Consider the Laplacian of a general graph,
$$ L = D - A $$ where $D$ is the degree matrix and $A$ is the adjacency matrix. Its eigenvectors are defined by
$$ L\mathbf{u}_i = \lambda_i \mathbf{u}_i. $$
Because $L$ is symmetric and positive semidefinite, the eigenvectors are real and orthonormal,
$$ \sum_k (\mathbf{u}_i)_k (\mathbf{u}_j)_k = \delta_{ij}. $$
I am wondering if something can be said about more complicated sums over the entries of the eigenvectors. In particular, I have some numerical experiments that suggest that most of the entries in
$$ \sum_k (\mathbf{u}_i)_k (\mathbf{u}_j)_k (\mathbf{u}_\ell)_k (\mathbf{u}_m)_k $$
are zero for generic graphs. Is it possible to find an expression for that sum?