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Research Article · Preprint Mathematics

A sharp lower bound for adjacent-edge correlations in random subtrees

Scott Schubr

  • Mathematics
  • Combinatorics and Graph Theory
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

Choose uniformly a nonempty subtree of the labeled complete graph, including the single-vertex subtrees. We prove that the conditional probability of one edge, given an adjacent edge, exceeds three quarters of its unconditional probability. This establishes the lower-bound conjecture stated by Luo, Xu and Tian in their author manuscript. The proof uses the classical fixed-forest extension counts, Jensen's inequality, and a count of four-vertex trees. The constant is optimal by the asymptotic limit already obtained by Luo, Xu and Tian.

Keywords random subtrees · adjacent-edge correlations · complete graphs · Cayley’s formula · Jensen’s inequality

Scope of the work

For a uniform nonempty subtree of the labelled complete graph and two distinct adjacent edges, the conditional inclusion probability exceeds three quarters of the unconditional one; the constant is asymptotically sharp.

Reproduction notes

The paper has 3 pages. The editable-source archive contains 5 files. Compile manuscript/main.tex. The standard-library verification/independent_check.py reproduces the finite rational and small-graph controls in a fresh output directory.

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Cite this edition

Scott Schubr. (2026). A sharp lower bound for adjacent-edge correlations in random subtrees (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-lower-bound-adjacent-edge-correlations-random-subtrees/v/2

BibTeX
@techreport{luminary_sharp_lower_bound_adjacent_edge_correlations_random_subtrees_v2,
  title = {A sharp lower bound for adjacent-edge correlations in random subtrees},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/sharp-lower-bound-adjacent-edge-correlations-random-subtrees/v/2}
}

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