ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Expander mixing lemma

Example

For the looped complete adjacency-slot graph $A = J_{n}$ , $d = n$ and $α = 0$ , with $e (S, T) = ∣ S ∣ ∣ T ∣$ for all sets. In contrast, for $K_{r, r}$ with $r \geq 2$ one has $μ_{2} = 0$ but $α = 1$ .

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

For any subsets $S, T$ of a finite $d$ -regular adjacency-slot graph on $n \geq 1$ vertices, let $e (S, T) = \sum_{u \in S, v \in T} A_{u v}$ count ordered slots. Then $∣ e (S, T) - \frac{d ∣ S ∣ ∣ T ∣}{n} ∣ \leq α d \sqrt{∣ S ∣ (1 - ∣ S ∣ / n) ∣ T ∣ (1 - ∣ T ∣ / n)} .$ Overlap and loop slots are allowed. (Expander mixing lemma).

Verification

1.1F1algebra

The normalized complete matrix sends every vector to its average constant vector, so it vanishes on the mean-zero space. There is one slot for every ordered pair, giving $e (S, T) = ∣ S ∣ ∣ T ∣$ even for overlapping sets. This is exact equality in mixing, including empty or full sets and $n = 1$ .

2.1step 1.1algebra∎

For $K_{r, r}$ , $M$ averages across the opposite side. Its constant vector has eigenvalue one; the vector $+ 1$ on one side and $- 1$ on the other has eigenvalue minus one. The $2 r - 2$ dimensional space with zero sum on each side has eigenvalue zero. These subspaces span, so $μ_{2} = 0$ for $r \geq 2$ , whereas the absolute nontrivial norm is one. A walk alternates sides, explaining why a positive algebraic gap alone does not give the absolute contraction used in mixing and walk estimates.

Depends on

Expander mixing lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hoory–Linial–Wigderson, Expander Graphs and Their Applications, May 2006 draft; §2.3 spectral properties and §2.4 mixing, pp20–21; finite computations. (standard reference, not scraped)