  The main result of this paper is that, if $\Gamma$ is a connected 4-valent
$G$-arc-transitive graph and $v$ is a vertex of $\Gamma$, then either $\Gamma$
is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or
$2|G_v|\log_2(|G_v|/2)\leq |\V\Gamma|$ and that this last bound is tight. As a
corollary, we get a similar result for $3$-valent vertex-transitive graphs.
