Degree of a regular graph to be connected

Question

Let $G$ be a graph of $\;n\;$ vertices in which every vertex has a degree equal to $\;d\;$.How large must $\;\;d\;\;$ be for $\;G$ to be connected?

Please provide a hint! (I should try at least from my side :) )

Thanks !

Answer 1

(Compiling information so the question can be removed from unanswered queue)

The graph $K_5\cup K_5$, the disjoint union of two complete graphs on five vertices, is an example of a $4$-regular graph on ten vertices which is not connected.

Generalizing, $K_{n/2}\cup K_{n/2}$ for $n$ even is an example of an $n/2 - 1$ regular graph which is not connected.

If you were to suppose that a graph wasn't connected, then there must be at least two connected components, and at least one of those must be the "smallest" or tied for the smallest in terms of number of vertices. (why?). Further, the vertices in that smaller component must have smaller than a certain degree (why?). This all implies something useful.

(Note: a bit of extra care needs to be taken for the argument for $n$ odd, but you can skirt around the issue by remembering the handshaking lemma implies you cannot have an odd number of odd degree vertices)