Neumann Domains on Quantum Graphs.

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the subgraphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove some foundational results on Neumann domains of quantum graph eigenfunctions: bounds on the number of Neumann domains and properties of the probability distributions of these numbers. We present fundamental geometric and spectral parameters of Neumann domains, such as the normalized isoperimetric ratio and the spectral position and prove the relevant bounds and properties of the probability distributions.