Nina White
The Laplacian on Graphs
The Laplace-Beltrami operator is important in several areas of physics
and mathematics. Usually it acts on smooth functions on a Riemannian
manifold. There is an analogous operator on graphs, also called the
Laplacian, with amazing similarities to its continuous counterpart.
The spectrum of this operator, in both categories, is an object of
active study. In the combinatorial setting, it tells us information
about the connectivity of our graph, leading to applications in
network theory. In this talk I'll introduce both Laplacians, and prove
some results in the combinatorial setting relating the spectrum of the
Laplacian to certain isoperimetric constants of a graph.