A graph $\Gamma$ is called prime with respect to the Cartesian product if
$\Gamma=\Gamma\_1\square\Gamma\_2$ implies that $\Gamma\_1=K\_1$ or $\Gamma\_2=K\_1$, where $\square$ denote the Cartesian product.
Is there any classification of connected and vertex-transitive prime graphs with respect to Cartesian product? Is there any information about the automorphism group of such graphs?



connected and vertex-transitive prime graphs with respect to Cartesian product
There almost certainly isn't a meaningful classification of vertex-transitive prime graphs. In fact, it's quite likely that almost all vertex-transitive graphs are prime. 


