This paper studies the Laplacian spectrum and the average effective resistance of (large) graphs that are sampled from graphons. Broadly speaking, our main finding is that the Laplacian eigenvalues of a large dense graph can be effectively approximated by using the degree function of the corresponding graphon. More specifically, we show how to approximate the distribution of the Laplacian eigenvalues and the average effective resistance (Kirchhoff index) of the graph. For all cases, we provide explicit bounds on the approximation errors and derive the asymptotic rates at which the errors go to zero when the number of nodes goes to infinity. Our main results are proved under the conditions that the graphon is piecewise Lipschitz and bounded away from zero.

中文翻译：

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从 Graphons 采样的大图的拉普拉斯谱

本文研究了拉普拉斯谱和从graphons 中采样的（大）图的平均有效电阻。从广义上讲，我们的主要发现是，可以通过使用相应图形的度函数来有效地近似大型密集图的拉普拉斯特征值。更具体地说，我们展示了如何近似图的拉普拉斯特征值和平均有效电阻（基尔霍夫指数）的分布。对于所有情况，我们提供了近似误差的明确界限，并推导出了当节点数趋于无穷大时误差变为零的渐近率。我们的主要结果是在图形是分段 Lipschitz 并且远离零的条件下证明的。