Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy-independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices.

引入量子图来使用区间网络上的自伴哈密顿量来模拟有机分子中的自由电子。第二个图量化通过指定顶点处的散射矩阵来描述图上的波传播。经常提出的一个问题是这些模型在多大程度上相同或互补。特别是，所有能量无关的酉顶点散射矩阵是否都与自伴哈密顿量相关？我们在这里回顾与此问题相关的结果。此外，我们观察到，当狄拉克算子描述零质量粒子且顶点条件满足时，具有四个分量旋量的自伴狄拉克算子会生成图谱的长期方程，该方程与与波在图上传播相关的长期方程相匹配。不允许在顶点自旋。