Quantum graphs which optimize the spectral gap

TL;DR

This paper investigates how assigning lengths to edges in quantum graphs affects the spectral gap, providing solutions for minimization and partial results for maximization, advancing understanding of spectral optimization in quantum graph structures.

Contribution

It fully solves the spectral gap minimization problem for all finite discrete graphs and develops methods for the maximization problem, addressing a key aspect of quantum graph spectral theory.

Findings

Complete solution for spectral gap minimization across all graphs.

Development of new tools for spectral gap maximization.

Partial solutions for maximizing spectral gap in certain graph families.

Abstract

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.