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 $q\equiv1\pmod4$0 when $q\equiv1\pmod4$1.

• $q\equiv1\pmod4$2 is an undirected, loopless, $q\equiv1\pmod4$3-regular graph.
• $q\equiv1\pmod4$4 is self-complementary, as for any quadratic nonresidue $q\equiv1\pmod4$5, the map $q\equiv1\pmod4$6 is an automorphism exchanging edges and non-edges (Elsawy, 2012, Jones, 2017).
• $q\equiv1\pmod4$7 is a strongly regular graph with parameters:

$q\equiv1\pmod4$8

where $q\equiv1\pmod4$9 and $\mathbb{F}_q$0 count common neighbors for adjacent and non-adjacent pairs, respectively (Elsawy, 2012, Kim et al., 2024).

2. Spectral and Quasi-Random Properties

Let $\mathbb{F}_q$1 be the adjacency matrix of $\mathbb{F}_q$2. The spectrum is explicit:

• Eigenvalues:

$\mathbb{F}_q$3

both with multiplicity $\mathbb{F}_q$4 (Mináč et al., 2022, Jones, 2017, Kim et al., 2024).

• $\mathbb{F}_q$5.

By Chung–Graham–Wilson (Kim et al., 2024), a sequence of graphs with edge-density $\mathbb{F}_q$6 is quasi-random if the edge distribution in each induced subgraph matches that of a random (density-$\mathbb{F}_q$7) graph to within $\mathbb{F}_q$8. The expander-mixing lemma shows that Paley graphs $\mathbb{F}_q$9 meet these criteria:

$q$0

for all $q$1, and the second-largest eigenvalue is $q$2, so $q$3 is a standard example of a quasi-random graph.

3. Automorphism Group and Symmetries

The full automorphism group of $q$4 is

$q$5

a semidirect product $q$6 (Jones, 2017). The action is vertex- and edge-transitive; every affine map with square multiplier is a graph automorphism.

$q$7 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 $q$8,

$q$9

where $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$0 is the clique number and $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$1 the independence number (Kim et al., 2024).

SDP relaxations: The clique number satisfies the classical upper bound $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$2 (Kobzar et al., 2023), with recent computational evidence (block-diagonal SDP relaxations, such as $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$3 and SOS-4) indicating actual growth may be sub-$$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$4: numerically, $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$5 (Kobzar et al., 2023).

• The Lovász $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$6 function equals $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$7 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 $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$8; 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 $$Q = \{\,x\in\mathbb{F}_q^* : x \text{ is a square} \,\}.$$9 (Gaar et al., 2024) | $P(q)$0 | $P(q)$1 | $P(q)$2 | $P(q)$3 | $P(q)$4 | |-----|--------------|------------------|------------|-------------| | 13 | 3 | 3.6056 | 3.6056 | 3.0000 | | 29 | 4 | 5.3852 | 5.3852 | 4.3177 |

5. Extremal and Combinatorial Properties

• Pancyclicity: For $P(q)$5, $P(q)$6, $P(q)$7 is pancyclic: it contains cycles of every possible length $P(q)$8 (Nishimura, 2023).
• Subgraph enumeration: For the number of triangles and 4-cliques, explicit formulas in terms of Jacobi sums are known. For $P(q)$9,

$V(P(q)) = \mathbb{F}_q$0

where $V(P(q)) = \mathbb{F}_q$1 with $V(P(q)) = \mathbb{F}_q$2, $V(P(q)) = \mathbb{F}_q$3 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 $V(P(q)) = \mathbb{F}_q$4 dividing $V(P(q)) = \mathbb{F}_q$5 (and $V(P(q)) = \mathbb{F}_q$6 when $V(P(q)) = \mathbb{F}_q$7 odd), define adjacency via $V(P(q)) = \mathbb{F}_q$8-th power residues:

$V(P(q)) = \mathbb{F}_q$9

Regular of degree $x, y$0, often not strongly regular for $x, y$1 (Elsawy, 2012, Schneider et al., 2013, Bonini et al., 2024).

• Automorphism group: For large $x, y$2 relative to $x, y$3, $x, y$4 (Ponomarenko, 23 Nov 2025).
• Paley graphs in characteristic $x, y$5: A distinct construction exists, using the trace map and Möbius transformations, resulting in a self-complementary, vertex-transitive pseudo-random graph on $x, y$6 points for $x, y$7 (Thomason, 2015).
• Paley graphs over $x, y$8: With $x, y$9 restricted to ensure $x-y\in Q$0 is a square in the unit group, a version exists for rings, where the underlying group is $x-y\in Q$1 and adjacency is defined by units that are squares modulo $x-y\in Q$2 (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 $x-y\in Q$3 (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 $x-y\in Q$4 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 $x-y\in Q$5 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 $x-y\in Q$6"
• (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"