The Paulus-Rozenfeld-Thompson graph on 26 vertices revisited and related combinatorial structures. (English) Zbl 1442.05248
Jones, Gareth A. (ed.) et al., Isomorphisms, symmetry and computations in algebraic graph theory. Selected papers based on the presentations at the workshop on algebraic graph theory, Pilsen, Czech Republic, October 3–7, 2016. Cham: Springer. Springer Proc. Math. Stat. 305, 73-154 (2020).
Summary: This paper deals with the strongly regular graph \(T\), having parameters (26, 10, 3, 4) and the largest possible automorphism group \(G=\operatorname{Aut}(T)\) of order 120. The group \(G\) in its action on the vertex set of \(T\) has two orbits of length 20 and 6. Many special features of the graph \(T\) and its group \(G\) make it a remarkable object in algebraic graph theory. The presentation, arranged in the style of a tutorial, describes the graph \(T\) from many viewpoints of mathematics and computer algebra. Special attention is paid to the links of \(T\) with such classical structures as the Petersen graph, semi-icosahedron, icosahedron, dodecahedron, two-graphs on 26 points, Paley graphs, inversive plane and generalized quadrangles of order 5 and others.
For the entire collection see [Zbl 1443.05003].
For the entire collection see [Zbl 1443.05003].
MSC:
| 05E30 | Association schemes, strongly regular graphs |
| 05B05 | Combinatorial aspects of block designs |
| 62K10 | Statistical block designs |
Keywords:
algebraic graph theory; strongly regular graph; distance regular graph; antipodal graph; two-graph; coherent configuration; association scheme; finite permutation group; block design; icosahedron; semi-icosahedron; dodecahedron; Paley graph; Weisfeiler-Leman stabilization; computer algebra; GAP; COCO; inversive plane; generalized quadrangleReferences:
| [1] | G.M. Adel’son-Vel’skii, B.Ju. Weisfeiler, A.A. Leman, I.A. Faradžev, An example of a graph which has no transitive group of automorphisms. (Russian) Dokl. Akad. Nauk SSSR 185, 975-976 (1969) |
| [2] | V.L. Arlazarov, A.A. Leman, M.Z. Rosenfeld, Computer aided construction and analysis of graphs with 25, 26 and 29 vertices. (Russian). Moscow, IPU (1975) |
| [3] | ATLAS of Finite Group Representations - Version 3.2., http://brauer.maths.qmul.ac.uk/Atlas/ |
| [4] | L. Babel, I.V. Chuvaeva, M. Klin, D.V. Pasechnik, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. II. Program implementation of the Weisfeiler-Leman algorithm, http://www.arxiv.org/abs/1002.1921 |
| [5] | J.C. Baez, Home page, http://math.ucr.edu/home/baez/. Accessed June 2018 |
| [6] | R.A. Bailey, Association Schemes. Designed Experiments, Algebra and Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 84. (Cambridge University Press, Cambridge, 2004), pp. xviii+387 · Zbl 1051.05001 |
| [7] | E. Bannai, T. Ito, Algebraic Combinatorics: I. Association Schemes (The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984) · Zbl 0555.05019 |
| [8] | A. Betten, M. Klin, R. Laue, A. Wassermann, Graphical \(t\)-designs via polynomial Kramer-Mesner matrices. Discrete Math. 197(198), 83-109 (1999) · Zbl 0936.05011 · doi:10.1016/S0012-365X(98)00225-8 |
| [9] | N.L. Biggs, Distance-regular Graphs with Diameter Three. Algebraic and geometric combinatorics, in North-Holland Mathematics Studies, vol. 65 (1982), pp. 69-80 · Zbl 0506.05057 |
| [10] | N.L. Biggs, The Icosian calculus of today. Proc. R. Irish Acad. 95A(Supplement), 23-34 (1995) · Zbl 1280.05058 |
| [11] | R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs. Pac. J. Math. 13, 389-419 (1963) · Zbl 0118.33903 · doi:10.2140/pjm.1963.13.389 |
| [12] | R.C. Bose, T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Am. Stat. Assoc. 47, 151-184 (1952) · Zbl 0048.11603 · doi:10.1080/01621459.1952.10501161 |
| [13] | A.E. Brouwer, home page: www.win.tue.nl/ aeb |
| [14] | A.E. Brouwer, Classification of small (0,2)-graphs. J. Combin. Theory A 113, 1636-1645 (2006) · Zbl 1105.05009 · doi:10.1016/j.jcta.2006.03.023 |
| [15] | A.E. Brouwer, W. Haemers, Spectra of Graphs (Springer, Berlin, 2012) · Zbl 1231.05001 · doi:10.1007/978-1-4614-1939-6 |
| [16] | A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance Regular Graphs (Springer, Berlin, 1989) · Zbl 0747.05073 · doi:10.1007/978-3-642-74341-2 |
| [17] | R.H. Bruck, Finite nets II. Uniqueness and imbedding. Pac. J. Math. 13, 421-457 (1963) · Zbl 0124.00903 · doi:10.2140/pjm.1963.13.421 |
| [18] | R.H. Bruck, Finite nets. I. Numerical invariants. Canad. J. Math. 3, 94-107 (1951) · Zbl 0042.38802 · doi:10.4153/CJM-1951-012-7 |
| [19] | F. Buekenhout, M. Parker, The number of nets of the regular convex polytopes in dimension \(\le 4\). Discrete Math. 186(1-3), 69-94 (1998) · Zbl 0956.52014 · doi:10.1016/S0012-365X(97)00225-2 |
| [20] | P.J. Cameron, blog, https://cameroncounts.wordpress.com/2018/07/03/ |
| [21] | P.J. Cameron, Permutation Groups. London Mathematical Society Student Texts, vol. 45 (Cambridge University Press, Cambridge, 1999), p. x+220 · Zbl 0922.20003 |
| [22] | P.J. Cameron, J.H. van Lint, Designs, Graphs, Codes and Their Links (Cambridge University Press, Cambridge, 1991), p. x+240 · Zbl 0743.05004 |
| [23] | P.J. Cameron, Two remarks on Steiner systems. Geom. Dedicata 4, 403-418 (1975) · Zbl 0324.05013 · doi:10.1007/BF00148772 |
| [24] | L.C. Chang, Association schemes of partially balanced block designs with parameters \(v = 28, n_1 = 12, n_2 = 15\) and \(p_{11}^2 = 4\). Sci. Record 4, 12-18 (1960) · Zbl 0093.32101 |
| [25] | Y. Chen, The Steiner systems \(S(3,6,26)\). J. Geom. 2, 7-28 (1972) · Zbl 0231.50019 |
| [26] | W.H. Clatworthy, Tables of Two-associate Partially Balanced Designs (National Bureau of Standards, Washington D.C., 1973) · Zbl 0289.05017 |
| [27] | C.J. Colbourn, A. Rosa, Triple Systems (Clarendon Press, Oxford, 1999) · Zbl 0938.05009 |
| [28] | Conference in Algebraic Graph Theory: Symmetry vs. Regularity, (the first 50 years since Weisfeiler-Leman stabilization), WL2018, July 1-7, 2018, Pilsen, Czech Republic, https://www.iti.zcu.cz/wl2018/index.html |
| [29] | J.H. Conway, A. Hulpke, J. McKay, On transitive permutation groups. LMS J. Comput. Math. 1, 1-8 (1998) · Zbl 0920.20001 · doi:10.1112/S1461157000000115 |
| [30] | D.G. Corneil, R.A. Mathon, Algorithmic techniques for the generation and analysis of strongly regular graphs and other combinatorial configurations. Algorithmic aspects of combinatorics (Conf., Vancouver Island, B.C., Ann. Discrete Math. 2(1978), pp. 1-32 (1976) · Zbl 0398.05054 |
| [31] | H.S.M. Coxeter, A symmetrical arrangement of eleven hemi-Icosahedra, in North-Holland Mathematics Studies, vol. 87(C) (1984), pp. 103-114 · Zbl 0569.52010 |
| [32] | H.S.M. Coxeter, Regular Polytopes, 3rd edn. (Dover Publications, USA, 1973) · Zbl 0031.06502 |
| [33] | H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, 4th edn. (Springer, Berlin, 1980) · Zbl 0422.20001 · doi:10.1007/978-3-662-21943-0 |
| [34] | P. Dembowski, Finite Geometries (Springer, Heidelberg, 1968) · Zbl 0159.50001 · doi:10.1007/978-3-642-62012-6 |
| [35] | R.H.F. Denniston, Uniqueness of the inversive plane of order 5. Manuscr. Math. 8, 11-19 (1973) · Zbl 0251.05018 · doi:10.1007/BF01317572 |
| [36] | J.D. Dixon, B. Mortimer, Permutation Groups. Graduate Texts in Mathematics, vol. 163 (Springer, New York, 1996), p. xii+346 · Zbl 0951.20001 · doi:10.1007/978-1-4612-0731-3 |
| [37] | A. Duval, A directed version of strongly regular graphs. J. Combin. Theory Ser. A 47, 71-100 (1988) · Zbl 0642.05025 · doi:10.1016/0097-3165(88)90043-X |
| [38] | D. Epple, M. Klin, Some Strongly Regular Graphs on \(p^2+1\) Vertices \(p\in \{3,5,7,11\}\) and Related Structures. Paper in preparation |
| [39] | M. Erickson, S. Fernando, W.H. Haemers, D. Hardy, J. Hemmeter, Deza graphs: a generalization of strongly regular graphs. J. Combin. Designs 7, 395-405 (1999) · Zbl 0959.05122 · doi:10.1002/(SICI)1520-6610(1999)7:6<395::AID-JCD1>3.0.CO;2-U |
| [40] | I.A. Faradžev, Constructive Enumeration of Combinatorial Objects. Algorithmic studies in combinatoric (Russian), vol. 185 (Nauka, Moscow, 1978), pp. 3-11 · Zbl 0412.05006 |
| [41] | I.A. Faradžev, M. Klin, Computer package for computation with coherent configurations, in Proceedings ISSAC-91, ed. by S.M. Watt (ACM Press, Bonn, 1991), pp. 219-223 · Zbl 0925.20006 |
| [42] | I.A. Faradžev, M.H. Klin, M. Muzichuk, Cellular rings and groups of automorphisms of graphs, in Investigations in algebraic theory of combinatorial objects, ed. by I.A. Faradžev, A.A. Ivanov, M.H. Klin, A.J. Woldar (Kluwer Academic Publishers, Dordrecht, 1994), pp. 1-152 · Zbl 0795.05073 · doi:10.1007/978-94-017-1972-8 |
| [43] | F. Fiedler, M. Klin, M. Muzychuk, Small vertex-transitive directed strongly regular graphs. Discrete Math. 255, 87-115 (2002) · Zbl 1009.05139 · doi:10.1016/S0012-365X(01)00391-0 |
| [44] | P.H. Fisher, T. Pentilla, Ch. Praeger, G. Royle, Inversive planes of odd order. Eur. J. Combin. 10, 331-336 (1989) · Zbl 0702.05022 · doi:10.1016/S0195-6698(89)80005-8 |
| [45] | L.L. Foster, On the symmetry group of the dodecahedron. Math. Mag. 63(2), 106-107 (1990) · Zbl 0742.20005 · doi:10.1080/0025570X.1990.11977496 |
| [46] | GAP - Groups, Algorithms, Programming—A System for Computational Discrete Algebra, www.gap-system.org |
| [47] | C.D. Godsil, G. Royle, Algebraic Graph Theory (Springer, New York, 2001) · Zbl 0968.05002 |
| [48] | R.W. Gray, home page, http://www.rwgrayprojects.com. Accessed December 2018 |
| [49] | K. Gray, A.P. Street, Two regular solids holding the twelve \((6,3,2)\)-designs. Bull. Inst. Combin. Appl. 57, 53-59 (2009) · Zbl 1214.05003 |
| [50] | R. Grimaldi, Discrete and Combinatorial Mathematics, 5th edn. (2007) |
| [51] | Ya.Yu. Gol’fand, A.V. Ivanov, M. Klin, Amorphic Cellular Rings, in Investigations in Algebraic Theory of Combinatorial Objects, ed. by I.A. Faradžev et al. (Kluwer Academic Publishers, Dordrecht, 1994), pp. 167-186 · Zbl 0794.20010 · doi:10.1007/978-94-017-1972-8_3 |
| [52] | J.L. Gross, T.W. Tucker, Topological Graph Theory (Wiley, New York, 1987) · Zbl 0621.05013 |
| [53] | Š. Gyürki, M. Klin, M. Ziv-Av, WL-Logo, https://www.iti.zcu.cz/wl2018/ |
| [54] | M. Hall Jr., The Theory of Groups. Reprinting of the 1968 edition (Chelsea Publishing Co., New York, 1976), p. xiii+434 · Zbl 0354.20001 |
| [55] | A. Hanaki, I. Miyamoto, Tables of Homogeneous Coherent Configurations, http://kissme.shinshu-u.ac.jp/as/ · Zbl 1014.05073 |
| [56] | F. Harary, Graph Theory (Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969), p. ix+274 · Zbl 0182.57702 |
| [57] | A. Heinze, M. Klin, Loops, Latin squares and strongly regular graphs: An algorithmic approach via algebraic combinatorics, in Algorithmic Algebraic Combinatorics and Gröbner Bases, ed. by M. Klin et al. (Springer, Berlin, 2009), pp. 3-65 · Zbl 1186.05036 · doi:10.1007/978-3-642-01960-9_1 |
| [58] | D.G. Higman, Coherent configurations. I. Rend. Sem. Mat. Univ. Padova 44, 1-25 (1970) · Zbl 0279.05025 |
| [59] | J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd edn., Oxford Mathematical Monographs (Clarendon Press, Oxford, 1998) · Zbl 0899.51002 |
| [60] | D.A. Holton, J. Sheehan, The Petersen Graph (Cambridge University Press, Cambridge, 1993) · Zbl 0781.05001 |
| [61] | T.C. Holyoke, On the structure of multiply transitive permutation groups. Am. J. Math. 4, 787-796 (1952) · Zbl 0049.15501 · doi:10.2307/2372226 |
| [62] | B. Hopkins, Hamiltonian paths on platonic graphs. Int. J. Math. Math. Sci. 29-32, 1613-1616 (2004) · Zbl 1064.05094 · doi:10.1155/S0161171204307118 |
| [63] | D.R. Hughes, F.C. Piper, Design Theory (Cambridge University Press, Cambridge, 1985) · Zbl 0561.05009 · doi:10.1017/CBO9780511566066 |
| [64] | Q.M. Husain, On the totality of the solutions for the symmetrical incomplete block designs: \( \lambda =2, k=5\) or \(6\). Sankhya 7, 204-208 (1945) · Zbl 0060.31407 |
| [65] | A.A. Ivanov, Construction with the aid of a computer of some new automorphic graphs, in “Air Physics and Applied Mathematics” (Moscow, MPhTI, 1981), pp. 144-146 (in Russian) |
| [66] | A.A. Ivanov, M.H. Klin, S.V. Tsaranov, S.V. Shpectorov, On the problem of computing subdegrees of a transitive permutation group. UMN 6, 115-116 (1983). (in Russian) |
| [67] | G.A. Jones, Paley and the Paley Graphs, arXiv:1702.00285 |
| [68] | G.A. Jones, Permutation Groups (Theory and Applications) (Belianum, Vydavatelstvo Univerzity Mateja Bela, Banská Bystrica, 2015), p. viii+84 · Zbl 1347.20001 |
| [69] | G.A. Jones, M. Ziv-Av, Petrie duality and the Anstee-Robertson graph. Symmetry 7, 2206-2223 (2015) · Zbl 1371.05054 · doi:10.3390/sym7042206 |
| [70] | A. Kerber, Applied Finite Group Actions, Algorithms and Combinatorics, vol. 19, 2nd edn. (Springer, New York, 1999) · Zbl 0951.05001 |
| [71] | A. Kerber, R. Laue, M. Meringer, Ch. Rücker, E. Schymanski, Mathematical Chemistry and Chemoinformatics: Structure Generation, Elucidation and Quantitative Structure-Property Relationships (De Gruyter, 2014) · Zbl 1309.92006 |
| [72] | M.Kh. Klin, I.A. Faradzhev, The method of V-rings in the theory of permutation groups and its combinatoric applications. Investigations in Applied Graph Theory (Russian) (Nauka Sibirsk. Otdel., Novosibirsk, 1986), pp. 59-97 · Zbl 0659.20004 |
| [73] | B. Kim, Mappings which preserve regular dodecahedrons. Int. J. Math. Math. Sci. 31(4), 515-522 (2006) · Zbl 1133.51005 |
| [74] | M. Klin, Š. Gyürki, Selected Topics from Algebraic Graph Theory (Belianum, Vydavatelstvo Univerzity Mateja Bela, Banská Bystrica, 2015), p. xi+209 · Zbl 1360.05001 |
| [75] | M. Klin, I. Kovács, Automorphism groups of rational circulant graphs. Electron. J. Combin. 19(1), Paper 35, 52 pp (2012) · Zbl 1244.05234 |
| [76] | M. Klin, A. Munemasa, M. Muzychuk, P.H. Zieschang, Directed Strongly Regular Graphs Via Coherent (cellular) Algebras. Preprint Kyushu-MPS-1997-12 (Kyushu University, 1997) · Zbl 1030.05119 |
| [77] | M. Klin, C. Pech, S. Reichard, A. Woldar, M. Ziv-Av., Examples of computer experimentation in algebraic combinatorics. Ars Math. Contemp. 3 (2), 237-258 (2010) · Zbl 1227.05274 · doi:10.26493/1855-3974.119.60b |
| [78] | M.H. Klin, S. Reichard, A.J. Woldar, Siamese Combinatorial Objects via Computer Algebra Experimentation, in Algorithmic algebraic combinatorics and Gröbner bases (Springer, Berlin, 2009), pp. 67-112 · Zbl 1200.05084 · doi:10.1007/978-3-642-01960-9_2 |
| [79] | M. Klin, C. Rücker, G. Rücker, G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras. MATCH 40, 7-138 (1999) · Zbl 1029.05151 |
| [80] | M. Klin, M. Ziv-Av, Enumeration of Schur rings over the group \(A_5\), in Computer algebra in scientific computing. Lecture Notes in Computer Science, vol. 8136 (Springer, Berlin, 2013), pp. 219-230 · Zbl 1353.05004 · doi:10.1007/978-3-319-02297-0_19 |
| [81] | MCh. Klin, R. Pöschel, K. Rosenbaum, Angewandte Algebra für Mathematiker und Informatiker (Vieweg and Sohn, Braunschweig, 1988) · Zbl 0648.20001 · doi:10.1007/978-3-322-84116-2 |
| [82] | M. Klin, A. Munemasa, M. Muzychuk, P.H. Zieschang, Directed strongly regular graphs obtained from coherent algebras. Linear Algebra Appl. 377, 83-109 (2004) · Zbl 1030.05119 · doi:10.1016/j.laa.2003.06.020 |
| [83] | M. Klin, S. Reichard, A. Woldar, Siamese objects, and their relation to color graphs, association schemes and Steiner designs. Bull. Belg. Math. Soc. Simon Stevin 12(5), 845-857 (2005) · Zbl 1137.05072 |
| [84] | M.H. Klin, D.M. Mesner, A.J. Woldar, A combinatorial approach to transitive extensions of generously unitransitive permutation groups. J. Combin. Des. 18(5), 369-391 (2010) · Zbl 1205.05250 · doi:10.1002/jcd.20250 |
| [85] | M. Klin, N. Kriger, A. Woldar, On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters. J. Geom. 107(2), 329-356 (2016) · Zbl 1360.05186 · doi:10.1007/s00022-015-0308-9 |
| [86] | J. Lauri, R. Scapellato, Topics in Graph Automorphisms and Reconstruction, 2nd edn. (Cambridge University Press, Cambridge, 2016) · Zbl 1347.05001 |
| [87] | F. Levstein, D. Penazzi, The Terwilliger algebra of the dodecahedron, (English summary). Rev. Un. Mat. Argentina 45(2), 11-20 (electronic) (2004) · Zbl 1082.05099 |
| [88] | E. Lord, Symmetry and Pattern in Projective Geometry (Springer, London, 2013) · Zbl 1262.51001 · doi:10.1007/978-1-4471-4631-5 |
| [89] | A. Malnič, D. Marušič, P. Šparl, On strongly regular bicirculants. Eur. J. Combin. 28(3), 891-900 (2007) · Zbl 1112.05052 · doi:10.1016/j.ejc.2005.10.010 |
| [90] | R. Mathon, 3-class association schemes, in Proceedings of the Conference on Algebraic Aspects of Combinatorics (University of Toronto, Toronto, Ont., 1975), pp. 123-155. Congressus Numerantium, No. XIII, Utilitas Math., Winnipeg, Man., 1975 · Zbl 0362.05004 |
| [91] | B.D. McKay, Nauty user’s guide, ver. 1.5, Technical Report TR-CS-90-02 (Computer Science Department, Australian National University, 1990) |
| [92] | M. Mulder, \((0,\lambda )\)-graphs and \(n\)-cubes. Disc. Math. 28, 179-188 (1979) · Zbl 0418.05034 · doi:10.1016/0012-365X(79)90095-5 |
| [93] | D.V. Pasechnik, Skew-symmetric association schemes with two classes and strongly regular graphs of type \(L_{2n-1}(4n-1)\). Interactions between algebra and combinatorics. Acta Appl. Math. 29(1-2), 129-138 (1992) · Zbl 0760.05095 |
| [94] | A.J.L. Paulus, Conference matrices and graphs of order 26, Technische Hogeschool Eindhoven. Report WSK 73/06, (Eindhoven, 1973), pp. I - IV, 0-89 · Zbl 0275.05123 |
| [95] | S.E. Payne, J.A. Thas, Finite Generalized Quadrangles, Research Notes in Mathematics, vol. 110. Pitman (Advanced Publishing Program, Boston, MA, 1984), p. vi+312 · Zbl 0551.05027 |
| [96] | M.Z. Rozenfeld, The construction and properties of certain classes of strongly regular graphs. Uspehi Mat. Nauk 28, 197-198 (1973) · Zbl 0284.05126 |
| [97] | A. Rudvalis, \((v, k,\lambda )\)-graphs and polarities of \((v, k,\lambda )\)-designs. Math Z. 120, 224-230 (1971) · Zbl 0202.51201 · doi:10.1007/BF01117497 |
| [98] | J.J. Seidel, Geometry and Combinatorics. Selected Works of J.J. Seidel Edited and with a preface by D. G. Corneil and R. Mathon (Academic Press, Inc., Boston, MA, 1991), p. xix+410 · Zbl 0770.05001 |
| [99] | J.J. Seidel, On two-graphs and Shult’s characterization of symplectic and orthogonal geometries over \(GF(2)\). T.H.-Report, No. 73-WSK-02 (Department of Mathematics, Technological University Eindhoven, Eindhoven, 1973), p. i+25 · Zbl 0273.05130 |
| [100] | S.S. Shrikhande, The uniqueness of the \(L_2\) association scheme. Ann. Math. Statist. 30, 781-798 (1959) · Zbl 0086.34802 · doi:10.1214/aoms/1177706207 |
| [101] | S.S. Shrikhande, V.N. Bhat, Nonisomorphic solutions of pseudo-(3, 5, 2) and pseudo-(3, 6, 3) graphs. Ann. N. Y. Acad. Sci. 175, 331-350 (1970) · Zbl 0229.05139 · doi:10.1111/j.1749-6632.1970.tb56489.x |
| [102] | S.S. Shrikhande, V.N. Bhat, Graphs derivable from \(L_3(5)\) graphs. Sankhya Ser. A 33, 315-350 (1971) · Zbl 0241.05117 |
| [103] | S.S. Shrikhande, V.N. Bhat, Seidel-equivalence in \(LB_3(6)\) graphs. Aequationes Math. 8, 271-280 (1972) · Zbl 0248.05122 · doi:10.1007/BF01844502 |
| [104] | E. Shult, Supplement to the “Graph Extension Theorem” (University of Florida, Gainesville, Fla. (mimeographic notes), 1971), pp. 1-100 |
| [105] | E. Shult, The graph extension theorem. Proc. Am. Math. Soc. 33, 278-284 (1972) · Zbl 0241.20005 · doi:10.1090/S0002-9939-1972-0294477-5 |
| [106] | L.H. Soicher, GRAPE: A system for computing with graphs and groups, Groups and computation (New Brunswick, 1991), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 11, 287-291; Am. Math. Soc. (Providence, RI, 1993) · Zbl 0833.05071 |
| [107] | E. Spence, Two-graphs, in Handbook of design and analysis of experiments, ed. by A. Dean, M. Morris, J. Stufken, D. Bingham (Chapman & Hall/CRC Handbooks of Modern Statistical Methods. CRC Press, Boca Raton, FL, 2015), p. xix+940; pp. 875-882 |
| [108] | H. Stachel, On the tetrahedra in the dodecahedron. KoG 4, 5-10 (1999) · Zbl 0949.53010 |
| [109] | Z.G. Sun, A strong \((10,3,4)\) regular graph. Kexue Tongbao (English Ed.) 28(3), 298-300 (1983) · Zbl 0513.05057 |
| [110] | D.B. Surowski, Stability of arc-transitive graphs. J. Graph Theory 38(2), 95-110 (2001) · Zbl 0989.05054 · doi:10.1002/jgt.1026 |
| [111] | D.B. Surowski, Automorphism groups of certain unstable graphs. Math. Slovaca 53(3), 215-232 (2003) · Zbl 1057.05042 |
| [112] | D.M. Thompson, Design constructibility: strongly regular graphs and block designs, Ph.D. dissertation (University of Arizona, 1979) |
| [113] | D.M. Thompson, Eigengraphs: constructing strongly regular graphs with block designs. Utilitas Math. 20, 83-115 (1981) · Zbl 0494.05043 |
| [114] | Boris Weisfeiler (ed.), On Construction and Identification of Graphs, vol. 558. Lecture Notes in Mathematics (Springer, Berlin, 1976) · Zbl 0366.05019 |
| [115] | B. Weisfeiler, 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) |
| [116] | H. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
