Baptista, Luís Carlos Nascimento2026-01-062026-01-062025http://hdl.handle.net/10400.5/116490Tese de Mestrado, Matemática, 2025, Universidade de Lisboa, Faculdade de CiênciasIn this thesis we will study operator theory in metric graphs and how it induces problems on partitions. We will start by formalizing the construction of metric graphs and Laplacians operators on them. Then by defining partitions of metric graphs we will be able to define minimization and maximization problems involving partitions. Starting by introducing the well studied spectral minimal partitions on graphs [12] (on subsets of Rd see [A. Henrot, De Gruyter Open, Warsaw, 2017], for example) we will then relate them to Neumann domains. The most important new results are Theorem 2.3.4 and Theorem 2.3.6. We will also characterize spectral minimal partitions to eigenvalues of the Laplacian heorem 2.3.10. We will then pose a new partition based problem using the mean distance function of a graph, ρ [Baptista, Kennedy and Mugnolo, J. Geom. Anal. 34 (2024), 137] and then study its asymptotic behaviour, Theorem 3.2.2, and how it relates with the spectral minimal partition problem, Corollary 3.2.5.application/pdfengMetric graphsLaplacianPartitionsNeumann DomainsMean Distancemaster thesis204122449