Maximal P-sets of matrices whose graph is a tree

The graph of a real n×nn×n symmetric matrix A  , G(A)G(A), is the undirected graph on n vertices whose edges correspond precisely with the nonzero off-diagonal entries of A  . Let mA(0)mA(0) denote the multiplicity of 0 as an eigenvalue of A  . It is possible that the multiplicity of 0 as an eigenvalue of an (n−1)×(n−1)(n−1)×(n−1) principal submatrix, An−1An−1, of A  , is larger than mA(0)mA(0). When this occurs the vertex in G(A)G(A) corresponding to the row and column of A   that was deleted to form An−1An−1 is called a P-vertex of A. If there is a list of principal submatrices of A  , A,An−1,An−2,…,An−kA,An−1,An−2,…,An−k, each one being obtained from the matrix preceding it in the list by deleting a row and column, such that mA(0)