Spectral Graph Theory  Part I
Org:
Sebastian Cioaba (University of Delaware) and
Michael Tait (Villanova University)
[
PDF]
 AIDA ABIAD, Eindhoven University of Technology
Neumaier graphs with few eigenvalues [PDF]

A Neumaier graph is a noncomplete edgeregular graph containing a regular clique. In this talk we will discuss some recent progress on Neumaier graphs with few eigenvalues. This is joint work with B. De Bruyn, J. D'haeseleer and J.H. Koolen
 KRYSTAL GUO, University of Amsterdam
Entanglement of free Fermions on distanceregular graphs [PDF]

Many physical processes evolving over time on an underlying graph have led to problems in spectral graph theory, including quantum walks. These problems provide new graph invariants and also new applications for theorems about the eigenspaces of graphs. In this talk, we will consider free Fermions on vertices of distanceregular graphs are considered. Using concepts from Terwilliger algebras, we study the entanglement Hamiltonian. This is based on joint work with Nicolas CrampĂ© and Luc Vinet.
 FERDINAND IHRINGER, Ghent University
Strongly regular graphs satisfying the $4$vertex condition [PDF]

A graph satisfies the $3$vertex condition if and only if it is strongly
regular. A graph of order $v$ satisfies the $v$vertex condition if and only if
it is rank 3, so the most symmetric kind of strongly regular graph. We show
that the family of strongly regular graphs $NO_n^\varepsilon(q)$, defined on
the nonzero square points of a nondegenerate quadric with two points
adjacent if they span a tangent, satisfies the $4$vertex condition. These
graphs have rank $(q+3)/2$. We also discuss some other families of strongly
regular graphs satisfying the $4$vertex condition.
Joint work with A. E. Brouwer.
 JEPHIAN LIN, National Sun Yatsen University
The strong spectral property for graphs [PDF]

A symmetric matrix $A$ is said to have the strong spectral property if $X=O$ is the only symmetric matrix satisfying $A\circ X=O$, $I\circ X=O$, and $AXXA=O$. Here the operation $\circ$ is the entrywise product. If a matrix has the strong spectral property, one may perturb the matrix slightly to create more nonzero entries without changing its spectrum. This behavior has been used widely for constructing matrices in the inverse eigenvalue problem of a graph. In this talk, we show that if the nonzero pattern of the matrix is described by certain graphs, then it always has the strong spectral property.