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

TL;DR

The paper analyzes the smooth critical points of the k-th eigenvalue λ_k on the magnetic perturbation torus 𝓜_h of a finite graph G. It shows that such critical points lie on finite-dimensional submanifolds F determined by critical data (V_N,h_N,ψ_N,λ), and that F is diffeomorphic to a torus times planar linkage manifolds, with an explicit Morse-index formula that combines nodal information with spectral data. An algorithm is provided to enumerate all critical submanifolds by examining a finite collection of signings and principal minors, and the Hessian analysis yields a precise index that explains when F yields band-edge extrema. The work connects to Floquet theory via Bloch-type decompositions on maximal abelian covers and to the nodal surplus problem through the nodal magnetic theorem, suggesting a universal nodal-distribution behavior in large β regimes. Numerical experiments on 3-regular graphs illustrate the abundance of non-symmetry critical points and support the proposed universality conjectures, highlighting both the richness of the critical set and its stability under perturbations.

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 $h_α$ obtained by varying the complex phases of individual matrix elements. This family is parametrized by a $β$-dimensional torus, where $β$ is the first Betti number of the underlying graph. The eigenvalues of each matrix are ordered, enabling us to treat the $k$-th eigenvalue $λ_k$ as a function on the torus. We classify the smooth critical points of $λ_k$, 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 $2^β$ real symmetric matrices $h_α$ in the family under consideration as well as their principal minors.

Figures (13)

• Figure 1: Plot of $\mathbb{E}(\sigma(h',k))/\beta(G)$, computed for a given $k$ and a random signing $h'$, against $k/n$. The matrix $h$ is the Laplacian of $G$ plus diagonal potential, and $G$ ranges over 35 randomly chosen 3-regular graphs of $n$ vertices, see \ref{['sec-experiments-3reg']}.
• Figure 2: Top row: examples of admissible support (vertices in $V_N$ are solid black circles, vertices in $V_{ZN}$ and $V_{ZZ}$ are empty blue and red vertices, correspondingly). Bottom row: examples of $V_N$ that are inadmissible (left: boundary degree is too small; right: the subgraph induced by $V_N$ is disconnected).
• Figure 3: An example of a graph with an admissible support $V_N$ shown as black vertices (the vertices in $V_{ZN}$ and $V_{ZZ}$ are cyan and red empty circles). The edges $E_N$, $E_{ZN}$, and $E_{ZZ}$ are shown in black, cyan, and red, respectively.
• Figure 4: A summary of the objects used in the paper and their relationships.
• Figure 5: A spanning tree (shaded edges) consistent with §\ref{['subsec:tree_choice']}

Theorems & Definitions (46)

• Proposition 2.3
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Corollary 3.6
• proof
• Remark 3.7
• Theorem 3.8
• Theorem 3.9