In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $\Omega\subset {\mathbb R}^d$, $d\ge 1$. When $\Omega=[0,1]$, such graphs include the standard Toeplitz graphs and, for $\Omega=[0,1]^d$, the considered class includes $d$-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol $\boldsymbol{\mathfrak{f}}$. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences (FDs), Finite Elements (FEs), and Isogeometric Analysis (IgA). Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.
翻译:在目前的工作中,我们所关注的是具有网格几何的图表序列,在封闭域中具有统一的本地结构,$\Omega\subset {mathbrb R ⁇ d$,$d\ge$1美元。当$\Omega=[0,1美元,这种图表包括标准的托普利茨图,对于美元=[0,1美元],所考虑的类别包括美元水平的托普利茨图。在一般情况下,相近矩阵的基本序列在Weyl 意义上具有卡通性电子价值分布,在这项工作的理论部分中已经显示这一点,我们可以将其用作一个符号$\boldsymbol_mathfrak{f ⁇ $1美元。对于符号及其基本分析功能的了解提供了本地化、光谱差距、集成和全球分布方面的关键信息。在本文中,许多不同的应用和各种数字例子都用来强调在任何时间里程值上实际使用已发展起来的网络,只要我们就可以将一个符号符号符号作为符号的符号。 测试和应用程序可以显示,例如:我们所研究的金融结构的数值的数值的数值和数据结构的模型的模型,可以反映。