We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, probability, harmonic functions, and boundary theory. The limit constructions, finite to infinite, and local to global, can be used in various applications.

我们介绍了无限图上谐波分析的最新进展。我们的方法将组合工具与希尔伯特空间，几何，边界构造和谱不变式中无界厄米算子理论的新结果相结合。我们专注于特定种类的无限图，包括在电网模型中出现的加权图，以及新的图解图表示形式。一方面，我们进一步强调了我们在无限图上的当前分析与另一方面，在势能理论，概率，调和函数和边界理论的特定领域之间的某些直接相似之处。极限构造（从有限到无限以及从局部到全局）可以用于各种应用中。