Maximum spread of Ks,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 Ks,t-minor. We show that for any t≥ s ≥ 2, there is an integer ξₜ such that the extremal n-vertex Ks,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 2n+ξₜ3t copies of Kₜ and n-s+1 - t 2n+ξₜ/3t isolated vertices. Furthermore, we give an explicit formula for ξₜ and an explicit description for the graph L for t ≥ 32(s-3) +4/s-1.