Connected cubic $s$-regular graph Let $X$ be connected cubic $s$-regular graph then $|Aut(X)| = 2^{s-1}\cdot 3\cdot |V(X)|$. I want a reference for proof.
 A: Biggs's "Algebraic Graph Theory". I would assume it's used in Tutte's paper on arc-transitive cubic graphs. 
But all this result says if that if $\mathrm{Aut}(X)$ is $s$-regular then $|\mathrm{Aut}(X)|$ is equal to the number of $s$-arcs. This would not normally need a reference.
A: By your assumption that the graph $X$ is cubic and $s$-regular, I assume you mean that $X$ is 3-regular and $s$-arc-regular; the latter means (by definition) that $Aut(X)$ acts regularly on the $s$-arcs of the graph.    (A permutation group is regular if it is transitive and semiregular.)  Thus, given any two $s$-arcs $S,S'$, there is a unique automorphism of the graph that maps $S$ to $S'$.  This implies $|Aut(X)|$ equals the number of $s$-arcs.  How many $s$-arcs $(v_0,v_1,\ldots,v_s)$ does a connected, 3-regular graph have?  The first vertex $v_0$ can be chosen in $|V|$ ways, then $v_1$ can be chosen in 3 ways to be any one of the three neighbors of $v_0$, and each of the remaining $v_i$'s $(i \ge 2)$ can be chosen to be any neighbor of $v_{i-1}$ except $v_{i-2}$, i.e. in 2 ways (by definition of $s$-arcs, every $s$-arc needs to satisfy the condition that $v_{i-2}$ cannot be the same vertex as $v_i$).  Thus, the number of $s$-arcs in a connected, cubic graph on $|V|$ vertices is $|V| \times 3 \times 2^{s-1}$.  
In addition, the statement can be strengthened since  in the case of cubic graphs the hypothesis that $X$ is $s$-arc-regular can be replaced by the hypothesis that $X$ is just $s$-transitive (i.e. $Aut(X)$ is transitive on the set of $s$-arcs but not transitive on the set of $(s+1)$-arcs).  More precisely, it can be shown that if a connected graph is cubic and symmetric, then it is necessarily $s$-arc regular for some $s \ge 1$. This is proved in the paper [Tutte, ''On the symmetry of cubic graphs,'' Canadian Journal of Mathematics, pp. 621-624, 1959], and a proof is also in the text [Biggs, Algebraic Graph Theory].  
