Affiliation: University of Waterloo, Ontario, Canada

**Title: New Characterizations of Distance-Biregular Graphs**

**Abstract:** Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, and were able to use this notion to come up with a characterization of when a graph is distance-regular. Subsequently, Fiol and Garriga were able to use pseudo-distance-regular vertices and a bound on the excess of a vertex to come up with another characterization of distance-regular graphs. We will present an overview of their results, as well as recent extensions to distance-biregular graphs.

**Title: A Spectral Moore Bound for Bipartite Semiregular Graphs**

**Abstract**: The Moore bound provides an upper bound on the number of vertices of a regular graph with a given degree and diameter, though there are disappointingly few graphs that achieve this bound. Thus, it is interesting to ask what additional information can be used to give Moore-type bounds that are tight for a larger number of graphs. Cioabǎ, Koolen, Nozaki, and Vermette considered regular graphs with a given second-largest eigenvalue, and found an upper bound for such graphs. When the bound is tight, the resulting graph is a distance regular graph of a particular form. Subsequently, Cioabǎ, Koolen, and Nozaki improved the bound for regular bipartite graphs. In this talk, we will give an overview of some of the techniques used for these proofs and show how they can be applied to give a new bound on the number of vertices in a semiregular graph with a given second-largest eigenvalue. We will also discuss distance biregular graphs, since this relaxation of distance regularity is necessary for the bound to be tight.

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

*This talk was given on April 26th, 2021*

Affiliations: University of Waterloo, Ontario & Clarkson University, New York, USA

**Title: Monogamy Violations in Perfect State Transfer**

**Abstract**: Continuous-time quantum walks on a graph are defined using a Hermitian matrix associated to a graph. For a quantum walk on a graph using either the adjacency matrix or the Laplacian, there can be perfect state transfer from a vertex to at most one other vertex in the graph. This monogamy property was proved by Kay for all real symmetric matrices. If a graph is associated with a Hermitian but not symmetric matrix, then we can still define a continuous-time quantum walk, but this monogamy property does not hold. In this talk, we will discuss graphs in violation with this property through examples, characterizations, and open questions.

This talk will largely be based on: https://arxiv.org/abs/1310.3885, https://arxiv.org/abs/1701.04145, and https://arxiv.org/abs/2002.04666.

You can watch the talk here.

*This talk was given on November 2nd, 2020*