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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The "less than or equal to" comparison operator in Java is __________.
a. < b. <= c. =< d. << e. !=
Answer the following statements true (T) or false (F)
1. If you set a group box control’s Visible property to False, the attached controls will still remain visible. 2. When a check box control has the focus, the spacebar can be used to invoke its CheckedChanged event. 3. As with the button and check box controls, an access key can be set up by placing an ampersand in front of a letter in the radio button’s Text property setting. 4. Suppose a form contains two group box controls with one radio button placed on each. In this case, both radio buttons can be selected (or turned “on”) at the same time.