The Lemma 6 in this paper mention the following spectral characterization of complete or complete bipartite graphs:

Let $G$ be a connected graph with $\ge 2$ vertices. Then $\lambda_2=...=\lambda_{n-1}$ iff $G\cong K_n$ or $G\cong K_{p,q}$. Here $\lambda_1\ge\lambda_2\ge...\lambda_{n-1}\ge\lambda_{n}=0$ is the normalized Laplacian spectrum, i.e. the spectrum of the matrix $L^{sym}_G=D^{-1/2}L_GD^{-1/2}$.

The reference they give is rather difficult to locate. Can anyone give a reference or a detailed proof? Thanks.