Maximum spread of Kₒ, ₓ-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 Kₒ, ₓ-minor. We show that for any t s 2, there is an integer ₓ such that the extremal n-vertex Kₒ, ₓ-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+ₓ3t copies of Kₜ and n-s+1 - t 2n+ₜ3t isolated vertices. Furthermore, we give an explicit formula for ₓ and an explicit description for the graph L for t 32 (s-3) +4s-1.