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PreprintCC BY 4.0arXiv

A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds

  • Zehua Liu
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Abstract

We develop a graph-theoretic approach to the expected-norm matrix arithmetic--geometric mean inequality. By encoding matrix interactions with a path-supported Gram matrix, we obtain a projection family whose product is nonzero exactly when the transition graph of its index sequence is a path. This gives an intuitive explanation for failure: repeated indices can create cycles or branches that annihilate products, an effect absent for positive scalars. We prove the universal lower bound (n)m/nm for the ratio of the with-replacement to the without-replacement mean and show that it is asymptotically sharp on the exponential scale when m=n. Specifically, our construction gives a ratio exp(−n+O(√n)), matching the lower bound's leading exponential rate over unrestricted dimensions. We also determine an explicit overlap threshold, derive exact six-factor counterexamples, and show that exponential failure can coexist with vanishing pairwise commutators.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026graphtheoretic,
  title = {{A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds}},
  author = {Zehua Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07649},
  eprint = {2611.07649},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07649},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds
AU  - Zehua Liu
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07649
UR  - https://arxiv.org/abs/2611.07649
AB  - We develop a graph-theoretic approach to the expected-norm matrix arithmetic--geometric mean inequality. By encoding matrix interactions with a path-supported Gram matrix, we obtain a projection family whose product is nonzero exactly when the transition graph of its index sequence is a path. This gives an intuitive explanation for failure: repeated indices can create cycles or branches that annihilate products, an effect absent for positive scalars. We prove the universal lower bound (n)_m/n^m for the ratio of the with-replacement to the without-replacement mean and show that it is asymptotically sharp on the exponential scale when m=n. Specifically, our construction gives a ratio exp(−n+O(√n)), matching the lower bound's leading exponential rate over unrestricted dimensions. We also determine an explicit overlap threshold, derive exact six-factor counterexamples, and show that exponential failure can coexist with vanishing pairwise commutators.
ER  -