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Spectrally indistinguishable pseudorandom graphs

Arthur Forey, Javier Fresán, Emmanuel Kowalski, Yuval Wigderson

TL;DR

The paper constructs explicit dense graphs whose spectra, after normalization, converge to the semicircle law, making them spectrally indistinguishable from random graphs while remaining structurally dissimilar (e.g., $K_{2,3}$-free). The authors leverage Cayley sum graphs on finite abelian groups with generating sets given by Kloosterman-type and Birch sums, bounding nontrivial eigenvalues via Weil bounds and proving semicircular limits through Deligne–Katz equidistribution, with an alternative self-contained route via Larsen's framework and Sidon sets. This yields near-Ramanujan graphs with continuous spectral measure, highlighting a separation between subgraph counts and spectral data and contributing to the study of explicit pseudorandom graphs and computational indistinguishability. The results illuminate how rich algebraic structures can produce graphs that are indistinguishable from random in spectrum yet have highly controlled subgraph densities, impacting both combinatorics and complexity theory.

Abstract

We construct explicit families of graphs whose eigenvalues are asymptotically distributed according to Wigner's semicircle law; in other words, that are spectrally indistinguishable from random graphs. However, in other respects they are strikingly dissimilar from random graphs; for example, they are $K_{2,3}$-free graphs with almost the maximum possible edge density.

Spectrally indistinguishable pseudorandom graphs

TL;DR

The paper constructs explicit dense graphs whose spectra, after normalization, converge to the semicircle law, making them spectrally indistinguishable from random graphs while remaining structurally dissimilar (e.g., -free). The authors leverage Cayley sum graphs on finite abelian groups with generating sets given by Kloosterman-type and Birch sums, bounding nontrivial eigenvalues via Weil bounds and proving semicircular limits through Deligne–Katz equidistribution, with an alternative self-contained route via Larsen's framework and Sidon sets. This yields near-Ramanujan graphs with continuous spectral measure, highlighting a separation between subgraph counts and spectral data and contributing to the study of explicit pseudorandom graphs and computational indistinguishability. The results illuminate how rich algebraic structures can produce graphs that are indistinguishable from random in spectrum yet have highly controlled subgraph densities, impacting both combinatorics and complexity theory.

Abstract

We construct explicit families of graphs whose eigenvalues are asymptotically distributed according to Wigner's semicircle law; in other words, that are spectrally indistinguishable from random graphs. However, in other respects they are strikingly dissimilar from random graphs; for example, they are -free graphs with almost the maximum possible edge density.

Paper Structure

This paper contains 8 sections, 11 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discussion and historical background
  3. Why should we care?
  4. History
  5. The independence number
  6. Proof of Theorem \ref{['th-main']}
  7. Larsen's alternative, Sidon sets and equidistribution
  8. Variants

Key Result

Theorem 1.1

Let $k$ be a finite field. SetThe letters $K$ and $B$ stand for "Kloosterman" and "Birch" respectively, for reasons that will be clear in the course of the proof in Section sec:proof. Define graphs $\Gamma_K(k)$ and $\Gamma_B(k)$ with vertex set $k\times k$ in both cases and with edges joining $x$ and $y$ if and only if $x+y\in K(k)$ or $x+y\in B(k)$, respectively. These are regular graphs of degr

Figures (2)

  • Figure 1: The normalized spectrum of the graphs $\Gamma_K(k)$ (first row) and $\Gamma_B(k)$ (second row), for $k = \mathbf{F}_{127}, \mathbf{F}_{251},\mathbf{F}_{601},\mathbf{F}_{1117}$. The dashed red curve is the density function of $\mu_{\mathrm{sc}}$, the semicircle distribution.
  • Figure 2: The spectrum of the graphs $\Gamma_B(\mathbf{F}_q)$ for $q = 625$ and $q=1024$. As predicted by Katz's theorem, the limiting spectrum is a finitely-supported atomic measure because the characteristic is less than $7$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof : Proof of \ref{['prop:symmetric sidon']}
  • Proposition 3.3
  • proof
  • proof : Proof of Theorem \ref{['th-main']}
  • Theorem 4.1
  • ...and 13 more