On the second eigenvalue of a graph (Q1182585)

Let \(G\) be a regular graph of valency \(d\) containing two edges at distance \(2k+2\). The author shows that the second largest eigenvalue of the adjacency matrix of \(G\) is at least \[ 2\sqrt{d-1}-{2\sqrt{d-1}-1\over k+1}. \] This implies an earlier bound of Alon and Boppana (stated without proof in: \textit{N. Alon} [Eigenvalues and expanders, Combinatorica 6, No. 2, 83-96 (1986; Zbl 0661.05053)]). The proof is short and direct.