Let $X=(V,E)$ be a finite, connected, regular graph with diameter $D$. Is it true that, for every $x\in V$, there exists $y\in V$ such that $d(x,y)=D$? (the answer is clearly yes if $X$ is vertex-transitive).



Realizing the diameter of a finite regular graph
[Counter-example http://www.freeimagehosting.net/uploads/9a4165cab4.png](http://www.freeimagehosting.net/uploads/9a4165cab4.png)
The diameter is 8, but 1 is centered with at most 5 as distance to every other.


