Extensions of $\mathit{GQ}(4,2)$, the completely regular case

The description of extensions of generalised quadrangles $\mathit{GQ}(s,t)$ with flag-transitive group of automorphisms is known. For $s=3$, in a series of researches a description of extensions was given, with no assumptions on the action of the group of automorphisms. While investigating extensions of $\mathit{GQ}(4,2)$, the former author has given the structure of hyperovals of $\mathit{GQ}(4,2)$ and proved that there exist no totally regular locally $\mathit{GQ}(4,2)$ graphs with $\mu=10$. In the present paper, we conclude the classification of totally regular locally $\mathit{GQ}(4,2)$-graphs.
This research was supported by the Russian Foundation for Basic Research, grant 99–01–00462.