Spectral flow and Robin domains on metric graphs

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.