On the second largest adjacency eigenvalue of trees with given diameter

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