Iterated Joining of Rooted Trees

Abstract

For a given pair of trees T ₁, T ₂, two vertices ${v_1\in T_1}$ and ${v_2\in T_2}$ are said to be path-congruent if, for any integer k ≥ 1, the number p _k (v ₁) of paths contained in T ₁, of length k and passing through v ₁, equals the number p _k (v ₂) of paths contained in T ₂, of length k and passing through v ₂. We first provide polynomial constructions, and related examples, of pairs of non-isomorphic rooted trees ${T_{v_1}, T_{v_2}}$ with path-congruent roots v ₁, v ₂. Then we employ a joining operation between ${T_{v_1}, T_{v_2}}$ to get a tree J ₂ where v ₁, v ₂ do not necessarily belong to a maximal path. For any integer number m, the joining can be made such that the set {v ₁, v ₂} has distance m from the center Z(J ₂) of J ₂. By iterating the idea, an s-fold joining J _s can be considered, where the roots v ₁, . . . , v _s , s ≥ 2, are consecutive vertices of J _s . For s = 3 we give an explicit general construction where ${\{v_1, v_2, v_3\} \cap Z(J_3)=\emptyset}$. On the other hand we prove that ${\{v_1,v_2,\ldots,v_s\} \cap Z(J_s)\neq\emptyset}$ for all s > 2, if ${T_{v_1}}$ and ${T_{v_s}}$ are isomorphic.