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Higmanian rank-5 association schemes on 40 points. (English) Zbl 1284.05339


MSC:

05E30 Association schemes, strongly regular graphs

Software:

GAP; nauty; GRAPE
Full Text: DOI

References:

[1] R. P. Anstee, An analogue of group divisible designs for Moore graphs, J. Combin. Theory Ser. B 30 (1981), 11–20. · Zbl 0407.05063 · doi:10.1016/0095-8956(81)90086-1
[2] E. Bannai, E. Bannai, and H. Bannai, Uniqueness of certain association schemes, European J. Combin. 29 (2008), 1379–1395. · Zbl 1200.05258 · doi:10.1016/j.ejc.2007.06.016
[3] E. Bannai and T. Ito, Algebraic combinatorics. I. Association schemes, Benjamin-Cummings, Menlo Park, CA, 1984. · Zbl 0555.05019
[4] A. Blokhuis, A. E. Brouwer, D. Buset, and A. M. Cohen, The locally icosahedral graphs, Finite geometries (Winnipeg, 1984), Lecture Notes in Pure and Appl. Math., 103, pp. 19–22, Dekker, New York, 1985. · Zbl 0587.05059
[5] J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976. · Zbl 1226.05083
[6] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin, 1989. · Zbl 0747.05073
[7] Y. Chang, Imprimitive symmetric association schemes of rank 4, Ph.D. thesis, Univ. of Michigan, 1994.
[8] Y. Chang and T. Huang, Imprimitive association schemes of low ranks and Higmanian graphs, Conference on combinatorics and physics (Los Alamos, 1998), Ann. Comb. 4 (2000), 317–326. · Zbl 0970.05044 · doi:10.1007/PL00001283
[9] H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413–455. · Zbl 0040.22803 · doi:10.1090/S0002-9904-1950-09407-5
[10] ——, The Pappus configuration and the self-inscribed octagon. I, II, III, Indag. Math. 39 (1977), 256–300. · Zbl 0382.51007
[11] A. Deza and M. Deza, The ridge graph of the metric polytope and some relatives, Polytopes: Abstract, convex and computational (Scarborough, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 440, pp. 359–372, Kluwer, Dordrecht, 1994. · Zbl 0809.52017
[12] M. Deza and T. Huang, A generalization of strongly regular graphs, Southeast Asian Bull. Math. 26 (2002), 193–201. · Zbl 1026.05106 · doi:10.1007/s100120200040
[13] M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, and J. Hemmeter, Deza graphs: A generalization of strongly regular graphs, J. Combin. Des. 7 (1999), 395–405. · Zbl 0959.05122 · doi:10.1002/(SICI)1520-6610(1999)7:6<395::AID-JCD1>3.0.CO;2-U
[14] C. W. Evans, Net structure and cages, Discrete Math. 27 (1979), 193–204. · Zbl 0407.05031 · doi:10.1016/0012-365X(79)90110-9
[15] S. Evdokimov, I. Ponomarenko, and G. Tinhofer, Forestal algebras and algebraic forests (on a new class of weakly compact graphs), Discrete Math. 225 (2000), 149–172. · Zbl 0962.05041 · doi:10.1016/S0012-365X(00)00152-7
[16] I. A. Faradžev and M. H. Klin, Computer package for computations with coherent configurations, Proc. ISSAC-91, pp. 219–223, ACM Press, Bonn, 1991. · Zbl 0925.20006
[17] I. A. Faradžev, M. H. Klin, and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, Investigations in algebraic theory of combinatorial objects (I. A. Faradžev et al., eds.), pp. 1–152, Kluwer, Dordrecht, 1994. · Zbl 0795.05073
[18] \(\langle\)http://www.gap-system.org\(\rangle.\)
[19] Ja. Ju. Gol’fand, A. V. Ivanov, and M. H. Klin, Amorphic cellular rings, Investigations in algebraic theory of combinatorial objects (I. A. Faradžev et al., eds.), pp. 167–186, Kluwer, Dordrecht, 1994.
[20] F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969. · Zbl 0182.57702
[21] M. D. Hestenes and D. G. Higman, Rank \(3\) groups and strongly regular graphs, Computers in algebra and number theory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970), SIAM-AMS Proc., vol. 4, pp. 141–159, Amer. Math. Soc., Providence, RI, 1971. · Zbl 0253.05127
[22] D. G. Higman, Coherent configurations, I, Rend. Sem. Mat. Univ. Padova 44 (1970), 1–25. · Zbl 0279.05025
[23] ——, Coherent configurations. I. Ordinary representation theory, Geom. Dedicata 4 (1975), 1–32. · Zbl 0333.05010 · doi:10.1007/BF00147398
[24] ——, Coherent algebras, Linear Algebra Appl. 93 (1987), 209–239. · Zbl 0618.05014 · doi:10.1016/S0024-3795(87)90326-0
[25] ——, Rank 5 association schemes and triality, Linear Algebra Appl. 226–228 (1995), 197–222. · Zbl 0832.05099 · doi:10.1016/0024-3795(95)00102-W
[26] A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters \(2\) and \(3,\) IBM J. Res. Develop. 4 (1960), 497–504. · Zbl 0096.38102
[27] M. Klin, M. Muzychuk, C. Pech, A. Woldar, and P.-H. Zieschang, Association schemes on 28 points as mergings of a half- homogeneous coherent configuration, European J. Combin. 28 (2007), 1994–2025. · Zbl 1145.05056 · doi:10.1016/j.ejc.2006.08.010
[28] M. Klin and M. Ziv-Av, A family of Higmanian association schemes on 40 points: A computer algebra approach, Algebraic combinatorics. Proceedings of an international conference in Honor of Eiichi Bannai’s 60th birthday (Sendai, 2006), pp. 190–203, Sendai International Center, Sendai, Japan.
[29] B. D. McKay, nauty user’s guide, ver. 1.5, Technical Report TR-CS-90-02, Computer Science Department, Australian National Univ., 1990.
[30] M. E. Muzychuk, On half-homogeneous coherent configurations, Unpublished manuscript. · Zbl 0917.05011
[31] N. Robertson, Graphs minimal under girth, valency and connectivity constraints, Ph.D. thesis, Univ. of Waterloo, 1969.
[32] M. Schönert et al., GAP—Groups, algorithms, and programming, 5th ed., Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, 1995.
[33] 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, pp. 287–291, Amer. Math. Soc., Providence, RI, 1993. · Zbl 0833.05071
[34] E. Spence, The strongly regular \((40,12,2,4)\) graphs, Electron. J. Combin. 7 (2000), R22. · Zbl 0940.05072
[35] B. Weisfeiler (ed.), On construction and identification of graphs, Lecture Notes in Math., 558, Springer-Verlag, Berlin, 1976. · Zbl 0366.05019
[36] H. Wielandt, Finite permutation groups, Academic Press, New York, 1964. · Zbl 0138.02501
[37] P.-H. Zieschang, An algebraic approach to association schemes, Lecture Notes in Math., 1628, Springer-Verlag, Berlin, 1996. · Zbl 0857.05100 · doi:10.1007/BFb0097032
[38] M. Ziv-Av, Two association schemes on 40 and 64 points: A supplement to the paper by Bannai–Bannai–Bannai, Poster presentation (jointly with M. Klin), Linz, 2006, \(\langle\)http://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/ZivAv_poster.pdf\(\rangle.\)
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