Spectral characterization of complete or complete bipartite graphs

Question

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.

Answer 1

Here is the link to paper in question: On the normalized Laplacian eigenvalues of graphs. Ars Comb.

Unfortunately only abstract seems to be available to the public

Let G = (V, E) be a simple connected graph with n vertices and m edges.
Further let λi(L), i = 1, 2, ., n, be the non-increasing eigenvalues of the normalized Laplacian matrix of the graph G. 
In this paper, we obtain the following result: 
For a connected graph G of order n, λ2(L) = λ3(L) = · = λn-1 (L)
if and only if G is a complete graph Kn or G is a complete bipartite graph Kp, q. 
Moreover, we present lower and upper bounds for the normalized Laplacian spectral radius of a graph and characterize graphs for which the lower or upper bounds is attained.