In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that quantum graph eigenvalues can be found as the solutions of a nonlinear eigenvalue problem, and in the special case of equilateral graphs, even as the solutions of a linear eigenvalue problem on the underlying combinatorial graph. The latter, remarkable relation to combinatorial graph spectra will be exploited to derive a solver for the general, non-equilateral case. Eigenvalue estimates from equilateral approximations will be applied as initial guesses in a Newton-trace iteration to solve the nonlinear eigenvalue problem.

翻译：在与量子图相关的广泛研究中，量子图谱作为一项备受关注的主题出现。它们在度量图上扩散类型问题的求解中具有重要意义。理论研究表明，量子图特征值可以作为非线性特征值问题的解求得，而在等边图的特殊情况下，甚至可以作为底层组合图上线性特征值问题的解求得。后者与组合图谱的显著关系将被利用来推导一般非等边情况下的求解器。来自等边近似估计的特征值将作为牛顿-迹迭代法中的初始值，用于求解非线性特征值问题。