Gluing of Graph Laplacians and Their Spectra

TL;DR

This paper introduces methods to compute the spectra of glued graphs' Laplacians, with applications in quantum mechanics and spectral bounds, advancing understanding of graph spectral properties after gluing operations.

Contribution

It provides explicit formulae for the Laplacians and spectra of graphs formed by interface and bridge gluing, a novel approach in spectral graph theory.

Findings

Derived formulae for even and odd Laplacians of glued graphs

Established bounds for the Fiedler value after gluing

Applied spectral results to quantum mechanics contexts

Abstract

We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing even and odd Laplacians of graphs obtained by gluing, as well as their spectra. We subsequently discuss applications to quantum mechanics and bounds for the Fiedler value of the gluing of graphs.