Maximum spread of $K_{s,t}$-minor-free graphs

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\ge s\ge 2$ and sufficiently large $n$, there is an integer ${\xi }_t$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+{\xi }_t}{3t}\rfloor$ copies of $K_t$ and $n-s+1-t\lfloor \frac{2n+{\xi }_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for ${\xi }_t$ and an explicit description for the graph $L$ for $t\ge \frac{3}{2}(s-3)+\frac{4}{s-1}$.