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 $δ$, and let $\mathcal{L}_{\mathcal{H}}^{2}(n,k,δ)\subseteq\mathcal{G}_{n,δ}$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $τ(G)\ge k$ is determined by maximizing $ρ(G)$ over this family. Set $h=δ-k$. For each fixed admissible pair $(k,δ)$ and all sufficiently large $n$, we determine the maximum and minimum adjacency spectral radii in $\mathcal{L}_{\mathcal{H}}^{2}(n,k,δ)$. In the core--placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\ge2$, the unique maximizer, up to isomorphism, satisfies $M\cong K_{1,h}\cup(h+1)K_1$, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang--Li--Zhang is not extremal. For every $h\ge1$, the unique minimizer satisfies $M\cong hK_2\cup2K_1$, with the $2h$ exceptional edges having distinct endpoints on the large-clique side. We also settle the cases $h=0,1$ and all equality cases. The proof combines an exact core--placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in $\sum_{x\in V(M)}d_M(x)^2$, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.