标题： 谱流与度量图上的罗宾域 显示英文标题

标题： Spectral flow and Robin domains on metric graphs

摘要： 本文致力于有限度量图上的诺依曼-基尔霍夫拉普拉斯算子。我们证明了一个指标定理，将特征函数的结点亏值与（1）Dirichlet到诺依曼映射的Morse指标、（2）其正指标和图的第一贝蒂数联系起来。然后我们将这一结果推广，用特征函数 f 的罗宾点代替其结点（这些点具有指定的 f'/f 值，也称为罗宾参数、狄拉克耦合或Prüfer角的余切）。这给出了罗宾计数，特征函数结点计数和诺依曼计数的推广。我们将罗宾计数亏值与罗宾映射的正指标（Dirichlet到诺依曼映射的推广）联系起来。此外，我们还表明，两个相关的指标独立于Prüfer角。我们的主要工具是带有特殊边界条件族的拉普拉斯算子的谱流。作为我们结果的应用，我们展示了这些族的谱流与图的拓扑性质（如其贝蒂数、相互作用顶点的数量以及它们相对于图循环的位置）有关。 显示英文摘要

摘要： This paper is devoted to the Neumann-Kirchhoff Laplacian on a finite metric graph. We prove an index theorem relating the nodal deficiency of an eigenfunction with (1) the Morse index of the Dirichlet-to-Neumann map, (2) its positive index and the first Betti number of the graph. We then generalize this result, replacing nodal points of an eigenfunction f with its Robin points (these are points with a prescribed value of f'/f, known as the Robin parameter, or delta coupling, or cotangent of Pr\"ufer angle). This provides the Robin count, a generalization of the nodal and Neumann counts of an eigenfunction. We relate the Robin count deficiency with the positive index of the Robin map (a generalization of the Dirichlet-to-Neumann map). In addition, we show that two of the relevant indices are independent of the Pr\"ufer angle. Our main tool is the spectral flow of the Laplacian with special families of boundary conditions. As an application of our results, we show that the spectral flow of these families is related to topological properties of the graph, such as its Betti number, the number of interaction vertices, and their positions with respect to the graph cycles.