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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
##### Keywords:
distance regular graphs; Johnson graphs; root lattice
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