  The minimum number of distinct eigenvalues, taken over all real symmetric
matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other
parameters related to $G$, bounds for $q(G)$ are proven and then applied to
deduce further properties of $q(G)$. It is shown that there is a great number
of graphs $G$ for which $q(G)=2$. For some families of graphs, such as the join
of a graph with itself, complete bipartite graphs, and cycles, this minimum
value is obtained. Moreover, examples of graphs $G$ are provided to show that
adding and deleting edges or vertices can dramatically change the value of
$q(G)$. Finally, the set of graphs $G$ with $q(G)$ near the number of vertices
is shown to be a subset of known families of graphs with small maximum
multiplicity.
