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

This talk answers the question "Can one count a tree?" which appears
in the following context: It is known that for all n, the n-th
eigenfunction on a tree graph has n-1 sign changes. Is the reverse
true? If yes, one can tell a tree just by counting the number of its
sign changes. We treat this question for both metric and combinatorial
graphs.  For the proof we introduce an auxiliary magnetic field and
use a very recent result initiated by Berkolaiko (with follow-up works
by Colin de-Verdiere and by Berkolaiko and Weyand) to connect the
spectrum and the number of sign changes. 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.