## Seminar:

Fall 2012, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** November 1 **

**" Spectral Theory on Graphs "**

**Tracy Weyand**

__Abstract__Graphs are commonly used as models by physics, chemists, and many other scientists. Applications range from networks to nanotechnology to quantum chaos. Given a graph that models a system, one usually wants to know what happens when some “action” takes place on the system. We try to represent such an action with an operator. Given an operator, the natural questions usually involve its eigenvalues and eigenfunctions.

I will begin by showing specific examples of operators on both discrete and metric graphs. Then I will introduce a few concepts, including quadratic forms on graphs and gauge invariance, which are common tools in finding the spectrum of a graph. Finally, I will state some important results about the spectrum of graphs, including one I have proven recently. This talk will be accessible to all graduate students; no prior background knowledge of graphs will be needed.