CanaDAM 2019
SFU Harbour Centre, May 29 - 31, 2019

Spectral graph theory
Chair: Florian Lehner (University of Warwick)

KATE LORENZEN, Iowa State University
Constructions of Distance Laplacian Cospectral Graphs  [PDF]

Graphs are mathematical objects that can be embedded into matrices. Two graphs are cospectral if they have the same set of eigenvalues with respect to a matrix. In this talk, we discuss constructions of cospectral graphs for the distance Laplacian matrix. The first uses a relaxation of vertex twins called cousins which have predictable eigenvectors and eigenvalues in the distance Laplacian. This allows us to almost classify all cosepctral distance Laplacian pairs on eight or fewer vertices.

KRIS VASUDEVAN, University of Calgary
Influence of signs on expander graphs: Ramanujan graphs*  [PDF]

Spectral theory for dynamics on undirected and directed graphs containing only attractive (or positive) interactions has been the subject of detailed research studies. However, in many applied problems, these graphs can carry interactions which are repulsive (or negative) to a certain degree. Here, we report results of the influence of signs on the spectral properties of and dynamics on expander graphs and highlight its importance. To this end, we consider Lubotzky-Phillips-Sarnak Ramanujan graphs and their edge-rewired analogs.

*Work jointly done with Dr. Michael S. Cavers at the University of Toronto Mississauga

XIAOJING WANG, University Waterloo
Constructing Comatching Graphs by $2$-Sums  [PDF]

The \emph{matching polynomial} is a graph polynomial that does not only have interesting mathematical properties, but also possesses meaningful applications in physics and chemistry. Two graphs are said to be \emph{comatching} if they have the same matching polynomial. In 1973, Schwenk proved almost all trees have a cospectral mate, which implies almost all trees have a comatching mate. In our talk, we give a construction for comatching graphs that are not necessarily trees. The main construction relies on a graph operation we call \emph{$2$-sums}. Our construction also includes useful formulas to compute the matching polynomials of graphs as results of edge contraction and vertex identification.

XIAOHONG ZHANG, University of Manitoba
Laplacian fractional revival in graphs  [PDF]

Let $G$ be a graph, and denote its Laplacian matrix by $L$. Let $U(t)=e^{itL}$, then $U(t)$ is a complex symmetric unitary matrix. We say that $G$ admits Laplacian fractional revival between vertices $j$ and $k$ at time $t=t_0$, if $U(t_0)e_j=\alpha e_j+\beta e_k$ for some complex numbers $\alpha$ and $\beta$ with $\beta\neq 0$. In the special case where $\alpha=0$, we say there is perfect state transfer between vertices $j$ and $k$ at time $t=t_0$. A lot of research has been done on perfect state transfer. Not many graphs are known to admit fractional revival. We prove that if there is Laplacian fractional revival between vertices $j$ and $k$, then the two vertices are strongly cospectral with respect to the Laplacian matrix $L$. We also give a characterization of threshold graphs that admit Laplacian fractional revival.