Geodesic transitive graphs of small valency

For a graph $\Gamma$, the {\em distance} $d_\Gamma(u,v)$ between two distinct vertices $u$ and $v$ in $\Gamma$ is defined as the length of the shortest path from $u$ to $v$, and the {\em diameter} $\mathrm{diam}(\Gamma)$ of $\Gamma$ is the maximum distance between $u$ and $v$ for all vertices $u$ and $v$ in the vertex set of $\Gamma$. For a positive integer $s$, a path $(u_0,u_1,\ldots,u_{s})$ is called an {\em $s$-geodesic} if the distance of $u_0$ and $u_s$ is $s$. The graph $\Gamma$ is said to be {\em distance transitive} if for any vertices $u,v,x,y$ of $\Ga$ such that $d_\Ga(u,v)=d_\Ga(x,y)$, there exists an automorphism of $\Gamma$ that maps the pair $(u,v)$ to the pair $(x,y)$. Moreover, $\Gamma$ is said to be {\em geodesic transitive} if for each $i\leq \mathrm{diam}(\Ga)$, the full automorphism group acts transitively on the set of all $i$-geodesics. In the monograph [Distance-Regular Graphs, Section 7.5], the authors listed all distance transitive graphs of valency at most $13$. By using this classification, in this paper, we provide a complete classification of geodesic transitive graphs with valency at most $13$. As a result, there are exactly seven graphs of valency at most $13$ that are distance transitive but not geodesic transitive.