2023/04/03 by Alon, Noga, Wei, Fan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.01281
We prove that for each d ≥ 3 the set of all limit points of the second largest eigenvalue of growing sequences of d-regular graphs is [2√(d-1),d]. A similar argument shows that the set of all limit points of the smallest eigenvalue of growing sequences of d-regular graphs with growing (odd) girth is [-d, -2 √(d-1)]. The more general question of identifying all vectors which are limit points of the vectors of the top k eigenvalues of sequences of d-regular graphs is considered as well. As a by product, in the study of discrete counterpart of the "scarring" phenomenon observed in the investigation of quantum ergodicity on manifolds, our technique provides a method to construct d-regular almost Ramanujan graphs with large girth and localized eigenvectors corresponding to eigenvalues larger than 2√(d-1), strengthening a result of Alon, Ganguly, and Srivastava.