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.

Nodal Statistics on Quantum Graphs

“…The nodal count of a tree graph is φ n = n − 1 which is a generalization of Sturm's oscillation theorem that was obtained in [50,53] (interestingly, the converse result has also been established [1]: if the nodal count is φ n = n − 1 then the graph is a tree). For graphs which are not trees n − 1 provides a baseline from which the actual number of zeros does not stray very far.…”

“…From here on we will also refer to k n ≥ 0 as the eigenvalue of the graph. The eigenfunctions of (1) can be chosen to be real and, if the eigenfunction does not vanish on entire edges (which is possible on graphs due to failure of unique continuation principle), one can count the number of the zeros of the n-th eigenfunction. This quantity will be denoted by φ n and will be the main object of our study.…”

“…Our approach to analyzing the nodal surplus sequence {σ n } is interpreting it as a sample of a certain function defined on a certain manifold endowed with a probability measure. The manifold and the measure go back to an idea of Barra and Gaspard [7]; the fact that the nodal surplus can be read off the manifold was shown by Band [1] who converted the nodal-magnetic connection of Berkolaiko and Weyand [15] into a function on the manifold. Existence of the limit in (5) follows from the ergodicity of the sampling process (which goes back to Weyl [59]).…”

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

“…Note that a number of other authors refer to our definition of the 'flip count' as the 'nodal (point) count', e.g. [17,35]. We prefer this terminology in this context to emphasise the distinction between flips and nodal domains.…”

Complex patterns in wave functions: drums, graphs and disorder

“…One focus theme of the seminars and the discussion during the meeting was the properties of nodal sets in graphs, drums and more general systems-and closely related topics such as spectral equipartitions. Five contributions presenting original research results summarize some aspects of this subject: Berkolaiko & Weyand [4] present their recent results on the effect of magnetic perturbations on the spectrum and on nodal counts in graphs, Band [5] uses related methods to prove that a graph with nodal count which is Courant-sharp 0, 1, 2, 3, . .…”