Higmanian rank-5 association schemes on 40 points

Michigan Mathematical Journal

Volumn 58, Issue 1, 2009, Pages 255-284

  Klin, Mikhail ^a   Muzychuk, Mikhail ^b   Ziv Av, Matan ^a  

^a BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

^b NETANYA ACADEMIC COLLEGE   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650825153     PISSN: 00262285     EISSN: 19452365     Source Type: Journal    
DOI: 10.1307/mmj/1242071692     Document Type: Article

References (38)

1. R. P. Anstee, An analogue of group divisible designs for Moore graphs, J. Combin. Theory Ser. B 30 (1981), 11-20.
2. E. Bannai, E. Bannai, and H. Bannai, Uniqueness of certain association schemes, European J. Combin. 29 (2008), 1379-1395.
3. E. Bannai and T. Ito, Algebraic combinatorics. I. Association schemes, Benjamin-Cummings, Menlo Park, CA, 1984.
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.
5. J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976.
6. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin, 1989.
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.
9. H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413-455.
10. -, The Pappus configuration and the self-inscribed octagon. I, II, III, Indag. Math. 39 (1977), 256-300.
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.
12. M. Deza and T. Huang, A generalization of strongly regular graphs, Southeast Asian Bull. Math. 26 (2002), 193-201.
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.
14. C. W. Evans, Net structure and cages, Discrete Math. 27 (1979), 193-204.
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.
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.
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.
18. http: //www.gap-system.org
19. Ja. Ju. Gol'fand, A. V. Ivanov, and M. H. Klin, Amorphic cellular rings, Investigations in algebraic theory of combinatorial objects (I. A. Faradzev et al., eds.), pp. 167-186, Kluwer, Dordrecht, 1994.
20. F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
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.
22. D. G. Higman, Coherent configurations, I, Rend. Sem. Mat. Univ. Padova 44 (1970), 1-25.
23. -, Coherent configurations. I. Ordinary representation theory, Geom. Dedicata 4 (1975), 1-32
24. -,Coherent algebras,Linear Algebra Appl.93(1987),209-239.
25. - Rank 5 association schemes and triality, Linear Algebra Appl.226-228 (1995), 197-222.
26. A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Develop. 4 (1960), 497-504.
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.
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.
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.
34. E. Spence, The strongly regular (40,12,2,4) graphs, Electron. J. Combin. 7 (2000), R22.
35. B. Weisfeiler (ed.), On construction and identification of graphs, Lecture Notes in Math., 558, Springer-Verlag, Berlin, 1976.
36. H. Wielandt, Finite permutation groups, Academic Press, New York, 1964.
37. P.-H. Zieschang, An algebraic approach to association schemes, Lecture Notes in Math., 1628, Springer-Verlag, Berlin, 1996.
38. M. Ziv-Av, Two association schemes on 40 and 64points: A supplement to the paper by Bannai-Bannai-Bannai, Poster presentation (jointly with M. Klin), Linz, 2006, (http://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/ ZivAv-poster.pdf).