Distance-regular Cayley graphs with small valency

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.

On Minimal Distance-Regular Cayley Graphs of Generalized Dihedral Groups

“…Remark 18. We would like to point out that the incidence graph of the Desarguesian affine plane of order 4 minus a parallel class of lines from the above proof is indeed a Cayley graph (see for instance [5,Proposition 3.2]). In fact, it is a Cayley graph of the generalized dihedral group GD(Z 4 × Z 4 ).…”

“…While in the references cited in the above paragraph the diameter of the graph was restricted to be 2, van Dam and Jazaeri [5] choose to restrict the valency of a distanceregular graph. They determined all distance-regular Cayley graphs with valency 3 and 4, and they characterized the Cayley graphs among all distance-regular graphs with valency 5 and with one of the known feasible intersection arrays.…”

On bipartite distance-regular Cayley graphs with diameter $3$

“…Furthermore, if a projective plane is Desarguesian, then its incidence graph is a Cayley graph on a dihedral group (cf. [4,Section 3.5]). Therefore we can conclude the following proposition.…”

On distance-regular Cayley graphs of generalized dicyclic groups

“…Abdollahi, van Dam and Jazaeri [1] determined all distance-regular Cayley graphs with least eigenval-ues −2. Very recently, van Dam and Jazaeri [5] determined 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. In addition, they also studied bipartite distance-regular Cayley graphs with diameter 3 or 4 [6].…”