Sharp Order Laws and Finite-Defect Stability for Domination versus Small Laplacian Eigenvalues in Trees
Abstract
For a tree T, let gamma(T) be its domination number and let mu(T) count the Laplacian eigenvalues in [0,1). The strict inequality gamma(T)/mu(T)<4/3 leaves open three linked questions: the exact finite-order defect, the structure near equality, and whether a fixed defect permits only finitely many local obstructions. We answer all three. For every n>=2, max_{|V(T)|=n} (7 gamma(T)-9 mu(T)) = floor((n-20)/9). Thus order twenty-nine is the first at which the ratio 9/7 can be exceeded. The proof is based on an exact decomposition of Phi(T)=|V(T)|-20-9(7 gamma(T)-9 mu(T)) into nonnegative integer terms, together with a clean-contraction identity. For subcubic trees, equality cases are canonical lifts of trees with perfect matchings. We classify every layer through Phi=42, locating successive phase transitions at 20, 21, 40, 41, and 42. The finiteness mechanism is simple to state: after the repeatable matched-skeleton pieces are removed, fixed slack leaves only a bounded exceptional core. Formally, a local compactness theorem bounds every nonordinary weighted deep component at fixed packing surplus and residual negative index. Consequently, for each fixed k, every subcubic tree with Phi(T)=k consists of a bounded ported defect kernel, an arbitrary compatible forest of ordinary tiles, and at most floor(k/21) clean expansions. All unbounded statements are proved symbolically; exact computation is used only for explicitly bounded atom lists and independent finite verification.
Related
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BibTeX
@misc{wang2026sharp,
title = {{Sharp Order Laws and Finite-Defect Stability for Domination versus Small Laplacian Eigenvalues in Trees}},
author = {Yufeng Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27602},
eprint = {2608.27602},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27602},
}
RIS
TY - UNPB
TI - Sharp Order Laws and Finite-Defect Stability for Domination versus Small Laplacian Eigenvalues in Trees
AU - Yufeng Wang
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27602
UR - https://arxiv.org/abs/2608.27602
AB - For a tree T, let gamma(T) be its domination number and let mu(T) count the Laplacian eigenvalues in [0,1). The strict inequality gamma(T)/mu(T)<4/3 leaves open three linked questions: the exact finite-order defect, the structure near equality, and whether a fixed defect permits only finitely many local obstructions. We answer all three. For every n>=2, max_{|V(T)|=n} (7 gamma(T)-9 mu(T)) = floor((n-20)/9). Thus order twenty-nine is the first at which the ratio 9/7 can be exceeded. The proof is based on an exact decomposition of Phi(T)=|V(T)|-20-9(7 gamma(T)-9 mu(T)) into nonnegative integer terms, together with a clean-contraction identity. For subcubic trees, equality cases are canonical lifts of trees with perfect matchings. We classify every layer through Phi=42, locating successive phase transitions at 20, 21, 40, 41, and 42. The finiteness mechanism is simple to state: after the repeatable matched-skeleton pieces are removed, fixed slack leaves only a bounded exceptional core. Formally, a local compactness theorem bounds every nonordinary weighted deep component at fixed packing surplus and residual negative index. Consequently, for each fixed k, every subcubic tree with Phi(T)=k consists of a bounded ported defect kernel, an arbitrary compatible forest of ordinary tiles, and at most floor(k/21) clean expansions. All unbounded statements are proved symbolically; exact computation is used only for explicitly bounded atom lists and independent finite verification.
ER -