Suppose I have a graph which is like this:
A--B--C--D
What is the diameter and radius of this graph?
Here r = 1 and d = 3 and r < d/2 ..right ?
Suppose I have a graph which is like this:
A--B--C--D
What is the diameter and radius of this graph?
Here r = 1 and d = 3 and r < d/2 ..right ?
The diameter is $3$, but the radius is $2$: the eccentricities of $A,B,C$, and $D$ are $3,2,2$, and $3$, respectively, and the radius is the minimum of the eccentricities.
Note that you can never have $r<d/2$. If $u$ is a vertex of eccentricity $r$, and $v$ and $w$ are any vertices, there must be paths of length at most $r$ from $v$ and $w$ to $u$, so there must be a path of length at most $2r$ from $v$ to $w$. Thus, $d\le 2r$.