How to prove vertex-transitivity in regular graphs.

I have problems to prove wheter a regular graph is vertex-transitive or not. For instance, consider the following examples: the generalized Petersen graphs $P_{2,7},\;P_{3,8}$ and the Folkman graph. For the three of them it is known wheter or not they are vertex-transitive, but it's never explained the reason. Of course you can always try to find an automorphism to prove wheter or not they're vertex-transitive, but I am curious if a more intuitive, and less time-consuming guideline exist.

One gets more confused because a priori the "general newbie rule" is that a regular graph is always vertex transitive.

• Not all regular graphs are vertex transitive. See SemisymmetricGraph at mathworld or wikipedia – JMoravitz May 21 '15 at 19:26