Eccentricity of vertices in a regular graph

Question

I was just trying to find out the eccentricity of the vertices in regular graphs, given in the link http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html#CRG. Surprisingly, eccentricity is the same in all graphs except http://www.mathe2.uni-bayreuth.de/markus/REGGRAPHS/GIF/08_3_3-2.gif. I checked up to order 8. How can we show that the eccentricity of all vertices of regular graphs are the same or not?

Answer 1

Take $K_4$ and subdivide one edge (put a vertex of degree two in the middle of it). Now take two copies of the subdivided graph and add an edge joining the two vertices of valency two. This new graph is cubic on 10 vertices and the eccentricities range from 3 to 5. This construction is easy to generalize; we can start with any pair of cubic graphs.

Answer 2

There are further counterexamples: the Folkman graph is a $4$-regular graph on $20$ vertices with radius $3$ and diameter $4$.

The Folkman graph is semi-symmetric, which means that it is regular and edge-transitive but not vertex-transitive. According to MathWorld, no graph on fewer than $20$ vertices is semi-symmetric. I don't know whether there are smaller examples of graphs that are regular but neither edge-transitive nor vertex-transitive.