Question I found in "Introduction to Graph Theory" by Douglas B. West:
Let $G$ graph on $n$ vertices with connectivity $\kappa(G)=k \geq 1$. Prove that $n \geq k(diam(G)-1)+2$, when $diam(G)$ is the the maximal (edge) distance between a pair of vertices in $G$
Seems easy and I've seen the answer, too, which goes the same as thought: by taking the vertices the end vertices of the path of length $diam(G)$, and applying Menger's Theorem. I'm having a hard time understanding one part - it seems like all the paths need to be of the same length for the $n \geq k(diam(G)-1)+2$ to apply. Am I missing some part here?
Thanks in advance.