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 $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular 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 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

Virtually Abelian Quantum Walks

We introduce quantum walks on Cayley graphs of non-Abelian groups. We focus on the easiest case of virtually Abelian groups, and introduce a technique to reduce the quantum walk to an equivalent one on an Abelian group with coin system having larger dimension. We apply the technique in the case of two quantum walks on virtually Abelian groups with planar Cayley graphs, finding the exact solution.Comment: 10 pages, 3 figure