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Asymptotic evaluation of a lattice sum associated with the Laplacian matrix

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.