  The spread of a graph $G$ is the difference between the largest and smallest
eigenvalues of the adjacency matrix of $G$. In this paper, we consider the
family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq
2$, there is an integer $\xi_t$ such that the maximum spread of an $n$-vertex
$K_{2,t}$-minor-free graph is achieved by the graph obtained by joining a
vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor$ copies of
$K_t$ and $n-1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. The
extremal graph is unique, except when $t\equiv 4 \mod 12$ and $\frac{2n+ \xi_t}
{3t}$ is an integer, in which case the other extremal graph is the graph
obtained by joining a vertex to the disjoint union of $\lfloor
\frac{2n+\xi_t}{3t}\rfloor-1$ copies of $K_t$ and $n-1-t(\lfloor
\frac{2n+\xi_t}{3t}\rfloor-1)$ isolated vertices. Furthermore, we give an
explicit formula for $\xi_t$.
