AI 中文总结
本文为加权图建立谱极小划分的存在性框架，引入典范紧化性与子图收敛概念，将无限图的谱极小问题转化为有限图研究，确保非紧情形下最优谱能量可实现。
AI 中文摘要
本文研究加权图的谱极小划分，从而扩展了当前关于区域、以及在较小程度上关于流形和度量图的大量结果。我们为分析图拉普拉斯算子在狄利克雷（Dirichlet）、诺伊曼（Neumann）和无边界能量形式下提供了严谨框架，研究的核心重点是建立极小划分的存在性定理。对于有限连通图，由于可容许划分类的有限性，存在性是显而易见的；而无限图则需要先进的拓扑和泛函分析工具。具体而言，我们引入了典范紧化性的概念，该概念与紧嵌入以及诺伊曼和无边界能量的一致庞加莱型常数相关；还引入了子图收敛的适当概念。通过这种方式，我们可以将无限图上的谱极小问题简化为有限图的研究，从而保证即使在非紧情形下，最优谱能量也能被适当的划分达到。
英文摘要
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.
