Prove that every graph with diameter d and girth 2d+1 is regular How to prove that every graph with diameter $d$ and girth $2d+1$ is regular.
I know just relation between diameter and girth. which is given by below formula
$girth(G)\leq 2diam(G)+1$

The proof of this fact that I know goes as follows. Suppose we have the following claim

Claim 1. If $G$ is a graph of diameter $d$ and girth $2d+1$ then  any two vertices $u,v$ at distance $d$ have the same degree.

Once you establish Claim 1 your claim follows easily. If $C$ is a $2d+1$ cycle in $G$ then by Claim 1 all vertices on $C$ have the same degree. For any vertex not on $C$ you can find a path of length $d$ to a vertex on $C$ and hence by Claim 1 you are done.
Hence it only remains to prove Claim 1, which I think you can do yourself. If you need any hints on that as well post a comment.
