diameter and radius in graph and disconnected graph

Question

We have an undirected Graph $G$.

Def $1$: The diameter of a graph is defined as the maximum of shortest paths between two vertices of $G$.

Def $2$: we define $L(S)$ as maximum length of shortest paths from $S$ to other vertices.

Def $3$: we define the radius of a graph as minimum value of $L(S)$ between all vertices in $G$.

If $\mathrm{diam}$ and $\mathrm{rad}$ be diameter and radius of graph which of the following was correct always (choose the best option):

1. $\mathrm{rad} \geq \frac{\mathrm{diam}}{2}$

2. $\mathrm{rad} \leq \mathrm{diam}$

Answer: 1 is the best option.

My question is why the author choose the 1 while 2 is also true. Then my idea is because in the question there is no assumption for connected graph so on disconnected graph option 1 can handle $\infty$ but option 2 cannot.

I think here by using best option words it means there is a case that we can support by one option and cannot support by another ones.

I know both of them is upper and lower bound but here there is a trick by the words "best option". is my conclusion about supporting disconnected graph is correct and reason for choosing $(1)$ is that support connect and disconnected graph at same time? if no is there any reasonable logic here?

Answer 1

Both are always true (with the convention that for a disconnected graph, radius and diameter are infinite).

Which is better really depends on context: what do you need the bound for? If you need a lower bound then the first is better, and if you need an upper bound the second is.

Without context, there's no reasonable way to say one bound is better than the other.

One thing which could apply, but doesn't here, is whether the bounds are best possible. For example, if the second said $\mathrm{rad}\leq 2\mathrm{diam}$, then that would be a worse bound, because although true it is weaker than it needs to be.

However, in this case both bounds are best possible (so equally good by this measure), since for any $k\geq 1$ there exist graphs $G,H$ with $\mathrm{rad}(G)=k,\mathrm{diam}(G)=2k$ and $\mathrm{rad}(H)=k,\mathrm{diam}(H)=k$ (e.g. $G$ is the path of length $2k$ and $H$ is the cycle of length $2k$).