Biplanes (56,11,2) with a fixed-point-free involutory automorphism

The aim of this article is to prove that exactly four biplanes with parameters (56,11,2) admit a fixed-point-free action of an involutory automorphism. These are: Hall\u27s biplane B20, Salwach and Mezzaroba\u27s biplane B22, Denniston\u27s biplane B24 and Denniston\u27s biplane B26

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