TITLE: Let $\text{rad}(G) = r$ and $\text{diam}(G)=d.$ Show that for each $k$ with $r\leq k \leq d,$ there exists $w \in G$ such that $e(w)=k.$
QUESTION [1 upvotes]: Let $G$ be a multigraph with $\text{rad}(G) = r$ and $\text{diam}(G)=d.$ Show that for each $k$ with $r\leq k \leq d,$ there exists vertex $w \in G$ such that $e(w)=\text{max}\{d(w,x): x \in G\}=k.$
Could anyone advise me on how to prove this? Hints will suffice, thank you.

REPLY [0 votes]: The difference in eccentricity between any two nodes is at most 1.  There exists a path from from a node with eccentricity $r$ to a node with eccentricity $d$.
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