We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.

我们在肯尼迪等人最近引入的框架内研究度量图的频谱最小分区的属性。（Calc Var 60：6，2021）。对于不同类别的分区中的最小分区能量，我们提供了较低的上限和较高的估算值；尽管下限使人联想到度量图的经典等距不等式，但上限更多地涉及到度量图的组合结构。结合它们，我们推论这些光谱最小能量还满足Weyl型渐近定律，该定律类似于众所周知的具有各种顶点条件的量子图拉普拉斯算子的特征值。通过两个示例，我们显示出一般而言，对于最小的分配能，在渐近展开中不存在第二项，但是表明各种行为都是可能的。