Is it true that any vertex-transitive graph $G$ has a vertex-transitive line graph $L(G)$? It seems to me that if I proved that for any $$e_1=\{x,y\},e_2=\{a,z\} \in E(G)$$ there exists an automorphism $\rho\in \text{Aut}(G)$ s.t. $$x=\rho(a) \wedge y=\rho(z) (1)$$ I would have a positive answer. There is probably a connection between the automorphisms' group $\text{Aut}(G)$ too (all are conjugate. Probably not a very helpful notice), but I think that there must be counterexample where $(1)$ doesn't hold. Could anyone help me clear this out?
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$\begingroup$ One has to be able to understand your question without its title, and in the case of yours that is impossible. $\endgroup$– Mariano Suárez-ÁlvarezMay 4 at 6:55
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$\begingroup$ By the way, no one is in such a hurry that is not able to write such that in full! $\endgroup$– Mariano Suárez-ÁlvarezMay 4 at 6:56
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$\begingroup$ I edited the post @MarianoSuárez-Álvarez $\endgroup$– Νικολέτα ΣεβαστούMay 4 at 6:58
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From https://mathworld.wolfram.com/Vertex-TransitiveGraph.html:
'A undirected connected graph is edge-transitive if its line graph is vertex-transitive'
Thus, if for any vertex-transitive graph $G$, the graph $L(G)$ is also vertex-transitive, it means that any vertex-transitive graph is edge-transitive.
This is false, see for a counter examples and more information: Graphs that are vertex transitive but not edge transitive