Spectral Graph Theory – Visiting and Early Research Scholar's Experiences in Mathematics (VERSEIM-REU)

Spectral Graph Theory

Fix an integer $q \ge 1$ . A homogeneous tree $X$ of degree q+1 is a tree (that is, a connected undirected graph without cycles) in which every vertex has exactly q+1 edges. Such a tree clearly has infinitely many vertices and edges, and its homogeneity allows us to study many structures on the tree from the perspective of harmonic analysis and group theory. Indeed, there have been numerous recent works on the subject (see, for example, Figa-Talamanca & Nebbia; Casadio Tarabusi–Gindikin–Picardello; Casadio Tarabusi–Picardello). These papers highlight the interesting similarities between homogeneous trees, Euclidean spaces, and rank-one noncompact Riemannian symmetric spaces from the perspective of geometric and harmonic analysis.

Let V be the set of all vertices of $X$ . Here, we will only consider functions defined on the vertices of $X$ so for convenience, we refer to functions $f: V \to \mathbb{C}$ as functions on $X$ . The vector space of all such functions $f$ is denoted by $\mathcal{F}(X)$ .

For each $f \in \mathcal{F}(X)$ and each integer $k$ , let $\mu_k f$ denote the radius $-k$ spherical mean of $f$ . That is, for any vertex $v \in X$ , $\mu_k f(v)$ is the average of $f$ over all vertices at distance $k$ from $V$ .

The objective of this REU research program is to analyze properties of the generalized Euler–Poisson–Darboux (EPD) equation on $X$ . This is a discrete differential equation on $X$ , defined as follows. Fix positive real numbers $s$ and $t$ such that $s+t=1$ . A sequence of functions $\{ f_k \}_{k=0}^\infty$ on $X$ is said to satisfy the generalized EPD equation if $\mu_1 f_k = t\,f_{k-1} + s\,f_{k+1}, \qquad k \ge 1.$

When $s = t = \frac{1}{2}$ , this equation reduces to the wave equation on $X$ . When $s = \frac{q}{q+1}$ , it becomes the classical EPD equation on $X$ , which characterizes the range of the mean value operator $\mu_k$ .

Desired Background: Students interested in this project must have taken a linear algebra course and at least one proof-intensive Mathematics course.

References

J. M. Cohen and M. Pagliacci, “Explicit solutions for the wave equation on homogeneous trees,” Advances in Applied Mathematics, 15 (1994), no. 4, 390–403.characterizes the range of the mean value operator $\mu_k$ .

A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Mathematical Society Lecture Note Series, vol. 162, Cambridge University Press, 1991.

E. Casadio Tarabusi, S. G. Gindikin, and M. A. Picardello, “The circle transform on trees,” Differential Geometry and its Applications, 19 (2003), no. 3, 295–305.

E. Casadio Tarabusi and M. A. Picardello, “Radon transforms in hyperbolic spaces and their discrete counterparts,” Complex Analysis and Operator Theory, 15 (2021), no. 1, Paper No. 13.