Abstract graph-like space and vector-valued metric graphs

TL;DR

This paper introduces a framework for constructing and analyzing graph-like spaces using boundary couplings, with applications to various types of graphs and a convergence theorem for shrinking spaces.

Contribution

It presents a unified approach to building and analyzing graph-like spaces, including spectral analysis and a convergence result for shrinking spaces with Dirichlet conditions.

Findings

Spectral analysis of graph-like spaces

Application to vector-valued quantum graphs

Convergence theorem for shrinking graph-like spaces

Abstract

In this note we present some abstract ideas how one can construct spaces from building blocks according to a graph. The coupling is expressed via boundary pairs, and can be applied to very different spaces such as discrete graphs, quantum graphs or graph-like manifolds. We show a spectral analysis of graph-like spaces, and consider as a special case vector-valued quantum graphs. Moreover, we provide a prototype of a convergence theorem for shrinking graph-like spaces with Dirichlet boundary conditions. (Dedicated to Pavel Exner's 70th birthday)