TITLE: Is a symmetric matrix characterized by the diagonal of its resolvent?
QUESTION [6 upvotes]: The resolvent of a square matrix $A$ is defined by $R(s) = (A-sI)^{-1}$ for $s \notin \operatorname{spect}(A)$. 
Is knowing the diagonal of $R(s)$ for all $s$ sufficient to recover $A$ when $A$ is symmetric?
edit: a counter-example of two matrices $A,B$ whose resolvent have the same diagonal has been found by Robert Israel. In the counter example, $A = P B P^T$ for some permutation matrix $P$. Now the question is, it is possible to recover $A$ up to permutations of rows and columns?

REPLY [2 votes]: If $A$ and $B$ are the adjacency matrices of non-isomorphic strongly regular graphs on $n$ vertices with the same parameters, then the diagonal entries of the resolvent are all equal to $1/n$ times the common characteristic polynomial. Such graphs on 16 vertices can be constructed from the two $4\times4$ Latin squares. Since the graphs are not isomorphic, the two matrices are not permutation equivalent.