I need to find the transition matrix for a hexagonal bipyramid. I believe it has something to do with the degree of the vertices. I know the degree of the top and bottom vertices are both 6, and the degree of all other vertices is 4. How do I put this together into a matrix?
This is to create a Markov chain of going from one vertex to another.
To calculate the Markov chain transition matrix entry $a_{ij}$ you need to know the probability that you visit vertex $j$ when starting from $i$.
If that probability is uniform on edges at each vertex then it's $1/6$ for $6$ of the $8$ vertices when you are leaving the top, since you don't stay where you are and can't go to the bottom.
You should be able to figure out the rest.