Speaker
Description
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 $i\neq j$, $a_{ij} \neq 0$ 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 $q(G) = {\rm min} \{ q(A) : A \in S(G)\}$. 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, we discuss some recent progress on this question for certain classes of graphs.