Spectral flow and Robin domains on metric graphs

TL;DR

This work analyzes the Neumann-Kirchhoff Laplacian on finite metric graphs and develops an index-theoretic framework for nodal deficiencies via Dirichlet-to-Neumann maps, extended to Robin data through the Robin map. Central to the approach is the spectral flow of specially designed operator families with mixed boundary conditions, which bridges spectral counts with topological invariants such as the first Betti number. The authors prove explicit relationships between nodal/Robin deficiencies and Morse indices of DtN/Robin maps, show independence from the Prüfer angle for key indices, and connect spectral flow to graph topology and cycle structure. The results yield topological interpretations of spectral data on graphs and offer inverse-spectral insights through Betti-number connections, with an appendix establishing a winding-number perspective on spectral flow.

Abstract

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 Prüfer 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 Prüfer 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.

Figures (6)

• Figure 1.1: The eigenfunction $f_{4}$ of the Neumann-Kirchhoff Laplacian on a star graph, partitioning it into $\nu_{0}\left(f_{4}\right)=4$ nodal domains.
• Figure 1.2: The orientation of edges at a vertex of degree two. The derivative $f'(v_{-})$ is directed into the vertex, while $f'(v_{+})$ is directed into the edge.
• Figure 1.3: Illustration of the spectral flow through a given horizontal cross-section. There are three positive intersections of the dashed line by spectral curves, accounting for the spectral flow through $\lambda$ equal to $3$.
• Figure 2.1: The lowest analytic eigenvalue branches of $H^{\mathcal{B}}_{\alpha}(t)$ with $|\mathcal{B}| = 2$ and $\alpha\neq 0$. Two spectral curves tend to $-\infty$ as $t\to +0$, showing that the family is not uniformly bounded from below.
• Figure 7.1: Demonstration of Theorem \ref{['thm:beta-beta']} for a two-cycle graph and two different choices of $\mathcal{B}$. In (A), the set $\mathcal{B}$ is chosen so that $\beta(\Gamma_{\mathcal{B}})=0$, and thus the spectral flow through the dashed line $\lambda=\varepsilon$ is $\beta(\Gamma)-\beta(\Gamma_{\mathcal{B}})=2$. In (B), the set $\mathcal{B}$ is chosen so that $\beta(\Gamma_{\mathcal{B}})=1$, and thus the spectral flow through the dashed line $\lambda=\varepsilon$ is $\beta(\Gamma)-\beta(\Gamma_{\mathcal{B}})=1$.

Theorems & Definitions (48)

• Remark
• Definition
• Theorem 1.1
• Corollary 1.3
• Definition
• Definition
• Remark
• Theorem 1.4
• Theorem 1.5
• Definition