We consider the problem of which distanceregular graphs with small valency are Cayley graphs. We determine the distanceregular Cayley graphs with valency at most 4, the Cayley graphs among the distanceregular graphs with known putative intersection arrays
for valency 5, and the Cayley graphs among all distanceregular graphs with girth 3 and valency 6 or 7. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distanceregular graphs that come from small generalized quadrangles or hexagons. Among some âexceptionalâ distanceregular graphs with small valency, we find that the ArmaniosWells graph and the Klein graph are Cayley graphs.
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