TITLE: Prove that the graph is connected
QUESTION [8 upvotes]: I was wondering if someone can help me understand how prove that this graph is connected.
Given a graph with n vertices, prove that if the degree of each vertex is at least $(n − 1)/2$ then the graph is connected.
So far I know that about connected graphs:
An undirected graph is called connected if there is a path between
every pair of distinct vertices of the graph
The distance between two vertices in a graph is the length of
the shortest path between them.
The diameter of a graph is the distance between the two vertices
that are farthest apart.

REPLY [7 votes]: Proof:
If each degree is at least that, then each component has at least one vertex and $\frac{n-1}{2}$ others, so at least $\frac{n-1}{2}+1=\frac{n+1}{2}$ vertices. Two components would have at least twice as many vertices, which is n+1, more than the graph has.
(I realize this is the same as David's, but I don't like unnecessary variables so I feel it's still worth posting.)
Edit: Different wording, appropriate for the OP's stated terminology/knowledge:
Alternative proof:
Take any two vertices. Each has at least $\frac{n-1}{2}$ neighbors. If there were no path between our two vertices, then the two neighborhoods must be separate and we'd have at least $2\cdot(1+\frac{n-1}{2})=n+1$ vertices, more than the graph has.