Minimum degree of a graph at least $\frac{1}{2}(n-1)$ implies connectedness

Assume there are at least $2$ connected components. Then there is a vertex in each component that has degree at least $\frac{1}{2}(n-1)$. What does that imply about the number of vertices in each component, and thus the total number of vertices in the graph?

Not only is the graph connected, its diameter is at most $2$.

Suppose to the contrary that there are $2$ or more components.

If $n$ is even, say $n=2k$, some component has $\le k$ vertices.

If $n$ is odd, say $n=2k+1$, some component has at most $k$ vertices.