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Characterization of a class of distance regular graphs. (English) Zbl 0552.05042
A distance regular graph of diameter \(d\geq 3\) with intersection array \(b_ i=(d-i)(a-ci),\) \(c_ i=i+c\binom{i}{2}\) is one of the graphs of Hamming, Egawa, Johnson, the half cubes, or the Gosset polytope \(3_{21}\). For \(c>0\), the proof proceeds by representing the graph inside the root lattices \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), or \(E_ 8\).

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
05C99 Graph theory
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