Adjacency matrix and probability matrix

[TheMathNoob]

Homework Statement

If Γ is a k-regular simple graph and Γ its directed double, show that the matrix ˜ S for Γ (as per the FEATURED ARTICLE ) is a multiple of the adjacency matrix ˜ for Γ. Find the multiple. Assume k > 1.
The matrix S is the probability matrix. The probability of going from one node to another node based on the number of lines coming out of each node.

R

Homework Equations

The Attempt at a Solution

So it looks like the factor that maps T->S is 1/k so T*1/k=S.
Proof
This makes sense because the idea of adjacency is preserved on both matrices and their nonzero entries differ by a factor of 1/k because in S each nonzero entry represents the probability of going from one vertex to another adjacent vertex. In this case, it will always be 1/k because the number of lines leaving each node is always k. In T we just assert the adjacency between vertices by putting a 1.

That's a good start. I don't how to set this up, so it proves that the factor is 1/k

 

[HallsofIvy]

The phrase "as per the featured article" indicates that there is more to this problem that you have not told us!