# eccentricity in vertex transitive graphs

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.

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Can you define what is ecentricity? Is it just the valency of each vertex? – Easy Apr 22 '13 at 12:22
@Easy sir.... eccentricity of a vertex v is defined as the greatest distance between v and any other vertex of the graph. That is, e(v) is the distance between v and a vertex farthest from v – monalisa Apr 22 '13 at 12:29

## 2 Answers

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.

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Sir i got that complement will be also vertex transitive. But can you explain briefly how we get that the eccentricity will be same for all vertices in a transitive graphs. Thanks – monalisa Apr 22 '13 at 12:36

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

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thanks sir, but can you please elaborate it more. about eccentricities specially. – monalisa Apr 22 '13 at 12:38
Not without handing you the entire solution, I'm afraid. And don't sir me! – Henning Makholm Apr 22 '13 at 12:40