05C50 Online

05C50 Online is a biweekly virtual seminar about graphs and matrices held on a Friday, 10AM Central Time via Zoom. This is organized by Stephen Kirkland (University of Manitoba), Hermie Monterde (University of Regina) and Homer de Vera (University of Manitoba). Kindly fill out THIS FORM to subscribe to our mailing list!

Feb 6, 10am CT

Speaker: Carlos Hoppen

Affiliation: Universidade Federal do Rio Grande do Sul (UFRGS), Brazil

Title: Recent progress on the inverse eigenvalue problem

Abstract: The inverse eigenvalue problem has played a central role in spectral graph theory in the last decade. In this talk, I shall be interested in a particular parameter, namely the minimum number of distinct eigenvalues of a graph G, i.e., in the minimum number of distinct eigenvalues of a real symmetric matrix whose underlying graph is G. To be precise, let M be a symmetric matrix of order n. A graph G is said to be the underlying graph of M if its vertex set has n elements and if any pair of distinct vertices i and j are adjacent if and only if the entry ij of M is nonzero. There is no constraint about diagonal entries.

It is well known that, if G is a tree, the value of this parameter is at least d(G)+1, where d(G) is the diameter of G. I will discuss instances where this bound is tight. In a different direction,  I will show that the number of eigenvalues of G is at most 4 whenever G is a cograph, that is, whenever G does not contain an induced path on four vertices.  

This includes joint work with L. Emilio Allem, Martin Fürer, Lucas Siviero Sibemberg, and Vilmar Trevisan.

Preprint 1, 2 and 3  |  This talk will be recorded  |  You may email the speaker for their slides

Feb 20, 10am CT

Speaker: John Urschel

Affiliation: Massachussetts Institute of Technology (MIT), USA

Title: Nodal Statistics for Graphs and Matrices

Abstract: Given a symmetric matrix with a given sign pattern, what can the sign patterns of its eigenvectors look like? This simple question is closely related to the study of discrete nodal statistics, and draws strong parallels with classical results in analysis for Laplacian eigenfunctions. In this talk, we will give an overview of the field, covering key results on nodal sets for graphs and their connection to known results and open problems in the continuous setting. In addition, we will discuss some recent progress towards a more complete understanding of the extremal properties of the nodal statistics of a matrix.

No preprint  |  This talk will be recorded  |  Slides will be shared

Mar 6, 10am CT

Speaker: Veronika Furst

Affiliation: Fort Lewis College (Colorado), USA

Title: The number of edges of graphs that admit two distinct eigenvalues

Abstract: The Inverse Eigenvalue Problem for Graphs (IEP-G) concerns determining all possible spectra of matrices in S(G), the set of real symmetric matrices described by a graph.  Specific subproblems involve studying the maximum nullity or minimum rank, the ordered multiplicity list of distinct eigenvalues, or the minimum number of distinct eigenvalues of a graph.  In this talk, we will share recent results on the last of these, known as the parameter q(G).  In particular, we will investigate bounds on the number of edges of a graph through the “allows problem,” which asks what sparsity allows q(G) = 2, and the complementary “requires problem,” which asks what density requires q(G) = 2.  We will answer the first question, present a characterization of certain regular graphs, and give some evidence in support of a conjectured answer to the second question.

Joint work with W. Barrett, E. Egolf, S. Fallat, F. Kenter, S. Nasserasr, B. Rooney, M. Tait, and H. van der Holst, and partially supported by NSF grant DMS-2331072.

Preprint 1, 2, 3 and 4  |  This talk will NOT be recorded  |  You may email the speaker for their slides

Mar 20, 10am CT

Speaker: JV Morales

Affiliation: De La Salle University, Philippines

Title: The S3-symmetric tridiagonal algebra and Q-polynomial distance-regular graphs

Abstract: TBA

Preprint  |  This talk will be recorded  |  You may email the speaker for their slides

Apr 3, 10am CT

Speaker: Mark Kempton

Affiliation: Brigham Young University, USA

Title: Non-backtracking Matrices of Graphs

Abstract: Famously, the adjacency matrix of a graph can be used to enumerate walks in graphs.  This makes the adjacency spectrum important to understanding graph structure.  I will talk about the non-backtracking matrix, which is used to enumerate walks that are not allowed to backtrack.  This matrix also have interesting spectral properties, which I will discuss.  We will also look at constructions of matrices that are cospectral with respect to the non-backtracking matrix.

No preprint  |  This talk will be recorded  |  Slides will be shared

Apr 17, 10am CT

Speaker: Lord Kavi

Affiliation: Concordia University of Edmonton, Canada

Title: TBA

Abstract: 

May 1, 10am CT

Speaker: Sarojini Mohapatra

Affiliation: NIT Rourkela, India

Title: Pair state transfer on graphs with isomorphic branches

Abstract: In this talk, we discuss pair state transfer on graphs with respect to the adjacency matrix. We show that the evolution of certain pair states in a quantum network with isomorphic branches depends solely on the local structure, and it remains unaffected even if the global structure is altered. Furthermore, all graphs that enable high-fidelity vertex state transfer can be considered as isomorphic branches of a quantum network to exhibit high-fidelity pair state transfer. This result provides a framework for constructing infinite families of graphs, such as trees and unicyclic graphs, that exhibit perfect pair state transfer. In particular, we characterize the existence of pair state transfer in various graphs, including paths and cycles.

Preprint  |  This talk will NOT be recorded  |  Slides will be shared

May 15, 10am CT

Speaker: Domingos Cardoso

Affiliation: University of Aveiro, Portugal

Title: TBA

Abstract: 

For titles and abstracts of previous talks, and links to recordings, please visit here .

Please contact Hermie Monterde  if you have inquiries about the seminar.