Maximum spectral gaps of graphs

The spread of a graph G is the difference λ₁ - λₙ 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 λᵢ₊₁ - λₙ₋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.