# zbMATH — the first resource for mathematics

The classification of distance-regular graphs of type IIB. (English) Zbl 0727.05050
Summary: The distance-regular graphs $$\Gamma$$ of type IIB in [E. Bannai and T. Ito, Algebraic combinatorics. I: Association schemes (1984; Zbl 0555.05019)] have intersection numbers of the form $k=-hxd(t-1)^{-1}$ $b_ i=h(i-t)(i-x)(i-d)(2i-t)^{-1}(2i-t+1)^{-1}(1\leq i\leq d-1)$ $c_ i=-hi(i-t+x)(i-t+d)(2i-t)^{-1}(2i-t-1)^{-1}(1\leq i\leq d-1)$ $c_ d=-hd(d-t+x)(2d-t-1)^{-1},$ where d is the diameter of $$\Gamma$$, and h,x, and t are complex constants. In this paper we show a graph of type IIB and diameter d(3$$\leq d)$$ is either the antipoldal quotient of the Hamming graph $$H(2d+1,2)$$, or has the same intersection numbers as the antipodal quotient of H(2d,2).

##### MSC:
 05C75 Structural characterization of families of graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text:
##### References:
  E.Bannai and T.Ito,Algebraic Combinatorics I: Association Schemes, Benjamin–Cummings Lecture Note Series 58, Menlo Park, CA, 1984. · Zbl 0555.05019  N. Biggs,Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974. · Zbl 0284.05101  Y. Egawa, Characterization ofH(n, q) by the parameters,J. Combinatorial Theory (A),31 (1981), 108–125. · Zbl 0472.05056 · doi:10.1016/0097-3165(81)90007-8  A. Gardiner, Antipodal covering graphs,J. Combinatorial Theory (B),16 (1974), 255–273. · Zbl 0275.05119 · doi:10.1016/0095-8956(74)90072-0  A. A.Ivanov, private communication.  D. Leonard, Orthogonal polynomials, duality, and association schemes,Siam. J. Math. Analysis,13 (1982), 656–663. · Zbl 0495.33006 · doi:10.1137/0513044  D. Leonard, Parameters of association schemes that are bothP- andQ-polynomial,J. Combinatorial Theory (A),36 (1984), 355–363. · Zbl 0533.05016 · doi:10.1016/0097-3165(84)90042-6  A.Neumaier, Characterisation of a class of distance-regular graphs,preprint. · Zbl 0552.05042  D. Stanton, Someq-Krawtchouk polynomials on Chevalley groups,Amer. J. Math.,102 (1980), 625–662. · Zbl 0448.33019 · doi:10.2307/2374091  P.Terwilliger, A class of distance-regular graphs that areQ-polynomial,Submitted to J. Combinatorial Theory (B). · Zbl 0568.05029  P.Terwilliger, Root systems and the Johnson and Hamming graphs,submitted to Europ. J. Combinatorics. · Zbl 0614.05048  P.Terwilliger, Towards a classification of distance-regular graphs with theQ-polynomial property,submitted to J. Combinatorial Theory (B).
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.