For PHPTR Only Chapter 8 Spectra of Complex Networks 1. Explain why the spectrum of a graph is independent of vertex labeling. Answer: When we change the vertex labeling, i.e., when we permute the vertex labels, the new adja- cency matrix can be written in the form Anew = P?1AP, where P is a permutation matrix and A is old adjacency matrix. Therefore, the new adjacency matrix is similar to the old one. Hence, the eigenvalues of the matrix does not change. 2. Prove
(a) The Laplacian of an undirected graph has non-negative eigenvalues.
(b) The eigenvalues of the normalized Laplacian of an undirected graph lie in the interval [0, 2].
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Fill in the blank(s) with the appropriate word(s).
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