TREE AND SHRUBSLeaf it to me

graph theory proof?

Let G be a connected graph with radius r (where rad=min eccentricity of the graph) and diameter d (where diam=max eccentricity). Knowing that r<=d<=2r, improve this result for trees by proving the thm:
Thm: Let T be a tree with radius r and diameter d. Then f(r)<=d<=g(r) where f and g are simple functions which you must determine.
Hint: Consider cases depending on the number of vertices in the center.

I’m not too familiar with the concept of the center of a graph, so I won’t be able to shed much light on the hint.

I can give you some help in determining the functions f(r) and g(r). The extreme cases for trees would be paths and stars, so start there. Then you might want to consider how adding edges (particularly to a path) will alter your radius and diameter.

As for your hint: it seems to me that in a tree the center is either a single vertex or two vertices. I’m not sure how to go about proving this, but if you could manage that (or if you’ve already got it as a theorem) then I think the two cases would be what your hint is getting at. Each case then would provide one of the two possible values for d (as a function of r) that I’ve gotten.

If you can’t get the f(r) and g(r), or if you get those and are having trouble putting together a proof, let me know.

Good luck!

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