The Good Will Hunting Problem

Here is a problem from the movie Good Will Hunting, shown in the screenshot below.



For the the graph G(V,E) shown above, find the following :

• The adjacency matrix A :

• The matrix giving the number of 3 step walks in G : [A^k]_ij is the number of paths of length k from i to j. So, the answer is A³.

• The generating function for walks from point i to j : The generating function is as follows. Here are more examples of generating functions.
  

• The generating function for walks from points 1 to 3 : Simplify the above formula using cramer's rule for i=1 and j=3.