论文标题
与拉普拉斯矩阵相关的晶格总和的渐近评估
Asymptotic evaluation of a lattice sum associated with the Laplacian matrix
论文作者
论文摘要
拉普拉斯矩阵在图，网络，晶格随机步行和曲线算术的研究中至关重要。在某些情况下，其伪源的痕迹似乎是计算某些固有图形不变式的唯一非平凡术语。在这里，我们研究了一个双和$ f_n $，该$与某些图形的laplacian矩阵的伪倒数相关。我们将此款项的渐近行为称为$ n \ to \ infty $。我们的方法基于经典分析与渐近和数值分析相结合，并利用了特殊功能。我们确定了$ n^2 \，\ log n $的主要订单项，并开发一般方法，以获取$ f_n $的渐近扩展中的次级主要术语，达到$ \ MATHCAL {o}（o log n）$和$ \ \ \ \ \ \ \ \ \ \ \ nmatcal {o}（o}（1）$ as $ n as $ n \ to \ to \ forn \ to \ forn \ tos \ forty \ to \ n \ to \ forty \ to \ for。我们提供了一些例子来证明我们的方法。
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.
