Consider that an unweighted graph has the Laplacian matrix $L$ which is $n \times n$.
The graph has nearest neighbors topology, so the elements on the main diagonal of $L$ are all the same.
Consider $\Lambda$ to be a diagonal matrix, consisting of the eigenvalues of $L$.
I was trying to find the explicit formula for $\text{trace}(\Lambda^{-2})$ in terms of $n$ and $m$ (the first is the number of nodes and the second is the number of neighbors for each node).
This trace is in fact sum of the squares of inverted eigenvalues of the Laplacian, i.e., $\frac{1}{\lambda_1^2}+\cdots+\frac{1}{\lambda_n^2}$.
Still I have nothing obtained for this problem that does not seem to be much difficult...
Sincerely,