4.1 Transitive graph: just definition

An isomorphism from a graph $X$ to itself is called an automorphism of $X$. An automorphism is therefore a permutation of the vertices of $X$ that maps edges to edges and nonedges to nonedges.

A graph $X$ is vertex transitive (or just transitive) if its automorphism group acts transitively on $V(X)$.

A graph $X$ is edge transitive if its automorphism group acts transitively on $E(X)$.

An arc in $X$ is an ordered pair of adjacent vertices, and $X$ is arc transitive if $Aut(X)$ acts transitively on its arcs. An arc transitive graph is necessarily vertex and edge transitive.

A graph $\Gamma$ is called $s$-arc-transitive with $s\ge 1$ if it has an $s$-route and if there is always a graph automorphism of $\Gamma$ sending each $s$-route onto any other $s$-route, where $s$-route is a sequence of vertices $(v_0,\cdots ,v_s)$ of $\Gamma$ such that $v_i{v}_{i+1}\in E(\Gamma )$ and $v_{i-1}\ne v_{i+1}$.