Spectral preorder and perturbations of discrete weighted graphs

TL;DR

This paper introduces geometric and spectral preorders on weighted graphs with magnetic potentials, providing tools to analyze how graph perturbations affect Laplacian spectra and applying these to classify graphs, study eigenvalue stability, and detect spectral gaps.

Contribution

It develops new preorder relations that relate graph structure and spectra, extending eigenvalue interlacing and enabling spectral analysis under various graph modifications.

Findings

Classified graphs using the introduced preorders.

Proved stability of certain eigenvalues in graphs with maximal d-cliques.

Established monotonicity of eigenvalues and Cheeger constants under graph modifications.

Abstract

In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.