Abstract
We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.
Highlights
The study of graphs and trees, i.e., objects with pairwise relations between them, has a long and distinguished history throughout most the sciences
Graph theory is in itself an accepted branch of mainstream research and graphs are a central part of the field of discrete mathematics [4]
We numerically study the case of incomplete random trees, which is closest related to the tree tensor networks considered in Ref. [12]
Summary
The study of graphs and trees, i.e., objects (or vertices) with pairwise relations (or edges) between them, has a long and distinguished history throughout most the sciences. Tree-like structures have recently become more prominent in quantum physics of interacting particles with the advent of so-called tensor network methods [11]. In a recent publication [12] we show that certain correlation functions and measures of quantum entanglement can be constructed by a holographic distance and connectivity dependence along a tree network connecting certain leaves [13] In these quantum systems, the leaves are ordered according to their physical position, for example, the location of magnetic ions in a quantum wire. The leaves are ordered according to their physical position, for example, the location of magnetic ions in a quantum wire This ordering imposes a new restriction on the tree itself, and the lengths that become important are leaf-to-leaf distances across the ordered tree. We numerically study the case of incomplete random trees, which is closest related to the tree tensor networks considered in Ref. [12]
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