I am trying to prove this theorem: Let $S = \{ \lambda_{1},\lambda_{2},\dots, \lambda_{n} \}$ be the spectrum of an undirected graph $G$, where $ \lambda_{1}\ge\lambda_{2}\ge \dots \ge\lambda_{n}$. Then:
If $G$ contains at least one edge, then $ 1 \le \lambda_{1} \le n-1 $ and $ -\lambda_{1} \le \lambda_{n} \le -1 $.
If $G$ is connected, then $ 2\cos \left (\dfrac{\pi}{n+1}\right)\le\lambda_{1} \le n-1 $.
Any idea?