Suppose that we have two matrices A and B. Matrix B is taken from A with one row and one column deleted. On the other hand A is n*n matrix and B is (n-1)*(n-1) matrix and is created by deleting last row and last column of B. If we are given eigenvalues of A, is it possible to use them to calculate eigenvalues of B? Thanks.

EDIT: I take the liberty of rewriting the question, based on information gleaned from the comments.

If we know the eigenvalues of (the adjacency matrix of) a graph $G$, can we use that information to calculate the eigenvalues of a graph $G'$ obtained from $G$ by deleting a vertex (and all edges adjacent to that vertex)?

Presumably, the answer is "no", as it will depend on which vertex we remove, so let me ask for less: what can we say about the eigenvalues of $G'$, from just knowing the eigenvalues of $G$?

an incredible amount of extra structure! Not every matrix behaves like the adjacency matrix of a graph! – Yemon Choi May 5 '14 at 21:23