Directed by Drs. Kathryn Beck & Kasso Okoudjou

Fourier analysis on graphs, built on the spectral properties of the graph Laplacian, is now a well-established framework for studying signals defined on graphs and networks. This theory has led to the development of vertex–frequency analysis on graphs, whose goal is to design mathematical tools that simultaneously capture both the spatial (vertex-based) and frequency-based structure of graph signals.

Within this framework, several analogues of classical signal-processing constructions have been developed, including wavelet systems and Gabor-type systems on graphs. More recently, the Graph Short-Time Fourier Transform (GSTFT) has been introduced, where the graph heat kernel serves as the window function. This construction yields a family of time-dependent Gabor frames on graphs, which can be analyzed with respect to their frame and approximation properties.

Previous work has focused on highly symmetric graph classes, such as strongly regular graphs and vertex-transitive graphs. In this project, we will extend these ideas to new settings, including Cayley graphs and random graphs. The main goals will be to determine when these time-dependent Gabor systems form tight frames and to identify special structural or spectral properties that arise in these graph families.

Prerequisites: Students interested in this project should have completed coursework in linear algebra and differential equations, and should be comfortable with basic computer programming, such as MATLAB or Python. A course in real analysis will be helpful, but is not required.  No prior background in graph signal processing is required.