A self-centered graph is a graph whose diameter equals its radius. I am looking for examples of families of self-centered graphs. Here are some examples I know:

- Disconnected graphs
- cliques
- bicliques
- cycles
- Take two vertices and connect them by $k\geq2$ disjoint paths of the same length
- take a self-centered graph and replace every edge $uv$ by a 4-cycle $uxvy$ (e.g. when you start with a single edge and repeat this procedure k times you have the so-called k-th diamond graph which btw. is a nice example of a graph metric which needs high distortion to embed into Euclidean space)

Do you know some further nice examples? Especially I am looking for examples with large radius (for my purposes that means radius of size $\omega(\log n)$ where $n$ is the number of vertices) which are nonplanar. Also other ways than 6. to systematically construct self-centered graphs would be interesting.

**edit:** More examples:

- There is a complete list of all self-centered graphs with diameter 2 that have minimum number of edges in: http://www3.interscience.wiley.com/journal/119436272/abstract
- For every finite group $\Gamma$ there is a self-centered graph whose automorphism group is isomorphic to $\Gamma$ (S.-M. Lee, P.-C. Wang, On groups of automorphisms of self-centered graphs,. Bull. Math. Soc. Sci. Math. Roumanie)