Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees

For a connected graph $G$, let $τ(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $ρ(G)$ be its adjacency spectral radius. For integers $n\ge1$, $k\ge2$, and $k\leδ\le2k-1$, let $\mathcal{G}_{n,δ}$ be the class of connected $n$-vertex graphs with minimum degree $δ$, a