The nodal count {0,1,2,3,...} implies the graph is a tree

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 (Pokorny et al. 1996 Mat. Zametki 60, 468-470 (doi:10.1007/BF02320380); Schapotschnikow 2006 Waves Random Complex Media 16, 167-178 (doi:10.1080/1745530600702535)) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278, 803-819 (doi:10.1007/S00220-007-0391-3); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54, 1346-1349 (doi:10.1103/PhysRevLett.54.1346); 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 (doi:10.2140/apde.2013.6.1213); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372, 20120522 (doi:10.1098/rsta.2012.0522); Colin de Verdière 2013 Anal. PDE 6, 1235-1242 (doi:10.2140/apde.2013.6.1235)). 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.