Maximum spread of K2,t-minor-free graphs

The spread of a graph Gis 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,K-t-minor. We show that for any t >= 2, there is an integer.tsuch that the maximum spread of an n-vertex K-2,K-t-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of [2n+xi(t)/3(t)] copies of K-t and n-1 - t [2n+xi(t)/3(t)] isolated vertices. The extremal graph is unique, except when t equivalent to 4 (mod12) and [2n+xi(t)/3(t)] is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of [2n+xi(t)/3(t)] - 1copies of K-t and n - 1 - t([2n+xi(t)/3(t)] - 1) isolated vertices. Furthermore, we give an explicit formula for xi(t). (c) 2023 Elsevier Inc. All rights reserved.