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Numerical and structural limits of the regular-graph method

PDF p. 29: Section 5 opening; PDF p. 30: Section 5 closing; PDF p. 39: Section 7, first paragraph

Statement

For a finite $d$ -regular Ramanujan graph in the article's convention, $d$ is a positive integer and its admissible adjacency-expansion parameter is $a (d) = 2 \sqrt{d - 1} / d$ (Ramanujan graph convention used in the article). A phase configuration assigns an angle modulo $2 π$ to each vertex; its order parameters are $ρ_{k} = n^{- 1} \sum_{x} e^{i k θ_{x}}$ for integers $k \geq 1$ (Daido order parameters). Global synchronization is the property specified in Globally synchronizing graph — Definition 1.1.

The article reports that its order-parameter and amplification method has a barrier at $α \leq 0.24585$ , because the authors lack an analogous order-parameter result beyond that range even with their regular-graph adaptations. They associate this threshold with Ramanujan degrees $d \geq 66$ and say that reaching $d \geq 3$ would require $α \leq 0.943$ , approximately. They report stopping their numerical improvements at $α = 0.0816$ , associated with $d \geq 600$ , and consider it unlikely that those computations would reach beyond $α \leq 0.1$ , associated with $d \geq 400$ . In Section 7 the authors say that they extensively used Lemmas 2.1 and 2.3 to start their amplification argument, and that these lemmas give no useful bounds unless $d \geq 66$ , limiting what can be achieved without new ideas.

These are attributed descriptions of the authors' method and expectations, not assertions that global synchronization is impossible beyond those parameter ranges. The degree conversions for integer $d \geq 3$ are verified below; this verification does not certify the numerical amplification or establish a universal barrier for other methods.

Facts & Assumptions

Given: An integer $d \geq 3$ and the parameter $a (d) = 2 \sqrt{d - 1} / d$ .

Goal: Verify the degree conversions stated above and the approximation at $d = 3$ .

[F1]

The article's Ramanujan convention uses the parameter $a (d) = 2 \sqrt{d - 1} / d$ for positive integer degree (Ramanujan graph convention used in the article).

Proof

1.1F1givenalgebra

For real $t > 2$ , extend the same formula by $a (t) = 2 \sqrt{t - 1} / t > 0$ . If $2 < s < t$ , direct subtraction gives $a (s)^{2} - a (t)^{2} = 4 (t - s) (s t - s - t) / (s^{2} t^{2}) > 0$ , because $s t - s - t = (s - 1) (t - 1) - 1 > 0$ . Dividing this difference by $a (s) + a (t) > 0$ gives $a (s) - a (t) > 0$ , so $a (t)$ is strictly decreasing on $t > 2$ . Direct rational comparisons give $4 \cdot 64 / 65^{2} > {0.24585}^{2} > 4 \cdot 65 / 66^{2}$ , $4 \cdot 598 / 599^{2} > {0.0816}^{2} > 4 \cdot 599 / 600^{2}$ , and $4 \cdot 397 / 398^{2} > {0.1}^{2} > 4 \cdot 398 / 399^{2}$ . Consequently, among integers $d \geq 3$ , the respective inequalities $a (d) \leq 0.24585$ , $a (d) \leq 0.0816$ , and $a (d) \leq 0.1$ first hold at $d = 66$ , $600$ , and $399$ . In particular, the article's association of $α \leq 0.1$ with $d \geq 400$ is sufficient, but $400$ is not the smallest admissible integer degree for this parameter inequality.

2.1F1step 1.1algebra∎

At $d = 3$ , $a (3) = 2 \sqrt{2} / 3 = 0.942809 \dots$ , which rounds to $0.943$ to three decimal places. This establishes the stated degree conversions and approximation; it makes no claim that the authors' method works at those endpoints.

Source notes

The statements about the authors' barrier, stopping point and expectations are attributed to Expander graphs are globally synchronizing, PDF pages 29, 30 and 39. They are not used as premises for the calculation above.

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Sources

Expander graphs are globally synchronizing · PDF p. 29, Section 5 opening; p. 30, Section 5 closing; p. 39, Section 7 first paragraph