What is actually a metric graph?

Abstract

Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an abstract, self-contained definition of metric graph. Being mostly topological, it doesn't require any knowledge from graph theory and already determines uniquely several concepts that are commonly and unnecessarily \textit{defined} in the literature. Nevertheless, many ideas mentioned here are folklore in the quantum graph community: we discuss them for later reference.

Cited by in corpus (5)

• Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
• Hidden symmetries in non-self-adjoint graphs
• Upper Eigenvalue Bounds for the Kirchhoff Laplacian on Embbeded Metric Graphs
• An existence theory for nonlinear equations on metric graphs via energy methods
• On fully supported eigenfunctions of quantum graphs