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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.)
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