eccentricity in vertex transitive graphs

Question

I am trying to prove the following..

If $G$ is a veretx transitive graph, then how can we prove that eccentricity of every vertex is same? Getting no idea from where to start? How to prove the same for its complement too. Any hint or suggestion will be helpful. Thanks for helping me.

Answer 1

Fix a vertex $v$. Suppose $u$ the vertex that has the greatest distance from $v$. Then an isomorphism $\varphi$ maps $u,v$ to $\varphi(u),\varphi(v)$ and preserves their distance. Assume for the vertex $\varphi(v)$, there exists a vertex $w$ has longer distance than $\varphi(u)$. Then $d(v,\varphi^{-1}(w))>d(v,u)$, a contradiction.

For the complement, simply notice that it is also vertex-transitive.

Answer 2

Hint: First prove that a graph isomorphism preserves distances, and therefore also eccentricities.