Alphabeta MathHow statement and proof provenance work
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The mean-square density, or energy, of a vertex partition
Definition
Let have order . For vertex sets and partitions of , of into nonempty sets, define their pair energy by For a vertex partition of (-regular vertex partitions, equitable partitions, and refinement), its energy, or mean-square density, is using the ordered-pair density of Edge counts and densities between nonempty vertex sets. The empty partition of the null graph has energy .
Depends on
Used by
- A prescribed finite vertex partition has a bounded ε-regular refinement, equitable when the initial partition is equitable Corollary
- The trivial partition has energy d(G)², while the singleton partition records every adjacency Example
- A small energy increment makes fine-pair densities close to their coarse densities almost everywhere Lemma
- An irregularity witness raises the pair energy by more than ε⁴|X||Y|/n² Lemma
- Energy lies in [0,1] and cannot decrease under refinement Lemma
- Szemerédi regularity lemma with an equitable partition and an explicit tower-type upper bound for graphs of order at least m₀ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 7 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Y. Zhao, Graph Theory and Additive Combinatorics, Definition 2.1.10 (standard reference, not scraped)
- D. Conlon and J. Fox, Graph removal lemmas, sec. 2.1 (standard reference, not scraped)