I know very little about graphs. I believe the following should be trivial, but I couldn't find an argument to prove it. Let $\Gamma$ be an undirected, connected, vertex-transitive graph on $n$ vertices.

What is known about an upper bound for $d(\Gamma)$ the diameter of $\Gamma$? It seems that the worst case scenario is when $\Gamma$ is a cycle and then $d(\Gamma)=\lfloor n/2\rfloor$, is that true? What is known if it is not a cycle?

Is it true that any two vertices in $\Gamma$ can be connected by two non-intersecting paths, that is, any two vertices are contained in a cycle? (That will show of course that $d(\Gamma) \leq n/2$.)