Eccentricity of vertices in a regular graph I was just trying to find out the eccentricity of the vertices in regular graphs, given in the link < Surprisingly, eccentricity is the same in all graphs except < I checked up to order 8. How can we show that the eccentricity of all vertices of regular graphs are the same or not?

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.