Math410 (Bayly) Last Homework (due self-graded on Monday 3 May)

6.1 1,2,(5), (7), 9

6.2 (1), 2, (3), 4, 5, (6), 7, (15)

6.3 (1), 2, 10, (11)

6.4 (1), 3, (11)

EXTRA CREDIT: Either ½-1 page on presentation by Dr. Graham, or to calculate the graph Laplacian and its eigenvalues for a couple of 3-node graphs. The graphs all have nodes A, B, C but different collections of arrows: (a) an arrow from A to B and an arrow from C to B, (b) an arrow from A to B and from B to C, (c) arrows from A to B, B to C, and C to A.

Besides answering these specific questions, also say whether it makes any difference what directions the arrows point along an edge. See if you can say what the Laplacian matrix would be directly from looking at the graph, without first finding E and doing the multiplication.

Recall that the edge-node matrix E has rows corresponding to arrows and columns corresponding to nodes. The row corresponding to an edge has a -1 at the “tail” node and a +1 at the “head” node, and zero in all other entries.

The Laplacian matrix is L = E^T E . I said that L always has (1,1,1)^T as an eigenvector belonging to eigenvalue 0; Say in your own words why you expect this always to be true.