  The Laplacian matrix is of fundamental importance in the study of graphs,
networks, random walks on lattices, and arithmetic of curves. In certain cases,
the trace of its pseudoinverse appears as the only non-trivial term in
computing some of the intrinsic graph invariants. Here we study a double sum
$F_n$ which is associated with the trace of the pseudo inverse of the Laplacian
matrix for certain graphs. We investigate the asymptotic behavior of this sum
as $n \to \infty$. Our approach is based on classical analysis combined with
asymptotic and numerical analysis, and utilizes special functions. We determine
the leading order term, which is of size $n^2 \, \log n$, and develop general
methods to obtain the secondary main terms in the asymptotic expansion of $F_n$
up to errors of $\mathcal{O}(\log n)$ and $\mathcal{O}(1)$ as $n \to \infty$.
We provide some examples to demonstrate our methods.
