Symmetry properties of subdivision graphs

The subdivision graph $S(\mathrm{\Sigma })$ of a graph $\mathrm{\Sigma }$ is obtained from $\mathrm{\Sigma }$ by `adding a vertex’ in the middle of every edge of $\text{Si}$. Various symmetry properties of $\mathrm{\S }(\mathrm{\Sigma })$ are studied. We prove that, for a connected graph $\mathrm{\Sigma }$, $S(\mathrm{\Sigma })$ is locally $s$-arc transitive if and only if $\mathrm{\Sigma }$ is $\lceil \frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(\mathrm{\Sigma })$ is $2d+\delta$, where $\mathrm{\Sigma }$ has diameter $d$ and $0\leqslant \delta\leqslant 2$,andlocal$s$-distancetransitivityof$\S(\Sigma)$ is defined for $1⩽s⩽2d+\delta$. In the general case where $s⩽2d-1$ we prove that $S(\mathrm{\Sigma })$ is locally $s$-distance transitive if and only if $\mathrm{\Sigma }$ is $\lceil \frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d⩽s⩽2d+\delta$, we classify the graphs $\mathrm{\Sigma }$ for which $S(\mathrm{\Sigma })$ is locally $s$-distance transitive in the cases, $s⩽5$ and $s⩾15+\delta$. The cases $\max{2d, 6}\leqslant s\leqslant \min{2d+\delta, 14+\delta}$ remain open.