We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain \({\Omega} {\subset} \mathbb{R}^{d}, d \geq 1\). 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 we show that we can associate to it a symbol \(\mathfrak{f}\). The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.

Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, 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.

我们主要关注具有网格几何图形的图序列，该图序列在有界域\（{\ Omega} {\ subset} \ mathbb {R} ^ {d}，d \ geq 1 \）中具有统一的局部结构。当\（\ Omega = [0，1] \）时，此类图包括标准Toeplitz图，并且对于\（\ Omega = [0，1] ^ {d} \），考虑的类包括d级Toeplitz图。在一般情况下，邻接矩阵的基础序列在Weyl意义上具有规范的特征值分布，并且我们证明了可以将其与符号\（\ mathfrak {f} \）相关联。该符号及其基本分析功能的知识可提供有关特征值结构，定位，谱隙，聚类和分布类型的许多信息。

很少考虑与广义局部Toeplitz序列有关的概化，并讨论了其应用，例如基于数值方案对差分算子的近似。但是，由于这里介绍的结果也可以用于研究一般大型图和网络的邻接矩阵和拉普拉斯算子的光谱特性，因此可以考虑更多的应用。