Maximum spectral gaps of graphs

The spread of a graph G is the difference λ 1 − λ n between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on n vertices with maximum spread for sufficiently large n . In this paper, we study a related question of maximizing the difference λ i + 1 − λ n − j for a given pair ( i , j ) over all graphs on n vertices. We give upper bounds for all pairs ( i , j ) , exhibit an infinite family of pairs where the bound is tight, and show that for the pair ( 1 , 0 ) the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on n vertices.