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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

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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
    
@dtldarek I agree with you. So, make it an answer so I can accept it. –  Graphth Mar 14 '12 at 14:21
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up vote 1 down vote accepted

I was asked to post the comment as an answer, so here it is:

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.

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