A sharp lower bound for adjacent-edge correlations in random subtrees
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Abstract
Give every nonempty subtree of the labeled complete graph Kₙ weight λ^|E(T)|, where λ > 0 and singleton subtrees are included. For every n ≥ 3, the conditional probability of an edge given an adjacent edge is strictly greater than three quarters of its unconditional probability. Thus the finite lower bound proposed by Luo, Xu and Tian for the uniform law holds throughout the positive activity family. Classical fixed-forest extension counts reduce the assertion to positivity of a polynomial in λ. Convexity makes every paired contribution nonnegative except one; the coefficient of λ² explicitly compensates for that exception. The constant is optimal even for each fixed n, since the law converges to the uniform spanning-tree law as λ tends to infinity. We also give a direct count-based derivation of the previously known uniform-law limit as n tends to infinity.
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Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory
- Keywords
- Manuscript date
- 2026-10-31
- Reader reuse
- none
Scope
A strict sharp lower bound of 3/4 for adjacent-edge conditional-to-marginal inclusion ratios in nonempty labeled-complete-graph subtrees weighted by λ^|E(T)|, for every n ≥ 3 and λ > 0, including singleton trees. Polynomial coefficient positivity gives the finite inequality and an explicit margin; the spanning-tree limit proves optimality for each fixed n, and the count-based uniform-law limit is derived with attribution.
Disclosures
No external funding was received for this work. OpenAI tools assisted literature comparison, proof exploration, drafting and consistency checks. The author takes responsibility for the statements, attribution and presentation.
Reproduction
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