The Michigan Mathematical Journal

Higmanian rank-5 association schemes on 40 points

Mikhail Klin, Mikhail Muzychuk, and Matan Ziv-Av

Source: Michigan Math. J. Volume 58, Issue 1 (2009), 255-284.

Primary Subjects: 05E30, 20B25
Secondary Subjects: 05E20, 20B04, 51E12

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.mmj/1242071692
Digital Object Identifier: doi:10.1307/mmj/1242071692
Mathematical Reviews number (MathSciNet): MR2526087
Zentralblatt MATH identifier: 05566082

References

R. P. Anstee, An analogue of group divisible designs for Moore graphs, J. Combin. Theory Ser. B 30 (1981), 11--20.
Mathematical Reviews (MathSciNet): MR609589
Digital Object Identifier: doi:10.1016/0095-8956(81)90086-1
Zentralblatt MATH: 0407.05063
E. Bannai, E. Bannai, and H. Bannai, Uniqueness of certain association schemes, European J. Combin. 29 (2008), 1379--1395.
Mathematical Reviews (MathSciNet): MR2423728
Digital Object Identifier: doi:10.1016/j.ejc.2007.06.016
Zentralblatt MATH: 05300774
E. Bannai and T. Ito, Algebraic combinatorics. I. Association schemes, Benjamin-Cummings, Menlo Park, CA, 1984.
Mathematical Reviews (MathSciNet): MR882540
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.
Mathematical Reviews (MathSciNet): MR826792
Zentralblatt MATH: 0587.05059
J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976.
Mathematical Reviews (MathSciNet): MR411988
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin, 1989.
Mathematical Reviews (MathSciNet): MR1002568
Y. Chang, Imprimitive symmetric association schemes of rank 4, Ph.D. thesis, Univ. of Michigan, 1994.
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.
Mathematical Reviews (MathSciNet): MR1811057
Digital Object Identifier: doi:10.1007/PL00001283
Zentralblatt MATH: 0970.05044
H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413--455.
Mathematical Reviews (MathSciNet): MR38078
Digital Object Identifier: doi:10.1090/S0002-9904-1950-09407-5
Project Euclid: euclid.bams/1183514923
------, The Pappus configuration and the self-inscribed octagon. I, II, III, Indag. Math. 39 (1977), 256--300.
Mathematical Reviews (MathSciNet): MR485468
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.
Mathematical Reviews (MathSciNet): MR1322070
Zentralblatt MATH: 0809.52017
M. Deza and T. Huang, A generalization of strongly regular graphs, Southeast Asian Bull. Math. 26 (2002), 193--201.
Mathematical Reviews (MathSciNet): MR2047798
Digital Object Identifier: doi:10.1007/s100120200040
Zentralblatt MATH: 1026.05106
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.
Mathematical Reviews (MathSciNet): MR1711897
Zentralblatt MATH: 0959.05122
C. W. Evans, Net structure and cages, Discrete Math. 27 (1979), 193--204.
Mathematical Reviews (MathSciNet): MR537475
Digital Object Identifier: doi:10.1016/0012-365X(79)90110-9
Zentralblatt MATH: 0407.05031
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.
Mathematical Reviews (MathSciNet): MR1798329
Digital Object Identifier: doi:10.1016/S0012-365X(00)00152-7
Zentralblatt MATH: 0962.05041
I. A. Faradžev and M. H. Klin, Computer package for computations with coherent configurations, Proc. ISSAC-91, pp. 219--223, ACM Press, Bonn, 1991.
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.
Mathematical Reviews (MathSciNet): MR1273366
$\langle$http://www.gap-system.org$\rangle.$
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.
Mathematical Reviews (MathSciNet): MR1273365
F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
Mathematical Reviews (MathSciNet): MR256911
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.
Mathematical Reviews (MathSciNet): MR340088
D. G. Higman, Coherent configurations, I, Rend. Sem. Mat. Univ. Padova 44 (1970), 1--25.
Mathematical Reviews (MathSciNet): MR325420
------, Coherent configurations. I. Ordinary representation theory, Geom. Dedicata 4 (1975), 1--32.
Mathematical Reviews (MathSciNet): MR398868
Digital Object Identifier: doi:10.1007/BF00147398
Zentralblatt MATH: 0333.05010
------, Coherent algebras, Linear Algebra Appl. 93 (1987), 209--239.
Mathematical Reviews (MathSciNet): MR898557
Digital Object Identifier: doi:10.1016/S0024-3795(87)90326-0
Zentralblatt MATH: 0618.05014
------, Rank 5 association schemes and triality, Linear Algebra Appl. 226--228 (1995), 197--222.
Mathematical Reviews (MathSciNet): MR1344562
Digital Object Identifier: doi:10.1016/0024-3795(95)00102-W
Zentralblatt MATH: 0832.05099
A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters $2$ and $3,$ IBM J. Res. Develop. 4 (1960), 497--504.
Mathematical Reviews (MathSciNet): MR140437
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.
Mathematical Reviews (MathSciNet): MR2344983
Digital Object Identifier: doi:10.1016/j.ejc.2006.08.010
Zentralblatt MATH: 1145.05056
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.
B. D. McKay, nauty user's guide, ver. 1.5, Technical Report TR-CS-90-02, Computer Science Department, Australian National Univ., 1990.
M. E. Muzychuk, On half-homogeneous coherent configurations, Unpublished manuscript.
N. Robertson, Graphs minimal under girth, valency and connectivity constraints, Ph.D. thesis, Univ. of Waterloo, 1969.
M. Schönert et al., GAP---Groups, algorithms, and programming, 5th ed., Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, 1995.
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.
Mathematical Reviews (MathSciNet): MR1235810
Zentralblatt MATH: 0833.05071
E. Spence, The strongly regular $(40,12,2,4)$ graphs, Electron. J. Combin. 7 (2000), R22.
Mathematical Reviews (MathSciNet): MR1754252
B. Weisfeiler (ed.), On construction and identification of graphs, Lecture Notes in Math., 558, Springer-Verlag, Berlin, 1976.
Mathematical Reviews (MathSciNet): MR543783
H. Wielandt, Finite permutation groups, Academic Press, New York, 1964.
Mathematical Reviews (MathSciNet): MR183775
Zentralblatt MATH: 0138.02501
P.-H. Zieschang, An algebraic approach to association schemes, Lecture Notes in Math., 1628, Springer-Verlag, Berlin, 1996.
Mathematical Reviews (MathSciNet): MR1439253
Zentralblatt MATH: 0857.05100
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$\rangle.$

2009 © The University of Michigan