Graduate Seminar

Location:  RI 209

Speaker: Johnna Parenteau

PhD Student supervised by Shaun Fallat and Jane Breen

Title: Scouting the Boundary by Scattering Eigenvalues

 Abstract:

: Given a simple graph, G, on n vertices, let \(S(G)\) be the set of \(n \times n\) real symmetric matrices, \(A = [a_{ij}]\), associated to G where, when \(i \neq j\), \(a_{ij} = 0\) if and only if \(ij\) is an edge in G and the main diagonal is free to be chosen. For any square matrix, A, let \(q(A)\) equal the number of distinct eigenvalues of A. The minimum number of distinct eigenvalues of G is defined as \(q(G) = min \{ q(A) : A \in S(G) \}\). A natural question that arises from this association is: Does there exist an \(A \in S(G)\) with \(q(A) = k\) for every integer k in the interval \([q(G), n]\)?

Since the inverse eigenvalue problem for graphs has been studied from many angles for years, a general solution seems difficult, so any progress made on subproblems, such as \(q(G)\), contributes to an understanding of the whole problem. In this talk, I discuss some recent progress on this question for certain classes of graphs.