Title: Maximum spectral gaps of graphs
Abstract:
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 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. (Joint work with William Linz and George Brooks)