On walk-regular graphs and graphs with symmetric hitting times
  Aldous [1] asked whether every graph in which the distribution of the return time of random is independent of the starting vertex must be transitive. We remark that this question can be reduced into a purely graph-theoretic one that had already been answered Godsil & McKay [6] and ask some questions motivated by this. 
Agelos Georgakopoulos
math.CO