NOVEMBER 5, 2020

TITLE AND ABSTRACT:

"Recent Trends on the Inverse Eigenvalue Problem for Graphs"

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Given a simple graph G=(V,E) with V = {1,2,...n}, we associate a collection of real n-by-n symmetric matrices governed by G, and defined as S(G) where the off-diagonal entry in position (i,j) is nonzero iff i and j are adjacent.

The inverse eigenvalue problem for G (IEP-G) asks to determine if a given multi-set of real numbers is the spectrum of a matrix in S(G). This particular variant on the IEP-G was born from the research of Parter and Wiener concerning the eigenvalue of trees and evolved more recently with a concentration on related parameters such as: minimum rank,maximum multiplicity, minimum number of distinct eigenvalues, and zero forcing numbers. An exciting aspect of this problem is the interplay with other areas of mathematics and applications. A novel avenue of research on so-called `strong properties' of matrices, closely tied to the implicit function theorem, provides algebraic conditions on a matrix with a certain spectral property and graph that guarantee the existence of a matrix with the same spectral property for a family of related graphs.

In this lecture, we will review some of the history and motivation of the IEP-G. Building, on the work Colin de Verdi\'ere, we will discuss some of these newly developed `strong properties' and present a number of interesting implications pertaining to the IEP-G.

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SPEAKER: 

Shaun M. Fallat, Professor at the University of Regina in the Department of Mathematics and Statistics 

(Learn more about the speaker here​)

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WHEN:

Thursday, November 5, 2020

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Meet and Greet:

• 2:00pm-2:30pm CST

​Talk:

• 2:30pm-3:20pm CST 

Questions:

• 3:20pm-3:30pm CST

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WHERE:

The event took place via Zoom due to Covid-19 restrictions.

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Watch recorded livestream here: