蜂窝图色散关系中的反常结点数和奇异性
Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
摘要：
我们研究了所谓的双密度图的节点数，并表明它表现出一个异常：节点盈余永远不等于0或$β$，图的第一个Betti数。根据磁-磁定理，这意味着这样的图的磁谱（色散关系）的带在磁参数的倒易空间的基本域的“通常”对称点处没有最大值或最小值。在寻找失踪的极值，我们证明了一个必要条件，光滑的临界点内发生的倒易基本域。利用这个条件，我们确定的极值的最大阿贝尔覆盖的图（蜂窝图是一个重要的例子）的色散关系中的奇点。特别是，我们的研究结果表明，异常的节点计数是一个迹象的存在的圆锥点的色散关系的最大通用覆盖。此外，我们发现，圆锥点是目前在色散关系的图形的对称性比以前的调查所需的少得多。
英文摘要：
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 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.
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