Spectral Geometry on Graphs

The talk's theme is the extraction of geometric information about graphs (metric or combinatorial) from the spectra of the graph's Schroedinger operators (continuous or discrete), and from the distribution of sign changes on the corresponding eigenfunctions. These include questions such as e.g., the ability to "hear the shape of the graph"; the extent to which the spectral sequence and the sequence of the number of sign changes (or number of nodal domains) complement or overlap each other; the derivation of topological information from the study of the response of the spectrum to variation of scalar or magnetic potentials on the graph, etc.

In the present talk I shall illustrate this research effort by reviewing several results I obtained recently. The first example answers the question "Can one count a tree?" which appears in the following context:  It is known that the number of sign changes of the eigenfunction on tree graphs equals to the position of the corresponding eigenvalue in the spectrum minus one. Is the reverse true? If yes, one can tell a tree just by counting the number of its sign changes. For the proof I shall introduce an auxiliary magnetic field and use a very recent result of Berkolaiko and Colin de Verdiere to connect the spectrum and the number of sign changes.  Next, I will discuss the band spectrum obtained by varying the magnetic phases on the graph. I will prove that the magnetic band-to-gap ratio (quality of conductance) is a universal topological quantity of a graph.  This result highlights the spectral geometric importance of this invariant and sheds a new light on previous works about periodic potentials on graphs.

The talk contains content of a work in progress with Gregory Berkolaiko.