Nodal statistics on quantum graphs
The number \(\phi_n\) of zeroes of the \(n\)-th eigenfunction of the Laplacian with standard (Neumann-Kirchhoff) vertex conditions on a metric graph is (for an eigenfunction which does not vanish on any vertex) larger than, or equal to, \(n - 1\), with equality \(\phi_n = n - 1\) for all \(n\) if and only if the graph is a tree; this can be considered a generalization of Sturm's oscillation theorem. In this paper the authors study the \textit{nodal surplus} \[ \sigma_n := \phi_n - (n - 1) \] for general finite metric graphs with \(\beta\) cycles. They focus on the \textit{nodal surplus distribution} of a metric graph. In the first main result it is shown that the latter is a well-defined probability distribution on the set \(\{0, 1, \dots, \beta \}\) and that this distribution is symmetric; as a consequence of this, the Betti number \(\beta\) of the graph can be recovered as twice the average nodal surplus. The second main result deals with the class of graphs whose cycles are disjoint. In this situation it is shown that the nodal surplus distribution is Binomial with parameters \(\beta\) and \(1/2\).
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