Graph theory question

(n)x,y = min{n|A^n(n,x)|} where A:Degree Matrix and A-D= real or more simply statded d(i,j):=deg(Vi, A^n)/sqrt(mXn) where i= a

its really a simple degree matrix, not much more to it.

I assume you are counting unlabelled graphs. Because labelled graphs would be a lot more.

I think that what you did is a good way to approach this. You need to draw the forests anyway, and this is seems like a reasonably systematic approach.

One thing you can do, is start with 0 edges. How many ways can you ad an edge? 1 of course.

Then go to all the forests with 1 edge (obviously there is only 1) and see how many ways you can add an edge. There should be 2, either you get a path of length 2 with 3 isolated vertices, or you get 2 disjoint paths with 2 isolated vertices.

Continue in this fashion, by considering each forest, and then seeing how many ways you can add an edge to each one.

I just spent a few minutes, and I seem to come up with 21. I may have missed a few. But see if you can look back and find what you are missing.

By the way, this is what I got...

0 edges: 1 forest

1 edge: 1 forest

2 edges: 2 forests

3 edges: 4 forests

4 edges: 6 forests

5 edges: 7 forests