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 β, 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 conical points in the dispersion relation of the maximal universal cover. We also discover that the conical points are present in the dispersion relation of graphs with much less symmetry than was required in previous investigations.
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December 2015
Research Article|
December 17 2015
Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs Available to Purchase
Ram Band;
Ram Band
1Department of Mathematics,
Technion - Israel Institute of Technology
, Haifa 32000, Israel
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Gregory Berkolaiko;
Gregory Berkolaiko
2Department of Mathematics,
Texas A&M University
, College Station, Texas 77843-3368, USA
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Tracy Weyand
Tracy Weyand
3Department of Mathematics,
Baylor University
, Waco, Texas 76798-7328, USA
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1Department of Mathematics,
Technion - Israel Institute of Technology
, Haifa 32000, Israel
2Department of Mathematics,
Texas A&M University
, College Station, Texas 77843-3368, USA
3Department of Mathematics,
Baylor University
, Waco, Texas 76798-7328, USA
J. Math. Phys. 56, 122111 (2015)
Article history
Received:
March 26 2015
Accepted:
November 22 2015
Citation
Ram Band, Gregory Berkolaiko, Tracy Weyand; Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs. J. Math. Phys. 1 December 2015; 56 (12): 122111. https://doi.org/10.1063/1.4937119
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