Spectral methods in graph theory
Organizer and Chair: Bill Martin
(Worcester Polytechnic Institute, USA) and Jason Williford
(University of Wyoming, USA)
- GABRIEL COUTINHO, University of Waterloo
Graph spectra and quantum walks [PDF]
In the last few years, we have seen many interesting applications of spectral tools to study quantum walks in graphs. In this talk I will survey some of these results and some of the open problems we have. For example, if perfect state transfer happens between two vertices of a graph, how far apart can they be?
- KRYSTAL GUO, Simon Fraser University
Eigenvalue interlacing in digraphs [PDF]
The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices $X$ and $Y$ such that there are no arcs from $X$ and $Y$. For a class of eulerian digraphs, we give an bound on the size of a separation in terms of the eigenvalues of the Laplacian matrix.
- TAKUYA IKUTA, Kobe Gakuin University
Complex Hadamard matrices attached to some association schemes. [PDF]
Recently, we classified type-II matrices attached to some association schemes. We have 6 infinite families of type -II matrices, among which are 4 families of our complex Hadamard matrices. To check whether our complex Hadamard matrices are inequivalent to the tensor product of known examples, we introduce the Haagerup set for type-II matrices and the Nomura algebra for our complex Hadamard matrices. In this talk, we mainly present how to compute the Haagerup set and the Nomura algebra for our complex Hadamard matrices. This is based on a joint work with A kihiro Munemasa.
- ANDRIY PRYMAK, University of Manitoba
Non-existence of (76,30,8,14) strongly regular graph and some structural tools [PDF]
Our main result is the non-existence of strongly regular graph with parameters $(76,30,8,14)$. We heavily use Euclidean representation of a strongly regular graph, and develop a number of tools that allow to establish certain structural properties of the graph. In particular, we give a new lower bound for the number of $4$-cliques in a strongly regular graph.
This is a joint work with A. Bondarenko and D. Radchenko.
- JASON WILLIFORD, University of Wyoming
The Maslov Index, Two-Graphs and Cometric Association Schemes [PDF]
In this talk we will discuss the so-called Maslov index of symplectic spaces over a finite field. The Maslov index can be viewed as an operation on triples (or more generally n-tuples) of maximal totally isotropic subspaces and has the following property: two such triples are equivalent under the action of the symplectic group if and only if they are equivalent under the general linear group and have the same Maslov index.
We will discuss some properties of this index in the context of some recently discovered association schemes
with the cometric property. No background in finite geometry is assumed.