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

For a connected graph $G$, let $\tau (G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $\rho (G)$ be its adjacency spectral radius. For integers $n\ge 1$, $k\ge 2$, and $k\le \delta \le 2k-1$, let ${\mathcal{G}}_{n,\delta }$ be the class of connected $n$-vertex graphs with minimum degree $\delta$, and let ${\mathcal{L}}_{\mathcal{H}}^2(n,k,\delta )\subseteq {\mathcal{G}}_{n,\delta }$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $\tau (G)\ge k$ is determined by maximizing $\rho (G)$ over this family. Set $h=\delta -k$. For each fixed admissible pair $(k,\delta )$ and all sufficiently large $n$, we determine the maximum and minimum adjacency spectral radii in ${\mathcal{L}}_{\mathcal{H}}^2(n,k,\delta )$. In the core–placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\ge 2$, 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 $h\ge 1$, the unique minimizer satisfies $M\cong h{K}_2\cup 2K_1$, 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.