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Find the number of closed walks of length` in the graph below

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L165100 - Fall 2010 - Homework 1

1. Prove that the distance between any two vertices of a connected graph G (that is, the length of a shortest walk between these vertices) is less than the number of distinct eigenvalues of G.

2. Find the eigenvalues of the complete bipartite graph Km,n in two different ways: by linear algebra and by counting closed walks.

3. Find the number of closed walks of length` in the graph below.

r r r

r r r

r r r

4. Show that a finite poset can be covered by k antichains if and only if it does not contain a(k + 1)-element chain.

5. How many antichains of maximal size are there in the Boolean poset Bn?

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