# Centre, diameter, and radius of graph

I have been thinking a lot on some questions related to centres, diameter ($D$), and radius ($R$) of an undirected connected graph, but couldn't find anywhere the answers, so am posting here.

Ques1. If $x,y$ are two vertices such that distance between them is $D$, then is it true that any shortest path between them will contain at least one centre?

Ques2. If $D=2R$, then does the graph contains only one centre?

Ques3. Also, can we say that if $D<<2R$, then graph would contain a lot of centres? (Intuitively I feel so because in figure below, $D$ is smaller than $2R$, and graph contains a lot of centres, the red coloured vertices.)

For Q1, if $x,y \in V$ such that $d(x,y) = D$, it is not true that any shortest path between $x$ and $y$ contains at least one central vertex (highlighted in red). Here, you can walk from 1 to 6 via 9, 5, and 8 avoiding the red vertex.