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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 ?

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The diameter is $3$, but the radius is $2$: the eccentricities of $A,B,C$ and $D$ are $3,2,2$, and $3$, and the radius is the minimum of these numbers. –  Brian M. Scott Apr 15 '12 at 20:11
    
Oh yes, you are right. I got the definition of eccentricity wrong. –  abc Apr 15 '12 at 20:21
    
@Brian, I guess that should be posted as an answer. –  Rahul Apr 15 '12 at 21:09

1 Answer 1

up vote 3 down vote accepted

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, $d\le 2r$.

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