I'm trying to prove this:
There are only three non isomorphic trees with 5 vertices.
The way it was done in my book is, they drew the three non isomorphic trees and they proved why every other tree has to be isomorphic to one of them.
I was thinking of a different approach however. I want to somehow use the adjacency matrix to solve this problem.
So far, the properties I know of adjacency matrices that represent are trees:
1) The diagonal has to be 0. (No loops)
2) Every element can be 1 or 0 only.
3) There has to be a total of 2(n-1) 1's. (Double the edges)
4) If there is a 1 from a to b, and 1 from b to c, there cant be a 1 from a to c, and so on for longer paths also.
5) The matrix is symmetric.
How do I put these properties together to prove the above? I'm looking for something along the lines of 'there are only three ways to arrange 8 1's in the matrix given the above constraints'
Also are there more properties that I might have missed?