The Fiedler Rose: On the extreme points of the Fiedler vector
  In this paper I present a counter-example to the conjecture: The Fiedler vector for the graph Laplacian of a tree has its most extreme values at the verticies which are the farthest apart. This counter-example looks roughly like a flower and so I have named it the "Fiedler rose". 
Lawrence Christopher Evans
math.GM