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 and Hermie Monterde from the University of Manitoba. If you wish to subscribe to our mailing list (and receive zoom links for upcoming talks), kindly fill out this subscription form.

September 13, 10am CT

Speaker: Hanmeng (Harmony) Zhan

Affiliation: Worcester Polytechnic Institute (USA)

Title: Epsilon-uniform mixing on strongly regular graphs via coined quantum walks

Abstract: Many quantum search algorithms are equivalent to coined quantum walks that start from a uniform superposition of arcs and get very close to a state that "concentrates on" some marked vertex. In this talk, we consider a related problem: if we start from a state that concentrates on a vertex, can the probability distribution get arbitrarily close to uniform over all arcs? We study this phenomenon on quantum walks where no vertices are marked. Using algebraic properties of graphs, we completely classify all strongly regular graphs on which this happens.

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

Preprint  |  Recording passcode: c&jbJf1d  |  No slides

September 27, 10am CT

Speaker: Geir Dahl

Affiliation: University of Oslo (Norway)

Title: Combinatorial Matrices and Frobenius theory

Abstract: In combinatorial matrix theory (CMT) one studies properties of combinatorial matrices, i.e., matrices with entries 0, 1 and -1. For instance, permutation matrices, i.e., (0,1)-matrices with exactly one 1 in every row and column, are interesting. There are basic and challenging questions for permutation matrices and related matrix classes. This is what this talk will be about. 

One of the roots of CMT is work done by Frobenius and, slightly later, by Konig. We present some of these ideas and results, including the classical Frobenius-Konig theorem, along with some connections to related mathematical areas. Also, some recent work with Richard Brualdi on an extension of the Frobenius-Konig theorem will be discussed, and there we will see a connection to a very powerful theorem by Hoffman. 

A main goal is to present this in a non-technical way, and hopefully the topic may be of interest in the 05C50-community and its neighborhood!

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

Preprint  |  Recording passcode: xC@D3@nS |  Slides

October 11, 10am CT

Speaker: Kevin Vander Meulen

Affiliation: Redeemer University (Canada)

Title: On counting eigenvalues of a matrix sign pattern

Abstract: The signs of the entries of a real matrix can give insight into the eigenvalues of the matrix. A  well-known example is the Perron Frobenius theorem for matrices that have no negative entries. As an inverse eigenvalue problem, one can ask for the possible spectra of a given sign pattern, where a sign pattern is a specified arrangement of the zero, positive and negative entries. We review combinatorial and analytic techniques employed to explore the possible spectra of a sign pattern. We present recent results, and open questions, on the number of distinct eigenvalues allowed by a sign pattern.

This talk will NOT be recorded. You may email the speaker for their slides.

No preprint  |  No recording  |  No slides

October 25, 10am CT

Speaker: Xiaohong Zhang

Affiliation: Université de Montréal (Canada)

Title: Continuous quantum walks and oriented Cayley graphs

Abstract: Let M be a Hermitian matrix associated to a graph X on n vertices. For any time t \geq 0, the transition matrix of the continuous quantum walk on X relative to M at time t is given by U(t)=exp(itM). For two vertices a and b of X, if there is some time t such that U(t)e_a=\gamma e_b for some scalar \gamma, then we say there is perfect state transfer from a to b at time t if a\neq b, and say the walk is periodic at vertex a at time t if a=b. In this talk, we will see how the field where the entries of M lie in influences state transfer properties. We will solve an interesting eigenvalue problem that arises from the study of quantum walks: which oriented Cayley graph has all its eigenvalues integer multiple of \sqrt{\Delta} for some square-free integer \Delta. This generalizes a result of Bridges and Mena on when a Cayley graph has only integer eigenvalues. 

This talk will be recorded and the speaker's slides will be made available in this website.

Preprint  |  Recording passcode: G4C#Ncj$  |  Slides

November 8, 10am CT

Speaker: Shaun Fallat

Affiliation: University of Regina (Canada)

Title: Sauntering through Strong Matrix Properties

Abstract: Let us ask: What is a "strong" matrix property and why is such a property useful for studying eigenvalues of graphs? During this talk we will attempt to address both questions above by first wandering through some of the developed strong matrix properties (SAP, SMP, SSP, SIPP, SNIP, SSVP, nSSP, ...) and highlight their primary purpose. Then, we will dive into some of their applications connecting spectra of graphs and perturbations.

This talk will be recorded and the speaker's slides will be made available in this website.

Preprint  |  Recording  |  Slides

November 22, 10am CT

Speaker: Carla Oliveira

Affiliation: National School of Statistical Sciences - ENCE, Rio de Janeiro (Brazil)

Title: Some results involving the Aα-matrix

Abstract: Let G = (V, E) be a simple graph of order n. The adjacency matrix of G is denoted by A(G) = [a_ij], where a_ij=1 if {v_i ,v_j} ∈ E and a_ij=0, otherwise. The diagonal matrix of degrees of G, D(G) = [d_ij], is defined by d_ii=d(v_i), and d_ij=0, ∀ i \neq j, and the signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2017, Nikiforov define the matrix Aα(G) as a convex linear combination of A(G) and D(G) as follows

Aα(G)= αD(G) + (1 − α)A(G),      α ∈ [0, 1]

It is easy to see that A_0(G) = A(G), A_1(G) = D(G) and 2A_{1/2}(G) = Q(G). This talk presents some results and problems involving the eigenvalues of the Aα-matrix.

This talk will be recorded and the speaker's slides will be shared on the website.

No preprint  |  Recording  |  Slides

December 6, 10am CT

Speaker: Edinah Gnang

Affiliation: Johns Hopkins University (USA)

Title: TBA

Abstract: TBA

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.