The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree

Let $\mathcal{S}(T)$ be the adjacency spectral center of a tree $T$, and let $\mathcal{C}(T)$ be its characteristic set. We determine the largest possible separation $d(T):={\mathrm{dist}}_T(\mathcal{S}(T),\mathcal{C}(T))$ among trees of every order $n\ge 3$. Writing ${\mathrm{\Delta }}_n:={\max }_{\mathrm{\mid }V(T)\mathrm{\mid }=n}d(T)$, we prove ${\mathrm{\Delta }}_n=0(3\le n\le 11),{\mathrm{\Delta }}_{12}=1$, and ${\mathrm{\Delta }}_n=\lfloor \frac{n-11}{2}\rfloor (n\ge 13)$. The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate $\mathrm{\mid }V(T)\mathrm{\mid }\ge 2d(T)+11(d(T)\ge 2)$. A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every $n\ge 12$.