×

zbMATH — the first resource for mathematics

Balanced sets and Q-polynomial association schemes. (English) Zbl 0644.05016
The author’s abstract: “The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. Let X be a d-class symmetric association scheme with Bose-Mesner algebra M and Krein parameters \(q^ h_{ij}\), and let \(E=E_ t\) (0\(\leq t\leq d)\) be any primitive idempotent of M. For each \(x\in X\) let \(x_ E\) denote the diagonal matrix with y, y entry \(E_{xy}\) (y\(\in X)\). Define the representation diagram \(D_ E\) on the nodes 0,1,...,d by drawing an undi (1987).
A graph is said to be k-neighbour connected if the removal of \(t<k\) closed neighbourhoods result neither in a disconnected nor in a complete graph. The authors give the sharp lower bound on the number of vertices of k-regular, k-neighbour connected graph whose every 5-cycle belongs to a clique.
Reviewer: P.Horák

MSC:
05B30 Other designs, configurations
05C40 Connectivity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin-Cummings Lecture Note Series 58. CA: Menlo Park 1984 · Zbl 0555.05019
[2] Bannai, E.: Personal communication
[3] Bannai, E., Damerell, R.M.: Tight spherical designs, I. J. Math. Soc. Japan (3)1, 199–207 (1979) · Zbl 0403.05022 · doi:10.2969/jmsj/03110199
[4] Bannai, E., Damerell, R.M.: Tight spherical designs, II. J. London Math. Soc. (2)21, 13–30 (1980) · Zbl 0436.05018 · doi:10.1112/jlms/s2-21.1.13
[5] Delsarte, P.: An algebraic approach to the association schemes of coding theory. Phillips Res. Rep. Supple.10, 1–97 (1973) · Zbl 1075.05606
[6] Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata6, 363–388 (1977) · Zbl 0376.05015 · doi:10.1007/BF03187604
[7] Egawa, Y.: Characterization ofH(n, q) by the parameters. J. Comb. Theory (A)31, 108–125 (1981) · Zbl 0472.05056 · doi:10.1016/0097-3165(81)90007-8
[8] Happel, D., Preiser, U., Ringel, C.M.: Binary polyhedral groups and Euclidean diagrams. Manuscripta Math.31, 317–329 (1980) · Zbl 0436.20005 · doi:10.1007/BF01303280
[9] McKay, J.: Graphs, singularities, and finite groups. Proc. of Symp. in Pure Math.37, 183–186 (1980) · Zbl 0451.05026
[10] Neumaier, A.: Characterization of a class of distance-regular graphs. J. Reine Angew. Math.357, 182–192 (1985) · Zbl 0552.05042 · doi:10.1515/crll.1985.357.182
[11] Seneta, E.: Nonnegative matrices, an Introduction to Theory and Applications. New York. Halsted Press 1973 · Zbl 0278.15011
[12] Terwilliger, P.: A characterization of theP- andQ-polynomial association schemes. J. Comb. Theory (A) (to appear)
[13] Terwilliger, P.: A noncommutative algebra for association schemes (submitted)
[14] Terwilliger, P.: Root systems and the Johnson and Hamming graphs. Europ. J. Comb. (to appear) · Zbl 0614.05048
[15] Terwilliger, P.: TheP- andQ-polynomial schemes withq = . J. Comb. Theory (B) (to appear)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.