On the second largest adjacency eigenvalue of trees with given diameter

For a graph G, let ₂ (G) denote the second largest eigenvalue of the adjacency matrix of G. We determine the extremal trees with maximum/minimum adjacency eigenvalue ₂ in the class T (n, d) of n-vertex trees with diameter d. This contributes to the literature on ₂-extremization over different graph families. We also revisit the notion of the spectral center of a tree and the proof of ₂ maximization over trees.