Abstract
By introducing a duality operator in coherent algebras (i.e. adjacency algebras of coherent configurations) we give a new interpretation to Delsarte's duality theory for association schemes. In particular we show that nonnegative matrices and positive semidefinite matrices, (0, 1)-matrices and distance matrices, regular graphs and spherical 2-designs, distance regular graphs and Delsarte matricesare pairs of dual objects. Several “almost dual” properties which are not yet fully understood are also reported.
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References
E. Bannai andR. M. Damerell, Tight spherical designs,I. J. Math. Soc. Japan,31 (1979), 199–207.
E. Bannai andR. M. Damerell, Tight spherical designs,II. J. London Math. Soc. (2),21 (1980), 13–30.
E. Bannai andT. Ito,Algebraic Combinatorics I: Association schemes. Benjamin/Cummings, Menlo Park, Calif. 1984.
N. L. Biggs,Algebraic Graph Theory. Cambridge University Press, Cambridge 1974.
R. C. Bose andD. M. Mesner, On linear associative algebras corresponding to association schemes of partially balanced designs,Ann. Math. Statist. 30 (1959), 21–38.
R. H. Bruck, Finite nets I: Numerical invariants,Canad. J. Math. 3 (1951), 94–107.
P. J. Cameron, J. M. Goethals andJ. J. Seidel, Strongly regular graphs having strongly regular subconstituents,J. Algebra 55 (1978), 257–280.
Ph. Delsarte, An algebraic approach to the association schemes of coding theory,Philips Res. Rep. Suppl. 10 (1973).
Ph. Delsarte, J. M. Goethals andJ. J. Seidel, Spherical codes and designs,Geom. Dedicata 6 (1977), 363–388.
D. G. Higman, Coherent configurations, part I: Ordinary representation theory,Geom. Dedicata 4 (1975), 1–32.
A. J. Hoffman andR. R. Singleton, On Moore graphs with diameters 2 and 3,IBM J. Res. Develop. 4 (1960), 497–504.
A. Neumaier, Distances, graphs, and designs,Europ. J. Combinatorics 1 (1980), 163–174.
A. Neumaier, Distance matrices andn-dimensional designs,Europ. J. Combinatorics 2 (1981), 165–172.
A. Neumaier, Distance matrices, dimension, and conference graphs,Indagationes Math. 43 (1981), 385–391.
A. Neumaier, Classification of graphs by regularity,J. Combin. Theory (Ser. B) 30 (1981), 318–331.
A.Neumaier, Combinatorial configurations in terms of distances,Memorandum 81-09 (Wiskunde), TH Eindhoven 1981.
D. E.Taylor and R.Levingston, Distance-regular graphs, in: Combinatorial Mathematics (D. A. Holton and J. Seberry, eds.),Springer Lecture Notes in Mathematics 686 (1978), 313–323.
H. Wielandt,Finite permutation groups, Academic Press, New York, 1964.
Ja. Ju. Golfand andM. H. Klin, Amorphic cellular rings I,Investigations in algebraic theory of combinatorial objects, Moscow, Inst. for Systems Studies, 1985, pp. 32–39.
J. L. Hayden, Association algebras for finite projective planes,J. Combin. Theory (Ser. A) 34 (1983), 360–374.
D. G. Higman, Coherent algebras,Linear Algebra Appl. 93 (1987), 209–240.