On a magneto-spectral invariant on finite graphs

TL;DR

This work introduces the magneto-spectral height $\nu(G)=\sup_{\sigma}\lambda_1^{\sigma}(G)$ for finite graphs via the magnetic Laplacian, establishing that $\nu(G)=0$ on forests and analyzing $\nu$ for key families (cycles, complete graphs, wheels, hypercubes, and complete bipartite graphs). It derives sharp combinatorial and subgraph bounds, connects maximal potentials to unit weighing matrices, and investigates structural operations (suspensions, products, and bridges) that preserve or bound $\nu$. The paper also demonstrates that $\nu$ can distinguish certain cospectral graphs and explores links to the spectral gap and magnetic Bakry-Émery curvature, while outlining extensive open problems and conjectures. Overall, it provides a rich framework for understanding how magnetic perturbations interact with graph cycle structure, with implications for spectral theory and graph constructions.

Abstract

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-Émery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.

Figures (12)

• Figure 1: Relation between a directed edge $(x,y) \in E^{or}(G)$ with $\sigma(x,y) = {\xi_5}^2$ and a matching between the fibers $\pi^{-1}(x)$ and $\pi^{-1}(y)$ in the $5$-lift $\widehat{G}$.
• Figure 2: Eigenfunctions $f_j$ of the magnetic Laplacian $\Delta^\sigma$ on $C_n$ with $j=0,1,\dots,n-1$. The red factors describe the influence of the magnetic potential $\sigma_t$.
• Figure 3: The two isospectral graphs $W_6$ and $\widehat{G}$
• Figure 4: Two non-isomorphic and cospectral trees $T_1$ and $T_2$
• Figure 5: A counterexample to Question \ref{['quest:nu_multiplicity2']} with potential $\sigma_t$

Theorems & Definitions (98)

• Definition 1.1
• Definition 1.2: Magneto-spectral height
• Lemma 2.1
• Example 2.2: The cycle $C_n$
• Corollary 2.3
• Example 2.4: The complete graph $K_n$
• Lemma 2.5
• Remark 2.6
• proof : Proof of Lemma \ref{['lem:upbdxdy']}
• Proposition 2.7