Consider the digraph G with three vertices P1,P2 and P3 and four directed edges, one each from P1 to P2, P1 to P3, P2 to P3 and P3 to P1.
a. Sketch the digraph.
b. Find the adjacency matrix M of G.
c. Use the adjacency formula for inverses to find g(z)=(I - zM)-1.
d. Expand the (1,3) entry of g(z) as a power series up to terms of order 6.
e. Find the number of walks of length 6 from P1 to P3.