge Univ. Press, Cambridge, 1990.
[107] D. R. Hughes, On t-designs and groups, Amer. J. Math. 87 (1965), 761–778. · Zbl 0134.03004
[108] D. R. Hughes, Semi-symmetric 3-designs, in: C. A. Baker, L. M. Batten (eds.), “Finite Geometries”, Lec. Notes in Pure and Appl. Math., Vol. 82, Marcel Dekker, New York, 1982, 223–235.
[109] D. R. Hughes and F. C. Piper, “Design Theory”, Cambridge Univ. Press, Cambridge, 1985.
[110] Q. M. Hussain, On the totality of the solutions for the symmetrical incomplete block designs with λ = 2, k = 5 or 6, Sankhya 7 (1945), 204–208.
[111] Q. M. Hussain, Structure of some incomplete block designs, Sankhya (1948), 381–383. · Zbl 0040.36202
[112] Y. J. Ionin and M. S. Shrikhande, “Combinatorics of Symmetric Designs”, Cambridge Univ. Press, Cambridge, 2006. · Zbl 1114.05001
[113] A. A. Ivanov, “Geometry of Sporadic Groups I. Petersen and Tilde Geometries”, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0933.51006
[114] A. A. Ivanov and S. V. Shpectorov, “Geometry of Sporadic Groups II. Representations and Amalgams”, Cambridge Univ. Press, Cambridge, 2002. · Zbl 0992.51006
[115] S. Iwasaki, An elementary and unified approach to the Mathieu-Witt systems II: The uniqueness of W22, W23, W24, Hokkaido Mat. J. 21 (1992), 239–250.
[116] F. Jaeger, Strongly regular graphs and spin models for the Kauffman polynomial, Geom. Dedicata 44 (1) (1992), 23–52. · Zbl 0773.57005
[117] T. B. Jajcayová and R. Jajcay, On the contributions of Dale Marsh Mesner, Bull. Inst ˙Combin. Appl. 36 (2002), 46–52. 58Mikhail H. Klin, Andrew J. Woldar
[118] T. B. Jajcayová, R. Jajcay and E .S. Kramer, The Life and Mathematics of Dale Marsh Mesner 1923 - 2009, Bull. Inst. Combin. Appl. 59 (2010), 9–30.
[119] Z. Janko and S. K. Wong, A characterization of the Higman-Sims simple group, J. Algebra 13 (1969), 517–534. · Zbl 0184.04602
[120] Z. Janko and T. van Trung, A new biplane of order 9 with a small automorphism group, J. Combin. Theory Ser. A 42 (2) (1986), 305–309. · Zbl 0656.05019
[121] V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math., 137 (1989), 311–334. · Zbl 0695.46029
[122] W. Jónsson, On the Mathieu groups M22, M23, M24and the uniqueness of the associated Steiner systems, Math. Zeitschrift 125 (1972), 193–214.
[123] L. K. Jørgensen and M. Klin, Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices. Electronic J. Combin. 10 (2003), Research Paper 17, 31 pp.
[124] J. G. Kalbfleisch and R. G. Stanton, On the maximal triangle-free edge-chromatic graphs in three colors, J. Combin. Theory 5 (1968), 9–20. · Zbl 0164.24702
[125] L. A. Kalužnin and M. H. Klin, On certain maximal subgroups of symmetric and alternating groups, Mat. Sbornik 87 (129) (1972), 91–121. (in Russian.)
[126] W. M. Kantor, Dimension and embedding theorems for geometric lattices, J. Combin. Theory A 17 (1974), 173–195. · Zbl 0302.06018
[127] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with k = 11, J. Combin. Designs 16 (2008), 117–127. · Zbl 1140.05012
[128] J. Key and V. Tonchev, Computational results for the known biplanes of order 9, in: J. W. P. Hirschfeld, S. S. Magliveras, M. J. de Resmini (eds.), “Geometry, Combinatorial Designs and Related Structures”, London Math. Soc. Lecture Note Ser. vol. 245, 1997, 113–122. · Zbl 0897.05011
[129] R. A. Kingsley and R. G. Stanton, A survey of certain balanced incomplete block designs, Proc. Third Southeastern Conf. Combin. (1971), 359–368.
[130] M. Klin and Š. Gyürki, “Selected Topics from Algebraic Graph Theory”, Lecture notes, Belianum, Vydavatel’stvo Univerzity Mateja Bela, Banská Bystrica, 2015, xi+209 pp.
[131] M. Klin, D. Mesner, and A. Woldar, A combinatorial approach to transitive extensions of generously unitransitive permutation groups, J. Combin. Designs 18 (5) (2010), 369–391. · Zbl 1205.05250
[132] M. Klin, D. Mesner and A. Woldar, On the combinatorics of certain classes of transitive extensions, preprint. · Zbl 1205.05250
[133] M. Klin, M. Meszka, S. Reichard and A. Rosa, The smallest non-rank 3 strongly regular graphs which satisfy the 4-vertex condition, Bayreuth. Math. Schr. 74 (2005), 145–205. · Zbl 1100.05102
[134] M. Klin, M. Muzychuk, C. Pech, A. Woldar, and P.-H. Zieschang, Association schemes on 28 points as fusions of a half-homogeneous coherent configuration, Europ. J. Combin. 28 (7) (2007), 1994–2025. · Zbl 1145.05056
[135] M. Klin, M. Muzychuk and M. Ziv-Av, Higmanian rank-5 association schemes on 40 points, Mich. Math. J. 58 (1) (2009), 255–284. · Zbl 1284.05339
[136] M. Klin, C. Pech, S. Reichard, A. Woldar and M. Ziv-Av, Examples of computer experimentation in algebraic combinatorics, Ars Mathematica Contemporanea 3 (2010), 237-258. · Zbl 1227.05274
[137] M. Klin, R. Pöschel and K. Rosenbaum, “Angewandte Algebra für Mathematiker und Informatiker. Einführung in gruppentheoretisch-kombinatorische Methoden”, VEB Deutscher Verlag der Wissenschaften. Berlin, 1988. (in German.)
[138] M. Klin, C. Rücker, G. Rücker and G. Tinhofer, “Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras”, Match (40) (1999), 7-138. Acta Univ. M. Belii, ser. Math. 25 (2017), 5–6259
[139] M. Klin and A. Woldar, Dale Mesner, Higman & Sims, and the strongly regular graph with parameters (100,22,0,6), Bull. ICA 63 (2011), 13–35. · Zbl 1251.05180
[140] M. Klin and M. Ziv-Av, Computer algebra experimentation with Higmanian rank 5 association schemes on 40 vertices and related combinatorial objects, Technical report, 2007.
[141] M. Klin and M. Ziv-Av, Computer algebra investigation of known primitive triangle-free strongly regular graphs, in: L. Kovacs and T. Kutsia (eds.) SCSS 2013 (EPIC Series, vol. 15), pp. 108–123.
[142] M. Klin and M. Ziv-Av, A non-Schurian coherent configuration on 14 points exists, Designs Codes and Cryptography 84 (2017), 203–221. · Zbl 1367.05216
[143] R. Kochendorfer, “Lehrbuch der Gruppentheorie unter besonderer BerÃijcksichtigung der endlichen Gruppen”, Akad. Verlagsges, Geest & Pertig, Leipsig, 1966.
[144] E. S. Kramer and D. M. Mesner, Intersections among Steiner systems, J. Combin. Theory Ser. A 16 (1974), 273–285. · Zbl 0282.05011
[145] C. W. H. Lam, L. Thiel and S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123. · Zbl 0691.51003
[146] J. Lauri and R. Scapellato, “Topics in Graph Automorphisms and Reconstruction”, London Math. Soc. Student Texts 54, Cambridge Univ. Press, Cambridge, 2003, Second edition 2016. · Zbl 1038.05025
[147] J. Leech, Notes on sphere packings, Canad. J. Math. 19 (1967), 251–267. · Zbl 0162.25901
[148] D. Leemans, Two rank six geometries for the Higman-Sims sporadic group, Discrete Math. 294 (1-2) (2005), 123–132. · Zbl 1080.51005
[149] J. H. van Lint and J. J. Seidel, Equilateral point sets in elliptic geometry, Indagationes Mathematicae 28 (1966), 335–348. · Zbl 0138.41702
[150] H. Lüneburg, Über die gruppen von Mathieu, J. Algebra 10 (1968), 194–210.
[151] H. Lüneburg, “Transitive Erweiterungen endlicher Permutationsgrupper”, Lecture Notes in Mathematics 84, Springer-Verlag, Berlin, 1969.
[152] M. Mačaj and J. Širáň, Search for properties of the missing Moore graph, Linear Algebra & Applic., 432 (9) (2010), 2381–2398. · Zbl 1217.05149
[153] S. S. Magliveras, The subgroup structure of the Higman-Sims simple group, Bull. Amer. Math. Soc. 77 (4) (1971), 535–539. · Zbl 0226.20012
[154] R. Mathon and A. P. Street, Partitions of sets of two-fold triple systems, and their relation to some strongly regular graphs, Graphs and Combin. 11 (1995), 347–366. · Zbl 0837.05021
[155] V. C. Mavron, and M. S. Shrikhande, On designs with intersection numbers 0 and 2, Arch. Math. 52 (1989), 407–412. · Zbl 0636.05015
[156] K. N. Majindar, On the parameters and intersection of blocks of balanced incomplete block designs, Ann. Math. Statist. 33 (1962), 1200–1205. · Zbl 0111.15603
[157] A. A. Makhnev and V. V. Nosov, On automorphisms of strongly regular graphs with λ= 0 and µ = 3, Algebra i Analiz, 21 (5) (2009), 138–154.
[158] B. D. McKay, “nauty User’s Guide (Version 1.5)”, Technical Report TR-CS-90-02. Computer Science Department, Australian National University, 1990.
[159] D. M. Mesner, “An investigation of certain combinatorial properties of partially balanced incomplete block experimental designs and association schemes, with a detailed study of designs of Latin square and related types”, Ph.D. thesis, Michigan State University, 1956.
[160] D. M. Mesner, Negative Latin square designs, Institute of Statistics, UNC, NC Mimeo series, 410, 1964.
[161] D. M. Mesner, A note on the parameters of PBIB association schemes, Ann. Math. Stat. 36 (1965), 331–336. 60Mikhail H. Klin, Andrew J. Woldar · Zbl 0128.13102
[162] D. M. Mesner, A new family of partially balanced incomplete block designs with some Latin square design properties, Ann. Math. Stat. 38 (1967), 571–581. · Zbl 0155.26901
[163] D. M. Mesner and P. Bhattacharya, Association schemes on triples and a ternary algebra, J. Combin. Theory, Ser. A 55 (2) (1990), 204–234. · Zbl 0761.05098
[164] D. M. Mesner, P. Bhattacharya, A ternary algebra arising from association schemes on triples, J. Algebra 164 (3) (1994), 595–613. · Zbl 0853.17001
[165] A. Mostowski, Axiom of choice for finite sets, Fund. Math. 33 (1945), 137–168. · Zbl 0063.04124
[166] M. Muzychuk, M. Klin and R. Pöschel, The isomorphism problem for circulant graphs via Schur ring theory, DIMACS Ser. Discrete Math. & Theoret. Comput. Sci. vol. 56 (2001), 241–264. · Zbl 0979.05079
[167] M. G. Oxenham and L. R. A. Casse, On a geometric representation of the subgroups of index 6 in S6, Discrete Math. 92 (1991), 251–259. · Zbl 0752.20002
[168] L. J. Paige, A note on the Mathieu groups, Canad. J. Math. 9 (1957), 15–18.
[169] D. Parrott and S. K. Wong, On the Higman-Sims simple group of order 44,352,000, Pacific J. Math. 32 (2) (1970), 501–516. · Zbl 0188.06502
[170] A. Pasini, “Diagram Geometries”, Clarendon Press, Oxford, 1994. · Zbl 0813.51002
[171] A. Pasini, A quarry of geometries, Rend. Sem. Matematico e Fisico di Milano 65 (1995), 179–247. · Zbl 0878.51003
[172] R. M. Pawale, Inequalities and bounds for quasi-symmetric 3-designs, J. Combin. Theory Ser. A 60 (2) (1992), 159–167. · Zbl 0773.05014
[173] R. M. Pawale, Non-existence of triangle free quasi-symmetric designs, Designs, Codes and Cryptography 37 (2) (2005), 347–353. · Zbl 1136.05303
[174] S. E. Payne and J. A. Thas, “Finite Generalized Quadrangles”, Research Notes in Mathematics 110, Pitman Publ. Inc., London, 1984. · Zbl 0551.05027
[175] B. Polster, “A Geometrical Picture Book”, Springer-Verlag, New York, 1998. · Zbl 0914.51001
[176] D. Raghavarao, “Constructions and Combinatorial Problems in Design of Experiments”, Wiley, New York, 1971. · Zbl 0222.62036
[177] D. K. Ray-Chaudhuri, On the application of the geometry of quadrics to the construction of PBIB designs and error correcting codes, Institute of Statistics, UNC, Chapel Hill, NC Mimeo Series 230, 1959.
[178] D. K. Ray-Chaudhuri, Some results on quadrics in finite projective geometry based on Galois fields, Canad. J. Math. 15 (1962), 129–138. · Zbl 0103.13701
[179] N. N. Rognelia and S. S. Sane, Classification of (16, 6, 2)-designs by ovals, Discrete Math. 51 (1984), 167–177.
[180] S. Reichard, Strongly regular graphs with the 7-vertex condition, J. Algebraic Combin. 41 (3) (2015), 817–842. · Zbl 1314.05235
[181] M. Ronan, “Symmetry and the Monster”, Oxford University Press, Oxford, 2006. · Zbl 1113.00002
[182] I. C. Ross and F. Harary, On the determination of redundancies in sociometric chains, Psychometrika 15 (1952), 195–208. · Zbl 0049.37803
[183] A. Rudvalis, (v, k, λ)-graphs and polarities of (v, k, λ)-designs, Math Z. 120 (1971), 224– 230. · Zbl 0202.51201
[184] C. J. Salwach and J. A. Mezzaroba, The four biplanes with k = 9, J. Combin. Theory Ser. A 24 (1978), 141–145. · Zbl 0374.05008
[185] S. S. Sane, An extremal property of the 4-(23, 7, 1) design, Proc. XVI Southeastern Internat. Conf. Combin., Graph Theory and Comp. (Boca Raton, Fla., 1985), Congr. Numer. 49 (1985), 191–193. Acta Univ. M. Belii, ser. Math. 25 (2017), 5–6261
[186] M. Schönert et al. “GAP – Groups, Algorithms, and Programming”, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, 1995.
[187] I. Schur, Zur Theory der einfach transitiven Permutationsgruppen, S.-B. Preus Akad. Wiss. Phys. Math. Kl. (1933), 598–623. · JFM 59.0151.01
[188] W. R. Scott, “Group Theory”, Prentice-Hall, Englewood Cliffs NJ, 1964 (Dover Publ., New York, 1987). · Zbl 0126.04504
[189] B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. 48 (4) (1959), 2–97.
[190] J. J. Seidel, Strongly regular graphs of L2type and of triangular type, Indagationes Mathematicae 29 (1967), 188–196. · Zbl 0161.20802
[191] J. J. Seidel, Strongly regular graphs with (−1, 1, 0) adjacency matrix having eigenvalue 3, Linear Algebra & Applic. 1 (1968), 281–298.
[192] J. J. Seidel, Strongly regular graphs, in: W. T. Tutte (ed.), “Recent Progress in Combinatorics”, Academic Press, New York, 1969, 185–197.
[193] M. S. Shrikhande, On a class of negative Latin square graphs, Utilitas Mathematica 5 (1974), 293–303. · Zbl 0294.05009
[194] M. S. Shrikhande and S. S. Sane, “Quasi-Symmetric Designs”, London Math. Soc. Lec. Notes Ser. 164, Cambridge Univ, Press, 1991. · Zbl 0746.05011
[195] S. S. Shrikhande, On the dual of some balanced incomplete block designs, Biometrics 8 (1952), 66–72.
[196] S. S. Shrikhande, The uniqueness of the L2association scheme, Ann. Math. Statist. 30 (1959), 781–798. · Zbl 0086.34802
[197] S. S. Shrikhande, Strongly regular graphs and symmetric 3-designs, Survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), 1973, 403–409.
[198] S. S. Shrikhande and D. Bhagwandas, Duals of incomplete block designs, J. Indian Statist. Assoc. Bull. 3 (1965), 30–37.
[199] C. C. Sims, Graphs and finite permutation groups, Math. Zeitschrift 95 (1967), 76–86. · Zbl 0244.20001
[200] C. C. Sims, On the isomorphism of two groups of order 44,352,000, in: “Theory of Finite Groups” (Symposium, Harvard Univ., Cambridge MA, 1968), Benjamin, New York, 1969, 101–108.
[201] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (3) (1938), 377–385. · Zbl 0019.00502
[202] M. S. Smith, On rank 3 permutation groups, J. Algebra 33 (1975), 22–42. · Zbl 0297.20002
[203] M. S. Smith, On the isomorphism of two simple groups of order 44352000, J. Algebra 41 (1976), 172–174.
[204] M. S. Smith, A combinatorial configuration associated with the Higman-Sims group, J. Algebra 41 (1976), 175–195. · Zbl 0367.05023
[205] L. H. Soicher, GRAPE: a system for computing with graphs and groups, in: L. Finkelstein, W. M. Kantor (eds.), “Groups and Computation”, Vol. 11 DIMACS Ser. Discrete Math. & Theoret. Comput. Sci., Amer. Math. Soc., 1993, 287–291. · Zbl 0833.05071
[206] E. Spence, Regular two-graphs on 36 vertices, Linear Algebra & Applic. 226/228 (1995), 459–497. · Zbl 0834.05009
[207] D. A. Sprott, Some series of partially balanced incomplete block designs, Canad. J. Math. 7 (1955), 369–381. · Zbl 0064.38503
[208] R. G. Stanton, The Mathieu groups, Canad. J. Math. 3 (1951), 164–174. 62Mikhail H. Klin, Andrew J. Woldar · Zbl 0042.25601
[209] R. G. Stanton and D. A. Sprott, Block intersections in balanced incomplete block designs. Canad. Math. Bull. 7, 1964, 539-548. · Zbl 0124.00904
[210] R. G. Stanton and J. G. Kalbfleisch, Quasi-symmetric balanced incomplete block designs, J. Combin. Theory 4 (1968), 391–396. · Zbl 0153.33003
[211] O. Tamaschke, Schur-Ringe, BI-Hochschulskripten 735, Bibliographisches Institut, Manheim, 1970.
[212] D. E. Taylor, “The Geometry of the Classical Groups”, Heldermann Verlag, Berlin, 1992. · Zbl 0767.20001
[213] G. Tinhofer and M. Klin, Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. Iii. Graph Invariants and Stabilization Methods, Technische Univerität München, Report TUM–M9902, 1999. · Zbl 1029.05151
[214] J. Tits, Sur les systèmes de Steiner associés aux trois ‘grands’ groupes de Mathieu, Rend. Mat. e Appl. 23 (5) (1964), 166–184. · Zbl 0126.26303
[215] J. A. Todd, On representations of the Mathieu groups as colineation groups, J. London Math. Soc. 34 (1959), 406–416.
[216] J. A. Todd, A representation of the Mathieu group M24as a colineation group, Ann. Mat. Pura Appl. 71 (4) (1966), 199–238.
[217] V. D. Tonchev, Embedding of the Witt-Mathieu system S(3, 6, 22) in a symmetric 2(78, 22, 6) design, Geom. Dedic. 22 (1987), 49–75.
[218] D. Wales, Uniqueness of the graph of a rank 3 group, Pacific J. Math. 30 (1969), 271–276. · Zbl 0206.30901
[219] B. Weisfeiler and A. A. Leman, A reduction of a graph to a canonical form and an algebra arising during this reduction, Nauchno-Techn. Inform. Ser. 2, 9, 1968.
[220] H. W. Wielandt, “Finite Permutation Groups”, Academic Press, New York, 1964. · Zbl 0138.02501
[221] H. Wielandt, Permutation groups through invariant relations and invariant functions, Ohio State Univ. Lecture Notes, Columbus, 1969.
[222] R. A. Wilson, Octonions and the Leech lattice, J. Algebra 322 (6) (2009), 2186–2190. · Zbl 1213.11143
[223] E. Witt, Die 5-Fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg 12 (1938), 256–264. · Zbl 0019.25105
[224] E. Witt, Uber Steinersche systeme, Abh. Math. Sem. Hansischen Univ. 12 (1938), 265– 275. · Zbl 0019.25106
[225] E. Witt, “Collected Papers”, Gesammelte Abhandlungen, Springer-Verlag, Berlin, New York, 1998. · Zbl 0917.01054
[226] A. Woldar, A combinatorial approach to the character theory of split metabelian groups, J. Combin. Theory Series A 50 (1) (1989), 100-109. · Zbl 0686.20005
[227] A. J. Woldar, The symmetric genus of the Higman-Sims group HS and bounds for Conway’s groups Co1, Co2, Ill. J. Math. 36 (1) (1992), 47–52. · Zbl 0742.20020
[228] S. Yoshiara, A locally polar geometry associated with the group HS, Europ. J. Combin. 11 (1990), 81–93. · Zbl 0697.51008
[229] H. Zassenhaus, Über transitive Erweiterungen gewisser Gruppen aus Automorphismen endlicher mehrdimensionaler Geometrien, Math. Ann. 111 (1935), 748–759. · Zbl 0012.24802
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