Graphs with $\operatorname{diam}(G)=2\operatorname{rad}(G)$

Question

When is the diameter of a graph equal to twice of radius? I am currently studying graph theory and have faced many questions related to graphs with the mentioned property. Is there any general class of graphs which follow this property?

I know path graphs with odd vertices are such graphs, but is there a more general graph?

Answer 1

As mentioned in the comments by munchhausen and me, there exist at least two classes of graphs with the "$d=2r$ property", generalizing your example of a path with an odd number of vertices:

• A tree has this property if and only if it has a center vertex.
• An interval graph has this property if and only if its diameter is even.

The intersection of these two classes is quite a small proportion of either, which suggests that the collection of graphs with the $d=2r$ property may be (otherwise) quite heterogeneous.