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

Abstract

Sturm's oscillation theorem states that the n th 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 ( 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 n th 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.