Preprint Mathematics
A sharp lower bound for adjacent-edge correlations in random subtrees
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Overview
Abstract
Choose uniformly a nonempty tree subgraph of the labeled complete graph, including single-vertex trees. For every n ≥ 3 and two distinct adjacent edges, we prove that the conditional probability of one edge, given the other, exceeds three quarters of its unconditional probability. The short proof combines classical fixed-forest extension counts, Jensen’s inequality and a count of four-vertex trees. The constant is sharp by the previously established asymptotic limit.
Keywords random subtrees · adjacent-edge correlations · complete graphs · Cayley’s formula · Jensen’s inequality
Scope of the work
Uniform nonempty tree subgraphs of labeled complete graphs, with singleton trees included. Subtrees need not be induced or spanning. The result makes no assertion for arbitrary host graphs, weighted models or monotonicity of the correlation ratio.
Reproduction notes
The complete package includes the three-page paper, LaTeX source, exact-arithmetic checker, review records and checksum manifest. Reproduction instructions are in README.txt. Finite exact checks supplement the analytic proof for all n.
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Scott Schubr. (2026). A sharp lower bound for adjacent-edge correlations in random subtrees (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/sharp-lower-bound-adjacent-edge-correlations-random-subtrees/v/1
BibTeX
@techreport{luminary_sharp_lower_bound_adjacent_edge_correlations_random_subtrees_v1,
title = {A sharp lower bound for adjacent-edge correlations in random subtrees},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Preprint},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/sharp-lower-bound-adjacent-edge-correlations-random-subtrees/v/1}
}Reader correspondence
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