Asymptotic Evaluation of a Lattice Sum Associated with the Laplacian Matrix

The Laplacian matrix is of fundamental importance in the study graphs, networks, random walks on lattices, and arithmetic curves. In certain cases, trace its pseudoinverse appears as only non-trivial term computing some intrinsic graph invariants. Here we a double sum Fn which associated with pseudo inverse for graphs. We investigate asymptotic behavior this n → ∞. Our approach based classical analysis combined numerical analysis, utilizes special functions. determine leading-order term, size n2 log n, develop general methods to obtain secondary main terms expansion up errors $$\mathcal {O}({\rm{log}} \; n)$$ {O}(1)$$ provide examples demonstrate our methods.