2,171 research outputs found
Distance-regular Cayley graphs with small valency
We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most , the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency , and the Cayley graphs among all
distance-regular graphs with girth and valency or . 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
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.Comment: 19 pages, 4 table
Quantum automorphisms of folded cube graphs
We show that the quantum automorphism group of the Clebsch graph is
. This answers a question by Banica, Bichon and Collins from 2007.
More general for odd , the quantum automorphism group of the folded -cube
graph is . Furthermore, we show that if the automorphism group of a
graph contains a pair of disjoint automorphisms this graph has quantum
symmetry.Comment: 19 pages, corrected a mistake in the proof of Theorem 2.2 + minor
change
- …