---
title: 'Paley Graphs: Algebraic and Spectral Insights'
url: https://www.emergentmind.com/topics/paley-graphs
type: topic
---

# Paley Graphs: Algebraic and Spectral Insights

A Paley graph is a fundamental object in algebraic combinatorics, defined over a finite field of odd characteristic, in which adjacency encodes quadratic residue structure. These graphs integrate arithmetic, algebraic, and spectral properties, serving as archetypes for strongly regular and quasi-random graphs, and they catalyze connections to coding theory, extremal combinatorics, spectral graph theory, and arithmetic Ramsey theory.

## 1. Definition and Construction

Let $q=p^e$ be a prime power with $q\equiv1\pmod4$, and let $\mathbb{F}_q$ denote the finite field of order $q$. Define the subgroup of nonzero quadratic residues,
$$
Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.
$$
The **Paley graph** $P(q)$ has vertex set $V(P(q)) = \mathbb{F}_q$, with two distinct vertices $x, y$ adjacent if and only if $x-y\in Q$. The adjacency relation is symmetric, because $-1$ is a square in $\mathbb{F}_q$ when $q\equiv1\pmod4$.

- $P(q)$ is an undirected, loopless, $(q-1)/2$-regular graph.
- $P(q)$ is self-complementary, as for any quadratic nonresidue $r\in\mathbb{F}_q^*$, the map $x\mapsto r x$ is an automorphism exchanging edges and non-edges [1203.1818], [1702.00285].
- $P(q)$ is a strongly regular graph with parameters:
  $$
  v=q, \qquad k=\frac{q-1}{2},\qquad \lambda=\frac{q-5}{4},\qquad \mu=\frac{q-1}{4}
  $$
  where $\lambda$ and $\mu$ count common neighbors for adjacent and non-adjacent pairs, respectively [1203.1818], [2405.09319].

## 2. Spectral and Quasi-Random Properties

Let $A$ be the adjacency matrix of $P(q)$. The spectrum is explicit:
- **Eigenvalues:**
  $$
  \lambda_1 = \frac{q-1}{2} \ \text{(simple)},
  \qquad
  \lambda_2 = \frac{-1+\sqrt{q}}{2},\ \lambda_3 = \frac{-1-\sqrt{q}}{2}
  $$
  both with multiplicity $(q-1)/2$ [2212.02005], [1702.00285], [2405.09319].
- $|\lambda_2| = |\lambda_3| = \frac{1}{2}\sqrt{q} + O(1) = o(q)$.

By Chung–Graham–Wilson [2405.09319], a sequence of graphs with edge-density $1/2$ is *quasi-random* if the edge distribution in each induced subgraph matches that of a random (density-$1/2$) graph to within $o(n^2)$. The expander-mixing lemma shows that Paley graphs $P(q)$ meet these criteria:
$$
e(S) = \frac{1}{4}|S|^2 + o(q^2)
$$
for all $S\subset V(P(q))$, and the second-largest eigenvalue is $o(q)$, so $P(q)$ is a standard example of a quasi-random graph.

## 3. Automorphism Group and Symmetries

The full automorphism group of $P(q)$ is
$$
\mathrm{Aut}(P(q)) = \{\, x\mapsto a x^\gamma + b : a\in(\mathbb{F}_q^*)^2,\, b\in\mathbb{F}_q,\, \gamma\in\mathrm{Gal}(\mathbb{F}_q/\mathbb{F}_p)\,\},
$$
a semidirect product $A\Delta L_1(q) \cong (\mathbb{F}_q^*)^2 \ltimes \mathbb{F}_q \rtimes \mathrm{Gal}(\mathbb{F}_q/\mathbb{F}_p)$ [1702.00285]. The action is vertex- and edge-transitive; every affine map with square multiplier is a graph automorphism.

$P(q)$ is self-complementary via multiplication by any quadratic nonresidue.

## 4. Extremal Subgraph Structure and SDP Bounds

**Clique and independence numbers**: By classic Fourier methods and subsequent quasi-random analysis, for $n=|V(P(q))|$,
$$
\omega(P(q)),\ \alpha(P(q)) \geq (1-o(1))\log_{3.008} q
$$
where $\omega$ is the clique number and $\alpha$ the independence number [2405.09319].

**SDP relaxations**: The clique number satisfies the classical upper bound $\omega(P(q))\leq\sqrt{q}$ [2304.08615], with recent computational evidence (block-diagonal SDP relaxations, such as $L^2$ and SOS-4) indicating actual growth may be sub-$\sqrt{q}$: numerically, $L^2(P(q)) = O(q^{0.456})$ [2304.08615].

- The Lovász $\vartheta$ function equals $\sqrt{q}$ and coincides with the value at the first level of the exact subgraph hierarchy (ESH).
- The ESH remains at the Lovász bound up to level $k_0\sim(\sqrt{q}+3)/2$; the local ESH, exploiting vertex-transitivity, gives strictly improved upper bounds already at low levels and is at least as tight as ESH [2412.12958].

Table: Numerical Comparison for Small $q$ ([2412.12958])
| $q$ | $\alpha(P_q)$ | $\vartheta(P_q)$ | $z_2(P_q)$ | $z'_2(P_q)$ |
|-----|--------------|------------------|------------|-------------|
| 13  | 3            | 3.6056           | 3.6056     | 3.0000      |
| 29  | 4            | 5.3852           | 5.3852     | 4.3177      |

## 5. Extremal and Combinatorial Properties

- **Pancyclicity**: For $q\equiv 1\pmod4$, $q\ne5$, $P(q)$ is pancyclic: it contains cycles of every possible length $3\leq k\leq q$ [2308.04759].
- **Subgraph enumeration**: For the number of triangles and 4-cliques, explicit formulas in terms of Jacobi sums are known. For $q\equiv 1\pmod4$,
  $$
  \mathcal{K}_3(P(q)) = \frac{q(q-1)(q-5)}{24\cdot3},\quad
  \mathcal{K}_4(P(q)) = \frac{q(q-1)((q-9)^2-4y^2)}{2^9\cdot3}
  $$
  where $q=x^2+y^2$ with $x\equiv 1\pmod 4$, $y$ even [2006.14716].

- **Even induced subgraphs**: The number of even induced subgraphs of Paley graphs matches that in random models for small sizes; the parity structure corresponds to the enumeration of MDS self-dual codes [2512.19312].

## 6. Generalizations of Paley Graphs

The Paley construction motivates several generalizations:
- **Generalized Paley graphs**: For $k$ dividing $q-1$ (and $q\equiv 1\bmod 2k$ when $q$ odd), define adjacency via $k$-th power residues:
  $$
  V = \mathbb{F}_q,\ \ \{x,y\} \in E \iff x-y \in (\mathbb{F}_q^*)^k.
  $$
  Regular of degree $(q-1)/k$, often not strongly regular for $k>2$ [1203.1818], [1306.6050], [2409.03631].
- **Automorphism group**: For large $q$ relative to $k$, $\mathrm{Aut}(\mathrm{GP}(q,(q-1)/k)) \leq \mathrm{A}\Gamma\mathrm{L}(1,q)$ [2511.18304].

- **Paley graphs in characteristic $2$**: A distinct construction exists, using the trace map and Möbius transformations, resulting in a self-complementary, vertex-transitive pseudo-random graph on $q+1$ points for $q=2^k$ [1509.05198].

- **Paley graphs over $\mathbb{Z}_n$**: With $n$ restricted to ensure $-1$ is a square in the unit group, a version exists for rings, where the underlying group is $\mathbb{Z}_n$ and adjacency is defined by units that are squares modulo $n$ [2012.09735].

## 7. Applications and Structural Invariants

- **Random models**: The multiplicative random-graph model more faithfully captures the clique-number fluctuations of Paley graphs than purely random Cayley graphs, matching the Graham–Ringrose phenomenon for cliques of size $\Omega(\log p\log\log\log p)$ [1603.00684].
- **Coding theory**: There is a tight connection between even/odd subgraph structure in Paley graphs and the existence and enumeration of MDS self-dual (extended) GRS codes [2512.19312].
- **Critical group / Smith normal form**: The critical (sandpile) group and Smith group of the adjacency matrix for $P(q)$ are described explicitly in terms of the field order; the primary decomposition involves detailed number-theoretic and character-sum data [1401.8260].

## 8. Infinite and Arithmetic Variants

- **Infinite Paley graphs**: The direct limits of Paley graphs over towers of extensions yield, up to isomorphism, the universal Erdős–Rényi–Rado random graph $R$ for any (locally finite, infinite) field of odd characteristic. This is established via the extension property and Weil’s character sum estimates [1912.02709].
- **Ramanujan and energy properties**: The spectrum of Paley and generalized Paley graphs provides explicit classes of Ramanujan graphs and infinite families of non-strongly regular, non-bipartite graphs which are equienergetic with their complements [2212.02005], [2204.08509].

---

References:
- [2405.09319]: "Paley-like quasi-random graphs arising from polynomials"
- [2212.02005]: "On the Paley graph of a quadratic character"
- [2308.04759]: "A new approach to pancyclicity of Paley graphs I"
- [1702.00285]: "Paley and the Paley graphs"
- [1203.1818]: "Paley Graphs and Their Generalizations"
- [1401.8260]: "The Smith and critical groups of Paley graphs"
- [1306.6050]: "Cliques and colorings in generalized Paley graphs and an approach to synchronization"
- [2409.03631]: "Condensed Ricci Curvature on Paley Graphs and their Generalizations"
- [1509.05198]: "A Paley-like graph in characteristic two"
- [2006.14716]: "Generalized Paley graphs and their complete subgraphs of orders three and four"
- [2512.19312]: "On induced subgraphs with degree parity conditions in Paley graphs and Paley tournaments"
- [2012.09735]: "On a Paley-type graph on $\mathbb{Z}_n$"
- [2304.08615]: "Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs"
- [2412.12958]: "The exact subgraph hierarchy and its local variant for the stable set problem for Paley graphs"
- [2511.18304]: "The automorphism groups and identification of some Generalized Paley Graphs"
- [2204.08509]: "Generalized Paley graphs equienergetic with their complements"
- [1603.00684]: "A random model for the Paley graph"
- [1912.02709]: "Infinite Paley graphs"

Source: https://www.emergentmind.com/topics/paley-graphs