Abstract
The distance-regular graphsΛ of type IIB in Bannai and Ito [1] have intersection numbers of the form
whered is the diameter of Λ, andh, x, andt are complex constants. In this paper we show a graph of type IIB and diameterd (3≦d) is either the antipodal quotient of the Hamming graphH(2d+1,2), or has the same intersection numbers as the antipodal quotient ofH(2d, 2).
Similar content being viewed by others
References
E.Bannai and T.Ito,Algebraic Combinatorics I: Association Schemes, Benjamin—Cummings Lecture Note Series 58, Menlo Park, CA, 1984.
N. Biggs,Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.
Y. Egawa, Characterization ofH(n, q) by the parameters,J. Combinatorial Theory (A),31 (1981), 108–125.
A. Gardiner, Antipodal covering graphs,J. Combinatorial Theory (B),16 (1974), 255–273.
A. A.Ivanov, private communication.
D. Leonard, Orthogonal polynomials, duality, and association schemes,Siam. J. Math. Analysis,13 (1982), 656–663.
D. Leonard, Parameters of association schemes that are bothP- andQ-polynomial,J. Combinatorial Theory (A),36 (1984), 355–363.
A.Neumaier, Characterisation of a class of distance-regular graphs,preprint.
D. Stanton, Someq-Krawtchouk polynomials on Chevalley groups,Amer. J. Math.,102 (1980), 625–662.
P.Terwilliger, A class of distance-regular graphs that areQ-polynomial,Submitted to J. Combinatorial Theory (B).
P.Terwilliger, Root systems and the Johnson and Hamming graphs,submitted to Europ. J. Combinatorics.
P.Terwilliger, Towards a classification of distance-regular graphs with theQ-polynomial property,submitted to J. Combinatorial Theory (B).