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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 [8] , and we use the magnetic potential as a spectral control parameter. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we apply the theory of stochastic decomposition for general metric spaces due to Lee and Noar to derive the existence of bounded Lipschitz random partition of complex projective spaces, by which we improve the high-order Cheeger inequalities for the discrete magnetic Laplacian. [ABSTRACT FROM AUTHOR]

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi–Shirai's conjecture for Z -periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence of spectral gaps. [ABSTRACT FROM AUTHOR]

In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on β ˜ . The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and nanoribbons in the presence of a constant magnetic field.