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The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior extremal points of f. The Neumann domains of f are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains.

In this talk, we present new results regarding the Neumann count (i.e number of Neumann points) and its probability distribution. In particular, the relation to the graph's topology and its application to the inverse problem. We also show a spectral local to global relation: The Kottos-Smilansky trace formula of the graph is equal to the sum of the trace formulas of the Neumann domains up to a small correction which is exactly the nodal surplus.

This is joint work with Ram Band (Technion).