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

Abstract

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 $d(T) := \operatorname{dist}_T(\mathcal{S}(T), \mathcal{C}(T))$ among trees of every order $n\ge3$. Writing $Δ_n:=\max_{|V(T)|=n}d(T)$, we prove $Δ_n=0\quad(3\le n\le11),\quad Δ_{12}=1$, and $Δ_n=\left\lfloor\frac{n-11}{2}\right\rfloor \quad(n\ge13)$. 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 $|V(T)|\ge 2d(T)+11 \quad(d(T)\ge2)$. A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every $n\ge12$.