# Prove that if $G$ is a graph with $n$ vertices, where $δ (G) ≥ \frac{n - 1}{2}$ , so $G$ is connected.

I would be very grateful for help with this proof:

"Prove that if G is a graph with n vertices, where $$δ (G) ≥ \frac{(n - 1)} {2}$$ , so $$G$$ is connected."

I know, that: $$δ (G)$$ ... is the minimum degree of the graph $$G$$

and I think, that maybe i can use this theorem:

If $$G$$ is a graph of order $$n$$ such that for every pair of nonadjacent vertices $$u$$ and $$v$$, $$deg$$ $$u$$ $$+$$ $$deg$$ $$v$$ is greater than or equal to $$n-1$$, then $$G$$ is connected

Could anyone please advise me how to prove this proof correctly, or if it can be done differently and better ?
Suppose for a contradiction that $$G$$ is not connected. Take a vertex $$v$$ on one connected component $$C_{1}$$. There must be at least $$(n-1)/2$$ additional vertices on the same component $$C_{1}$$ as $$v$$, for a total of $$1 + (n-1)/2$$ vertices on $$C_{1}$$. Now let $$u$$ be on some other component $$C_{2}$$. By a similar argument, $$C_{2}$$ must have at least $$1 + (n-1)/2$$ vertices.
Now at least how many vertices are in $$C_{1} \cup C_{2}$$? Derive your contradiction.