Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs

Abstract

We study the nodal count of the so-called bi-dendral graphs and show that it exhibits an anomaly: the nodal surplus is never equal to 0 or $\beta$, the first Betti number of the graph. According to the nodal-magnetic theorem, this means that bands of the magnetic spectrum (dispersion relation) of such graphs do not have maxima or minima at the "usual" symmetry points of the fundamental domain of the reciprocal space of magnetic parameters. In search of the missing extrema we prove a necessary condition for a smooth critical point to happen inside the reciprocal fundamental domain. Using this condition, we identify the extrema as the singularities in the dispersion relation of the maximal abelian cover of the graph (the honeycomb graph being an important example). In particular, our results show that the anomalous nodal count is an indication of the presence of the conical points in the dispersion relation of the maximal universal cover. Also, we discover that the conical points are present in the dispersion relation of graphs with much less symmetry than was required in previous investigations.

arXiv comment22 pages, 6 figures; corrections suggested by a referee; expanded interlacing lemma 4.2

References in corpus (19)

• The electronic properties of graphene
• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• On the spectra of carbon nano-structures
• Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations
• Quantum wires with magnetic fluxes
• On occurrence of spectral edges for periodic operators inside the Brillouin zone
• Nodal domains on quantum graphs
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• A lower bound for nodal count on discrete and metric graphs
• Nodal count of graph eigenfunctions via magnetic perturbation
• Magnetic interpretation of the nodal defect on graphs
• On the location of spectral edges in $\mathbb{Z}$-periodic media
• Universality of the momentum band density of periodic networks
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• Nodal domains on graphs - How to count them and why?
• Sharp eigenvalue bounds on quantum star graphs

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