A classification of graphs whose subdivision graphs are locally $G$-distance transitive

Abstract

The subdivision graph $S(\hat{I}\pounds )$ of a connected graph $\hat{I}\pounds$ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs $\hat{I}\pounds$ such that $S(\hat{I}\pounds )$ is locally $(G,s)$-distance transitive for $s\le 2diam(\hat{I}\pounds )-1$ and some $G\le Aut(\hat{I}\pounds )$. In this paper, we solve the remaining cases by classifying all the graphs $\hat{I}\pounds$ such that the subdivision graphs is locally $(G,s)$-distance transitive for $s\ge 2diam(\hat{I}\pounds )$ and some $G\le Aut(\hat{I}\pounds )$. In particular, their subdivision graph are always locally $G$-distance transitive, except for the complete graphs.

10 pages