Distance-transitive graphs admitting elations

A. A. Ivanov

Izvestiya. Mathematics , Tome 35 (1990) no. 2, pp. 307-335

Résumé

A graph $\Gamma$ is called distance-transitive if, for every quadruple $x,y,u,v$ of its vertices such that $d(x,y)=d(u,v)$, there is an automorphism in the group $\operatorname{Aut}(\Gamma)$ which maps $x$ to $u$ and $y$ to $v$. The graph $\Gamma$ is called $s$-transitive if $\operatorname{Aut}(\Gamma)$ acts transitively on the set of paths of length $s$ but intransitively on the set of paths of length $s+1$ in the graph $\Gamma$. A nonunit automorphism a $\operatorname{Aut}(\Gamma)$ is called an elation if for some edge $\{x,y\}$ it fixes elementwise all the vertices adjacent to either $x$ or $y$. In this paper a complete description of distance-transitive graphs which are $s$-transitive for $s\geqslant2$ and whose automorphism groups contain elations is obtained. Bibliography: 30 titles.