I am having a hard time with this question. I understood it when the graph was a complete cycle but I'm not sure if I am right with this problem.

Find the number of paths of length n between any two adjacent vertices in K3,3 for these values of n:

a) 2

b) 3

c) 4

d) 5

I drew my K3,3 graph (A complete bipartite graph).

So three points in one line and then three points on a different line. All three verticies on top connect to each verticy on the bottom. Okay got that.

Now we were taught that we can find the number of paths by using the adjacency matrix.

My matrix for K3,3 is:

A

0 0 0 1 1 1

0 0 0 1 1 1

0 0 0 1 1 1

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

Okay, so now if I do A^2, then I get 3 for the number of paths for length 2.

I get 9 paths for length 3

27 paths for length 4

and 81 paths for length 5.

Am I doing this way wrong? Thanks for the help!