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

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.

Answer 1

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 $d$.