Papers › Maximum spectral gaps of graphs
Maximum spectral gaps of graphs
George Brooks, William Linz, Linyuan Lu
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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.
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