I'm having some trouble understanding a homework question. The question:
"Consider the full binary tree of depth 6. Show that if $\lambda$ is an eigenvalue of the tree, then there is a function $f(r)$ which only depends on the depth of a vertex with eigenvalue $\lambda$. Write an algorithm to compute numerically the second eigenvalue of the tree"
What do you think is meant by "a function $f(r)$ which only depends on the depth of a vertex with eigenvalue $\lambda$ ". I think the function is on the vertices of the graph, but what does $\lambda$ have to do with a vertex?
edit: an eigenvalue of the tree, I think is an eigenvalue of the adjacency matrix of the tree.
Thanks!