# Radius and diameter of a line graph

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

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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Hi Brian, When I read this answer, I ran into a challenge. I knows $D$ Diameter is maximum of minimum paths between two vertex of a graph. $L(S)$ is maximum length of minimum paths from $s$ to other vertexes and $R$ Radius of graph is minimum number of $L(s)$ between all vertex of a graph. so in undirected graph can we conclude always $R \leq D$ and $R \geq D/2$ is hold? – user153695 Mar 27 '15 at 19:13
@user153695: Yes. – Brian M. Scott Mar 28 '15 at 0:31
Prof. Brian, my last question is, if you want choose one of these condition for undirected graph G, which one is better to select? – user153695 Mar 28 '15 at 7:02
@user153695: I would not say that either is better than the other: they express two different facts about the radius and diameter. – Brian M. Scott Mar 28 '15 at 21:59
Prof. Brian thanks so much, I found a new link on math.stackexchange.com/questions/1209847/… that exactly mentioned this question,would you please see it and share your experience? thanks. – user153695 Mar 29 '15 at 7:28