On the eigenvalues of the graphs D(5, q)

Let q = pe, where p is a prime and e is a positive integer. The family of graphs D(k, q), defined for any positive integer k and prime power q, were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs D(k, q), provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D(k, q) is always less than or equal to 2N/q. If true, this would imply that for a fixed q and k growing, D(k, q) would define a family of expanders that are nearly Ramanujan. In this paper we prove the smallest open case of the conjecture, showing that for all odd prime powers q, the second largest eigenvalue of D(5, q) is less than or equal to 2N/q. (c) 2023 Elsevier Inc. All rights reserved.