Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Partições espetrais ótimas: análise geométrica e numérica, Centro de Investigação e Desenvolvimento em Matemática e Aplicações, Grupo de Física Matemática, G2936814011, G3069120617, Centro de Investigação e Desenvolvimento em Matemática e Aplicações, Analysis auf Graphen Begegnungen von Mathematischer Physik, Diskreter Mathematik und Theoretischer Informatik, Grupo de Física Matemática, G7049327362, G7318052850, G7318643082, G8321563775, G8473570623
OpenAlex ID: W7203922633
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