  The smallest possible number of distinct eigenvalues of a graph $G$, denoted
by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the
graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number
of vertices of a unique shortest path joining any pair of vertices in $G$.
Thus, if $n$ is the number of vertices of $G$, then $n-q(G)$ is bounded above
by the size of the complement (with respect to the vertex set of $G$) of the
vertex set of the longest unique shortest path joining any pair of vertices of
$G$. The purpose of this paper is to commence the study of the minor-monotone
floor of $n-k$, which is the minimum of $n-k$ among all graphs of which $G$ is
a minor. Accordingly, we prove some results about this minor-monotone floor.
