Distance-regular Cayley graphs with small valency

TL;DR

This paper classifies which small valency distance-regular graphs are Cayley graphs, identifying specific cases and providing new examples like the Armanios-Wells and Klein graphs.

Contribution

It determines all Cayley graphs among distance-regular graphs with valency up to 4 and identifies new Cayley graph instances among certain known classes.

Findings

Cayley graphs with valency at most 4 are fully classified.

Incidence graphs of Desarguesian affine planes minus a parallel class are Cayley graphs.

The Armanios-Wells and Klein graphs are identified as Cayley graphs.

Abstract

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.