graph problem

[spyroy]

if G is graph with 2k + 1 vertices and the degree of every vertex is k so G is connected.
what is the diameter?
so i figured out that k has to be even otherwise there is no way to draw the graph, and that the graph must be a circle.
but how do i prove that G is connected??
thx from advance

 

[spyroy]

So i realised that because the graph have 2k+1 vertices if we take two separate vertices and say that the graph is not connected so we will get 2k+2 vertices bacause every vertex have degree k which is contradiction, and the diameter is 2k because we have at least two edges between every vertex, but is this enough for a proof?