We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains \(\nu _n\) of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with \(L^1\)-potentials and a variety of vertex conditions as well as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence \((\frac{\nu _n}{n})_{n\in \mathbb {N}}\), which are shown always to form a finite subset of (0, 1]. This extends the previously known result that \(\nu _n\sim n\) generically, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which \({\nu _n}\not \sim {n}\); but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

我们在紧凑度量图上的一大类算子的第n个特征函数的节点域数\(\nu _n\)的渐近上建立了 Pleijel 定理的度量图对应物，包括具有\(L ^1\) -势和各种顶点条件以及具有自然顶点条件的p -拉普拉斯算子，并且对边的长度、图的拓扑结构或顶点处特征函数的行为没有任何假设. 除其他外，这些结果表征了序列\((\frac{\nu _n}{n})_{n\in \mathbb {N}}\)的累积点, 总是显示形成 (0, 1] 的有限子集。这扩展了先前已知的结果\(\nu _n\sim n\) 一般地，对于拉普拉斯算子的某些实现，在几个方向上。特别是，在具有自然条件的拉普拉斯算子的特殊情况下，我们证明对于具有成对可公度边长和至少一个圈的任何图，人们可以找到其上的特征函数，其中\({\nu _n}\not \sim {n} \) ; 但在这种情况下，即使是累积点的集合也可能取决于特征基础的选择。