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.)