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Root systems and the Johnson and Hamming graphs. (English) Zbl 0614.05048
Let $$\Gamma$$ be any distance regular graph with diameter d, $$c_ 2\geq 0$$ and $c_ i-c_{i-1}+b_{i-1}-b_ i-a_ i-2=0.$ Then it is shown that $$\Gamma$$ is a graph of Hamming type one of the Johnson graphs, the halved graph of the n-cube, a cocktail party graph, the Shrikhande graph or one of a finite number of exceptional graphs with $$d\leq 8.$$
This result improves an earlier result of the author. The proof is built round the idea that any $$\Gamma$$ satisfying the equation above can be represented by a set of equi-length vectors spanning a Euclidean space which also contains a root system consisting of an orthogonal union of root systems of type $$A_ n$$, $$D_ n$$, $$E_ 6$$, $$E_ 7$$, $$E_ 8$$. The graphs listed above appear as consequences of the form of the root system.
Reviewer: D.A.Holton

##### MSC:
 05C75 Structural characterization of families of graphs 05C99 Graph theory
##### Keywords:
distance regular graph; root systems
Full Text:
##### References:
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