# Prairie Mathematics Colloquium

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Thu., Nov. 5, 2020 2:30 p.m.
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Location: Live Stream
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**Time: **Social tea break at 2:00 p.m., Talk at 2:30 p.m.

**Speaker: ** Shaun Fallat, University of Regina

**Title: **Recent Trends on the Inverse Eigenvalue Problem for Graphs (523 kB)

**Zoom: **https://zoom.us/j/95446214700

**Abstract**: 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 Verdière, we will discuss some of these newly developed "strong properties" and present a number of interesting implications pertaining to the IEP-*G*.

See: Prairie Mathematics Colloquium

This event is supported by PIMS.