Affiliation: University of Waterloo, Ontario, Canada

**Title: Laplacian States**

**Abstract**: It is customary to assume that the initial state of a continuous quantum walk on a graph $X$ is a vertex. However the Laplacian matrix of a graph with vertex set $V(X)$ is positive semidefinite, and can be scaled to produce a density matrix, and so provides an initial state for a walk on $X$. Qiuting Chen and I have studied this set up, and this talk will present some of what we have learned, and some of the lot that we don’t know.

You can watch the talk here, and you can access the slides here1, here2 and here3.

*This talk was given on January17th, 2021*

**Title: Tails and Chains**

**Abstract**: Physicists are interested in “graphs with tails”; these are constructed by choosing a graph $X$ and a subset $C$ of its vertices, then attaching a path of length $n$ to each vertex in $C$. We ask what is the spectrum of such graph? What happen if $n$ increases? We will see that the answer reduces to questions about the matrix

You can watch the talk here.

*This talk was given on September 13th, 2021*

**Title: The Matching Polynomial**

**Abstract: ** A $k$-matching in a graph is a matching of size $k$, and $p(X,k)$ denotes the number of $k$-matchings in $X$. The matching polynomial of a graph is a form of generating function for the sequence $(p(X,k))_{k\ge0}$.

It is closely related to the characteristic polynomial of a graph. I will discuss some of the many interesting properties of this polynomial and some of the related open problems.

You can watch the talk here and the slides can be viewed here: slides1, slides2, slides3, slides4

*This talk was given on January 25th, 2021*

**Title: Continuous Quantum Walks on Graphs**

**Abstract**: A quantum walk is a (rather imperfect analog) of a random walk on a graph. They can be viewed as gadgets that might play a role in quantum computers, and have been used to produce algorithms that outperform corresponding classical procedures. Physical questions about these walks lead to problems in spectral graph theory, and they also provide interesting new graph invariants. In my talk I will present some of the background, and some of the open problems of interest.

You can watch the talk here and the slides are available here.

*This talk was given on July 27th, 2020*