from $u$ to $v$, and this geodesic is called an $i$-geodesic if the distance between $u$ and $v$ is $i$.

The graph $\Gamma$ is said to be \emph{$s$-geodesic transitive} if the graph automorphism group is transitive

on the set of $s$-geodesics. In this talk, I will compare the $s$-geodesic transitivity with other two

well-known transitive properties, namely $s$-arc transitivity and $s$-distance transitivity, and determine the local structure

of $2$-geodesic transitive graphs, and also give some results about the family of locally disconnected 2-geodesic transitive but not 2-arc transitive graphs.