Paley Graphs: Algebraic and Spectral Insights

• Paley graphs are finite graphs constructed over finite fields where vertices represent field elements and edges exist if their difference is a quadratic residue.
• They are strongly regular and quasi-random, with explicit spectral properties that bridge arithmetic and combinatorial graph theory.
• Their unique structure supports applications in coding theory, extremal combinatorics, and enables advanced SDP relaxations for bounding clique numbers.

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 (Elsawy, 2012, Jones, 2017).
• $P(q)$ 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 (Elsawy, 2012, Kim et al., 2024).

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 $(q-1)/2$ (Mináč et al., 2022, Jones, 2017, Kim et al., 2024).

• $$|\lambda_2| = |\lambda_3| = \frac{1}{2}\sqrt{q} + O(1) = o(q)$$.

By Chung–Graham–Wilson (Kim et al., 2024), 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 $$A\Delta L_1(q) \cong (\mathbb{F}_q^*)^2 \ltimes \mathbb{F}_q \rtimes \mathrm{Gal}(\mathbb{F}_q/\mathbb{F}_p)$$ (Jones, 2017). 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 (Kim et al., 2024).

SDP relaxations: The clique number satisfies the classical upper bound $\omega(P(q))\leq\sqrt{q}$ (Kobzar et al., 2023), with recent computational evidence (block-diagonal SDP relaxations, such as $L^2$ and SOS-4) indicating actual growth may be sub-$\sqrt{q}$: numerically, $L^2(P(q)) = O(q^{0.456})$ (Kobzar et al., 2023).

• 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 (Gaar et al., 2024).

Table: Numerical Comparison for Small $q$ (Gaar et al., 2024) | $q$ | $\alpha(P_q)$ | $\vartheta(P_q)$ | $z_2(P_q)$ | $z'_2(P_q)$ | |-----|--------------|------------------|------------|-------------| | 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 $3\leq k\leq q$ (Nishimura, 2023).
• 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 (Dawsey et al., 2020).

• 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 (Li et al., 22 Dec 2025).

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 $k>2$ (Elsawy, 2012, Schneider et al., 2013, Bonini et al., 2024).

• Automorphism group: For large $q$ relative to $k$, $$\mathrm{Aut}(\mathrm{GP}(q,(q-1)/k)) \leq \mathrm{A}\Gamma\mathrm{L}(1,q)$$ (Ponomarenko, 23 Nov 2025).
• 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 $q=2^k$ (Thomason, 2015).
• 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 $n$ (Bhowmik et al., 2020).

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 $\Omega(\log p\log\log\log p)$ (Mrazović, 2016).
• 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 (Li et al., 22 Dec 2025).
• 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 (Chandler et al., 2014).

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 (Jones, 2019).
• 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 (Mináč et al., 2022, Podestá et al., 2022).

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References:

• (Kim et al., 2024): "Paley-like quasi-random graphs arising from polynomials"
• (Mináč et al., 2022): "On the Paley graph of a quadratic character"
• (Nishimura, 2023): "A new approach to pancyclicity of Paley graphs I"
• (Jones, 2017): "Paley and the Paley graphs"
• (Elsawy, 2012): "Paley Graphs and Their Generalizations"
• (Chandler et al., 2014): "The Smith and critical groups of Paley graphs"
• (Schneider et al., 2013): "Cliques and colorings in generalized Paley graphs and an approach to synchronization"
• (Bonini et al., 2024): "Condensed Ricci Curvature on Paley Graphs and their Generalizations"
• (Thomason, 2015): "A Paley-like graph in characteristic two"
• (Dawsey et al., 2020): "Generalized Paley graphs and their complete subgraphs of orders three and four"
• (Li et al., 22 Dec 2025): "On induced subgraphs with degree parity conditions in Paley graphs and Paley tournaments"
• (Bhowmik et al., 2020): "On a Paley-type graph on $\mathbb{Z}_n$"
• (Kobzar et al., 2023): "Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs"
• (Gaar et al., 2024): "The exact subgraph hierarchy and its local variant for the stable set problem for Paley graphs"
• (Ponomarenko, 23 Nov 2025): "The automorphism groups and identification of some Generalized Paley Graphs"
• (Podestá et al., 2022): "Generalized Paley graphs equienergetic with their complements"
• (Mrazović, 2016): "A random model for the Paley graph"
• (Jones, 2019): "Infinite Paley graphs"