On finite 2-path-transitive graphs

Hua Zhang (University of Western Australia, Australia)

Let $\Gamma=(V,E)$ be a graph with vertex set $V$ and edge set $E$. A $2$-arc is a triple of distinct vertices $(a,b,g)$ such that $b$ is adjacent to both $a$ and $g$. Identifying the two 2-arcs $(a,b,g)$ and $(g,b,a)$, we obtain a $2$-path, denoted by $[a,b,g]$. Let $G\le\mathrm{Aut}\Gamma$. Then $\Gamma$ is called $(G,2)$-path-transitive if $G$ acts transitively on the set of 2-paths of $\Gamma$.The class of 2-path-transitive graphs is slightly larger than the class of 2-arc-transitive graphs. In this talk we present a classification of vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. This leads to new constructions of half-transitive graphs.