Symmetry properties of subdivision graphs

Abstract

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