The nodal count {0123…} implies the graph is a tree

摘要

Sturm's oscillation theorem states that the nth eigenfunction of a Sturm–Liouville operator on the interval has n−1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. >1, 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki >60, 468–470 (); Schapotschnikow 2006 Waves Random Complex Media >16, 167–178 ()) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. >278, 803–819 (); Dhar & Ramaswamy 1985 Phys. Rev. Lett. >54, 1346–1349 (); Fiedler 1975 Czechoslovak Math. J. >25, 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n, the nth eigenfunction of the graph has n−1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE >6, 1213–1233 (); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A >372, 20120522 (); Colin de Verdière 2013 Anal. PDE >6, 1235–1242 ()). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.