Graphs with diameter equal to two times the radius

In basic graph theory books, we learn that the radius (rad) and diameter (diam) satisfy $$rad(G) \leq diam(G) \leq 2 rad(G)$$ I have seen books talk about graphs for which $rad(G) = diam(G)$. These graphs are called self-centered graphs. But, I have never seen anything on graphs satisfying the other bound, $diam(G) = 2 rad(G)$. I have googled for papers on the topic many times and never come up with anything. Do any of you know of any such papers/books that might mention this?

Thanks

• Maybe that property is uninteresting? For any graph of size $n$ with single connected component, pick any vertex, and then create 2 paths of length $n$ from this vertex by adding $2n$ new vertices (and $2n$ edges). Not much has changed, but $diam(G) = 2rad(G)$ now. – dtldarek Mar 13 '12 at 15:34

Maybe that property is uninteresting? For any graph of size $n$ with single connected component, pick any vertex, and then create 2 paths of length $n$ from this vertex by adding $2n$ new vertices (and $2n$ edges). Not much has changed, but now $\mathrm{diam}(G)=2\mathrm{rad}(G)$ now.