Matching number, Hamiltonian graphs and discrete magnetic Laplacians

TL;DR

This paper explores how the spectrum of the discrete magnetic Laplacian on a finite graph reveals structural properties like perfect matchings and Hamiltonian cycles, providing spectral obstructions influenced by magnetic potential.

Contribution

It introduces spectral obstructions parametrized by magnetic potential that relate the DML spectrum to matchability and Hamiltonian cycles in graphs.

Findings

Spectral obstructions for perfect matchings.

Spectral obstructions for Hamiltonian cycles.

Use of magnetic potential as a spectral control parameter.

Abstract

In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the underlying graph. In particular, we give a family of spectral obstructions parametrised by the magnetic potential for the graph to be matchable (i.e., having a perfect matching) or for the existence of a Hamiltonian cycle. We base our analysis on a special case of the spectral preorder introduced in [FCLP20a] and we use the magnetic potential as a spectral control parameter.