The routes for a small airline are shown in Figure 4.11. In terms of a network, the five cities are the vertices or nodes, and the six air routes are the connections, in a similar manner to those...

The routes for a small airline are shown in Figure 4.11. In terms of a network, the five cities are the vertices or nodes, and the six air routes are the connections, in a similar manner to those shown in Figure 4.11.

(a) Number the cities from 1 to 5, and from this write down the corresponding adjacency matrix.

(b) Compute the eigenvalues of the matrix you found in part (a). Also, it can be proved that if you sum up the eigenvalues of an adjacency matrix, you get zero (see Section 4.3.4). Do your values satisfy this condition? If not, provide a reason why.

(c) Number the cities in a different order than you used in part (a) and compute the eigenvalues of the resulting adjacency matrix. How do these differ from what you computed in part (b)?