The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
This work is concerned with the spectral structure of the potential-less Laplace operator acting on a compact graph with either Kirchhoff and continuity conditions at interior vertices, or so called anti-Kirchhoff conditions, and suitable boundary conditions at terminal vertices. The authors show that the spectrum (eigenvalues along with their multiplicities) determine uniquely: the number of vertices of the graph; the number of components of the graph; the number of edges of the graph and numbers of bipartite and non-bipartite components.
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