If $G$ is connected then $\lambda_2 < \lambda_1$.

Question

Let $G=(V,E)$ be an $n$-vertex , undirected graph with maximum degree $d$, then how to prove the following result.

If $G$ is connected then $\lambda_2 < \lambda_1$. where $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$ are eigen values of adjacency matrix of G.

Can someone help me. Thanks.

Answer 1

A basic question in Spectral Graph Theory. Please refer to Dr. Spielman's notes. (link is here)