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> A $k$-regular graph is a graph with all vertices having degree k.
>  A graph $X$ is called vertex-transitive if it's automorphism group acts transitively on the vertex set.
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> We know that all the vertex-transitive graphs are regular graphs but my question is that whether the reverse of this statement true with the additional condition that graph is connected. 
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Is connected k-regular graphs are always vertex-transitive?
No. For example, the [Frucht graph](https://en.wikipedia.org/wiki/Frucht_graph) is 3 regular but not vertex transitive.


