Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs

TL;DR

This paper analyzes the smooth critical points of eigenvalues in a family of Hermitian matrices derived from graph structures, classifying their nature and structure in relation to graph topology and eigenvector nodal properties.

Contribution

It introduces a classification of critical points of eigenvalues on the torus of magnetic perturbations, revealing their structure, Morse index, and the prevalence of critical submanifolds.

Findings

Critical points often form submanifolds, not isolated points.

The structure and Morse index depend on eigenvector nodal count.

An algorithmic method is provided to identify all critical submanifolds.

Abstract

Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of Hermitian matrices $h_{\alpha}$ obtained by varying the complex phases of individual matrix elements. This family is parametrized by a $\beta$-dimensional torus, where $\beta$ is the first Betti number of the underlying graph. The eigenvalues of each matrix are ordered, enabling us to treat the $k$-th eigenvalue $\lambda_k$ as a function on the torus. We classify the smooth critical points of $\lambda_k$, describe their structure and Morse index in terms of the support and nodal count, that is, the number of sign changes between adjacent vertices of the corresponding eigenvector. In general, the families under consideration exhibit critical submanifolds rather than isolated critical points. These critical manifolds appear frequently and cannot be removed through perturbations. We provide an algorithmic way of determining all critical submanifolds by investigating finitely many eigenvalue problems: the $2^\beta$ real symmetric matrices $h_\alpha$ in the family under consideration as well as their principal minors.