Exercises on graphs, networks, and incidence matrices
Problem 12.1: (8.2 #1. Introduction to Linear Algebra: Strang) Write down the four by four incidence matrix A for the square graph, shown below.
(Hint: the first row has -1 in column 1 and +1 in column 2.) What vectors (x1, x2, x3, x4) are in the nullspace of A? How do you know that (1,0,0,0) is not in the row space of A?
Problem 12.2: (8.2 #7.) Continuing with the network from problem one, suppose the conductance matrix is
0
C 0 .
Multiply matrices to find ATCA. For f = (1, 0, −1, 0), find a solution to
ATCAx = f. Write the potentials x and currents y = −CAx on the square graph (see above) for this current source f going into node 1 and out from node 3.
