1212.6710 (Ram Band)

The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree    [PDF]

Ram Band

Sturm's oscillation theorem states that the n-th eigenfunction of a Sturm-Liouville operator on the interval has n-1 zeros (nodes). Recent works generalized this result for all metric tree graphs and Fiedler proved a similar result for discrete tree graphs. 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. The proof also shows that when the graph is supplied with a magnetic field 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 show that their nodal counts are strongly connected to this of the metric graph.

View original: http://arxiv.org/abs/1212.6710