Publication Date
Summer 2026
Degree Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematics and Statistics
Advisor
Wasin So; Sogol Jahanbekam; Yan Zhang
Abstract
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 = 2 was settled independently by Hong and by Powers in 1988, and k ≥ 4 was shown to fail by Nikiforov and Linz. This thesis surveys the resolution of the remaining case, k = 3, achieved in March 2026 by Quanyu Tang via a new matrix inequality. We give a careful, self-contained account of Tang’s proof, contextualized with the history of the conjecture. Additionally we provide a detailed analysis of a gap in Powers’ original 1989 argument, previously noted only in passing by Nikiforov and by Tang but never carefully written up. We also show that the connectivity hypothesis in Powers’ original statement can be removed for k = 3. We close with a discussion of the conjecture’s status for k ≥ 4 and related open problems.
Recommended Citation
Schmidt, Kevin, "Powers' Conjecture on the Third Largest Eigenvalue of a Graph" (2026). Master's Theses. 5831.
DOI: https://doi.org/10.31979/etd.kkej-tu27
https://scholarworks.sjsu.edu/etd_theses/5831
