论文标题
听到图表的特征
Hearing Euler characteristic of graphs
论文作者
论文摘要
Euler特征$χ= | V | - | e | $和总长度$ \ Mathcal {l} $是公制图的最重要的拓扑和几何特性。在这里，$ | v | $和$ | e | $表示图形的顶点和边缘的数量。 Euler特征确定图中独立周期的数量$β$，而总长度通过Weyl定律决定了能量特征值的渐近行为。我们从理论上表明并实验确认可以从最低特征力的有限序列（听到）确定欧拉的特性$λ_1，\ ldots，λ_n$的简单量子图，而无需在视觉上检查系统。在实验中，量子图由微波网络模拟。我们证明，具有$β\ leq 3 $的微波网络的最低共振序列可以直接用于确定网络是否是平面，即可以嵌入平面中。此外，我们表明测得的欧拉特征$χ$可以用作完全连接图的敏感揭示者。
The Euler characteristic $χ=|V|-|E|$ and the total length $\mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $β$ of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyl's law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies $λ_1, \ldots, λ_N$ of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with $β\leq 3$ can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic $χ$ can be used as a sensitive revealer of the fully connected graphs.
