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. For each fixed positive activity, we quantify the approach to this bound: the correlation ratio is 3/4 + 3/(4eλn) + o(n⁻¹) as n tends to infinity. The proof uses the classical Poisson law for the number of missing vertices.
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Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory
- Keywords
- Manuscript date
- 2026-10-31
- Reader reuse
- none
Scope
Strict sharp 3/4 lower bound for adjacent-edge conditional-to-marginal inclusion ratios in nonempty labeled-complete-graph subtrees weighted by positive edge activity. Polynomial positivity yields a finite margin; the spanning-tree limit establishes optimality at fixed graph order. For each fixed positive activity λ, the first-order correlation margin is 3/(4eλn) + o(n⁻¹) as n grows, using classical forest counts and the missing-vertex Poisson limit.
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
The 4-page searchable manuscript is accompanied by complete editable sources, reader instructions, citation metadata and integrity checksums. The manuscript supplies the analytic arguments; accompanying computational materials, where included, check their specified finite certificates.