Abstract
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 K2,t-minor. We show that for any t≥2, there is an integer ξt such that the maximum spread of an n-vertex K2,t-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of ⌊2n+ξt3t⌋ copies of Kt and n−1−t⌊2n+ξt3t⌋ isolated vertices. The extremal graph is unique, except when t≡4(mod12) and 2n+ξt3t is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of ⌊2n+ξt3t⌋−1 copies of Kt and n−1−t(⌊2n+ξt3t⌋−1) isolated vertices. Furthermore, we give an explicit formula for ξt.
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