Powers' Conjecture on the Third Largest Eigenvalue of a Graph

In 1989, David Powers conjectured that for any connected graph G on n vertices, the k-th largest eigenvalue of the adjacency matrix satisfies λk(G) ≤ ⌊n/k⌋ for every 1 ≤ k ≤ n/2. The conjecture has since been resolved case by case, in the sharper form λk(G) ≤ n/k − 1: the case k = 1 is classical, k