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Paley-like quasi-random graphs arising from polynomials

Seoyoung Kim, Chi Hoi Yip, Semin Yoo

TL;DR

The paper constructs Paley-like graphs X_{f,q} from admissible polynomials over finite fields and proves they form quasi-random families, even when not Cayley-related. It develops an admissibility framework, uses Weil-type character sums to bound clique numbers, and establishes a robust expander-like mixing lemma for QR(θ) graphs. By combining these quasi-random properties with modern Ramsey bounds, the authors derive nontrivial lower bounds on clique and independence numbers for X_{f,q} and its derived families. The results unify Paley graphs, Paley sum graphs, and Diophantine-graph analogues within a polynomial-based construction, and demonstrate QR(3/4) or QR(1/2) for key families, including explicit lower-bounds on ω(X_{f,q}) in terms of q. This advances the understanding of deterministic graphs with random-like edge distribution and provides concrete lower bounds via Ramsey theory.

Abstract

Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs arising from polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs arising from Diophantine tuples and their generalizations. We also obtain lower bounds on the clique and independence numbers of the graphs in these families.

Paley-like quasi-random graphs arising from polynomials

TL;DR

The paper constructs Paley-like graphs X_{f,q} from admissible polynomials over finite fields and proves they form quasi-random families, even when not Cayley-related. It develops an admissibility framework, uses Weil-type character sums to bound clique numbers, and establishes a robust expander-like mixing lemma for QR(θ) graphs. By combining these quasi-random properties with modern Ramsey bounds, the authors derive nontrivial lower bounds on clique and independence numbers for X_{f,q} and its derived families. The results unify Paley graphs, Paley sum graphs, and Diophantine-graph analogues within a polynomial-based construction, and demonstrate QR(3/4) or QR(1/2) for key families, including explicit lower-bounds on ω(X_{f,q}) in terms of q. This advances the understanding of deterministic graphs with random-like edge distribution and provides concrete lower bounds via Ramsey theory.

Abstract

Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs arising from polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs arising from Diophantine tuples and their generalizations. We also obtain lower bounds on the clique and independence numbers of the graphs in these families.
Paper Structure (17 sections, 25 theorems, 82 equations)

This paper contains 17 sections, 25 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. More backgrounds and motivations
  3. Known explicit constructions of quasi-random graphs
  4. Clique number of Paley graphs
  5. More on Diophantine tuples and Diophantine graphs
  6. Preliminaries
  7. A necessary condition for $X_{f,q}$ to be Paley-like
  8. Basic properties of admissible polynomials
  9. Upper bound on the clique number of $X_{f,q}$
  10. quasi-random graphs
  11. Equivalent definitions of quasi-random graphs
  12. No family of quasi-random graphs has property $QR(\theta)$ with $\theta<1/2$
  13. An expander mixing lemma for quasi-random graphs
  14. Clique number and independence number of quasi-random graphs
  15. Quasi-randomness and clique number of $X_{f,q}$
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $d \geq 1$. Let $\mathcal{X}_d$ be the family consisting of all graphs $X_{f,q}$, where $q$ is an odd prime power and $f \in {\mathbb F}_q[x,y]$ is an admissible polynomial with degree $d$. Then $\mathcal{X}_d$ is a family of quasi-random graphs with property $QR(3/4)$.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Corollary 2.2
  • Corollary 2.3
  • ...and 45 more