2026/07/17 by Akash De, Kamal Lochan Patra
#math.CO
Let T be a tree. For a vertex v∈ V(T), the eccentric subtree number εT(v) is defined as εT(v)=min\fT(v,u): u∈ V(T)\ where fT(v,u) denotes the number of subtrees of T containing both v and u. A core vertex of T is a vertex with the maximum eccentric subtree number, and the set of all the core vertices of T is called the core center of T. The core center of T consists of either a single vertex or two adjacent vertices. There are other central concepts in a tree, such as the center, centroid, subtree core, and characteristic center, and these may all be different. By dT(C, \mathfrakC), dT(Cd, \mathfrakC) and dT(Sc, \mathfrakC) we mean the distance between center and core center, distance between centroid and core center and distance between subtree core and core center in T, respectively. We show that for any tree T on n≥ 6 vertices, (i)]dT(C,\mathfrakC)≤ \lfloor (n-g0-4)/(2) \rfloor; (ii)]dT(Cd,\mathfrakC)≤ \lfloor (n-5)/(2) \rfloor; (iii)] dT(Sc,\mathfrakC)≤\ 1, · if n=7, n-g0-3, · if n≠ 7; . where g0≥ 2 be the smallest positive integer such that 2g0-1+g0≥ n-3. Moreover, we show that these bounds are best possible by obtaining a tree which attains these bounds. We also obtain a tree which maximizes the distance between characteristic center and core center over all trees on n≥ 6 vertices. The asymptotic behaviour of all these distances are also studied.