论文标题
频谱成长的图
Spectrally grown graphs
论文作者
论文摘要
量子图吸引了数学家的关注一段时间。量子图是通过在度量图的每个边缘上的laplacian并在顶点施加边界条件以获取特征值问题来定义的。研究这种量子图的问题是，频谱是在手动计算的时间数量，并且很难找到具有指定光谱的量子图的反问题。我们使用先前开发的计算机程序解决了前进问题，以找到特征值。我们可以通过分析获得所有特征值，因为它的图形不太大，具有合理依赖的边缘。我们使用“光谱成长的图”解决了反问题。频谱成长的图是从启动（父）图演变而来的，因此子图具有特征值接近某些标准。我们的实验表明该方法有效，我们通常可以找到具有数值接近规定光谱的光谱的图。自然有例外，例如没有图形的频谱。选择标准（目标）强烈影响进化图的形状。我们的实验使我们能够对量子图光谱做出新的猜想。我们在https://github.com/meapistol/spectra-of-graphs上开放信息。
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A problem studying such quantum graphs is that the spectrum is timeconsuming to compute by hand and the inverse problem of finding a quantum graph having a specified spectrum is difficult. We solve the forward problem, to find the eigenvalues, using a previously developed computer program. We obtain all eigenvalues analytically for not too big graphs that have rationally dependent edges. We solve the inverse problem using "spectrally grown graphs". The spectrally grown graphs are evolved from a starting (parent) graph such that the child graphs have eigenvalues are close to some criterion. Our experiments show that the method works and we can usually find graphs having spectra which are numerically close to a prescribed spectrum. There are naturally exceptions, such as if no graph has the prescribed spectrum. The selection criteria (goals) strongly influence the shape of the evolved graphs. Our experiments allows us to make new conjectures concerning the spectra of quantum graphs. We open-source our software at https://github.com/meapistol/Spectra-of-graphs.
