I have the following generalized eigenvalue problem
$\det[P_i^TLP_j+zP_i^TP_j] = 0$
$L$ is a positive-semidefinite matrix with 1 eigenvalue at 0. More precisely, it is the combinatorial Laplacian matrix for a connected graph. $P_i$ ($P_j$) is the identity matrix with the i-th (j-th) column removed.
What, if anything, can be said about the generalized eigenvalues $z$ of this problem?