Suppose $G$ is a graph with vertex set
$V=\{1,2,\ldots,n\}$. Associate to $G$
the collection of real symmetric matrices defined by
\[
S(G) = \{ A: A=A^{T}, \; {\rm for} \; i \neq j, \; a_{ij} \neq 0 \Leftrightarrow \{i,j\}\textrm{ is an edge in $G$}\}.
\]
If we let $q(A)$ denote the number of
distinct eigenvalues of $A$, then for a graph $G$ we define
\[ q(G) = \min \{ q(A) : A \in S(G) \}.\]
In this talk I will present some preliminary work that was done with
the Discrete Math Research Group at the University of Regina on
$q(G)$.