On the second largest adjacency eigenvalue of trees with given diameter

For a graph G, let $ \lambda _2(G) $ lambda 2(G) denote the second largest eigenvalue of the adjacency matrix of G. We determine the extremal trees with maximum/minimum $ \lambda _2 $ lambda 2 in the class $ \mathcal {T}(n,d) $ T(n,d) of n-vertex trees with diameter d. This contributes to the literature on $ \lambda _2 $ lambda 2-extremization over different graph families. We also revisit the notion of the spectral centre of a tree and the proof of $ \lambda _2 $ lambda 2-maximization over trees.