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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 $G$?

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MO is intended for questions at the mathematics PhD student level and above. I've voted to close. –  Andy Putman May 5 at 18:03
    
@AndyPutman Although I think the question could be improved, it doesn't seem trivial to me. (E.g. Cauchy interlacing in the self-adjoint case) –  Yemon Choi May 5 at 18:07
    
@YemonChoi : Hmm...if you think it is interesting then I'll withdraw my objection. –  Andy Putman May 5 at 18:08
    
Sorry for my English, but I think it's not that trivial it looks. Any help would be appreciated. –  Farid Ala May 5 at 18:12
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@FaridAla I don't understand your thought processes: you say "there are no extra conditions on matrices" but then you say "A is the adjacency matrix of a graph" -- this is an incredible amount of extra structure! Not every matrix behaves like the adjacency matrix of a graph! –  Yemon Choi May 5 at 21:23

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