Maximum spread of $K_{2,t}$-minor-free graphs

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 $t\ge 2$, 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 $t\equiv 4\mathrm{m}\mathrm{o}\mathrm{d}12$ 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 ${\xi }_t$.