Generalized Paley Graphs: Theory and Applications

• Generalized Paley graphs are finite graphs constructed from finite fields where vertices are adjacent if their differences are k-th power residues, unifying classical Paley graphs.
• They exhibit strong regularity, well-defined spectral properties through Gaussian periods, and automorphism groups that often reduce to the affine semilinear group.
• These graphs are pivotal in applications ranging from spectral graph theory and coding theory to combinatorial designs and permutation group analysis.

A generalized Paley graph is a finite graph constructed from the additive group of a finite field, with adjacency defined by membership in a multiplicative subgroup (typically k-th power residues) of the field. These graphs unify and generalize classical Paley graphs (quadratic-residue graphs) and are central to the interplay between graph theory, combinatorics, finite fields, and algebraic number theory. They exhibit strong regularity, deep connections to spectral graph theory, automorphism group structure, and provide natural platforms for exploring extremal properties such as clique numbers, chromatic numbers, and Ramanujan properties.

1. Definition and Basic Construction

Given $q = p^d$ a prime power and $k \ge 2$ with $k \mid q-1$, let $$S = \{ x \in \mathbb{F}_q^* : x \text{ is a } k\text{-th power in } \mathbb{F}_q^* \}$$; then $S$ is a cyclic subgroup of $\mathbb{F}_q^*$ of index $k$. The generalized Paley graph $\mathrm{GPaley}(q, \frac{q-1}{k})$ is the Cayley graph

$\mathrm{Cay}\left(\mathbb{F}_q^+, S\right)$

with vertex set $\mathbb{F}_q$ and an (undirected) edge between $x,y$ if $x-y \in S$. To ensure undirectedness for $q$ odd, $(q-1)/k$ is taken to be even so that $-1 \in S$. For $k=2$, one recovers the classical Paley graph $P(q)$.

Key structural facts:

• Vertex set: $\mathbb{F}_q$
• Edge set: $\{\{x, y\} \mid x-y \in S\}$
• Regularity: Each vertex has degree $n = (q-1)/k$
• Arc- and vertex-transitivity: Induced by field addition and the multiplicative subgroup's action

2. Automorphism Groups and Coherent Configurations

The automorphism group structure is a fundamental topic for generalized Paley graphs. For sufficiently large $q$ compared to $k$, the main result is: $$\operatorname{Aut}\,\mathrm{GPaley}(q,\frac{q-1}{k}) \leq \mathrm{A\Gamma L}(1,q)$$ where $$\mathrm{A\Gamma L}(1,q) = \mathbb{F}_q^+ \rtimes \Gamma L(1,q)$$ is the affine semilinear group. The proof employs two key combinatorial lemmas:

• Distinguishing lemma: Non-neighboring vertices outside the neighborhood $\Delta$ of 0 are distinguishable via some vertex in $\Delta$.
• Half-distinguish lemma: For $k$ disjoint edges inside $\Delta$, more than half of $\Delta$ distinguishes endpoints of at least one edge.

These enable reduction to normal circulant structure on the induced subgraph $X_\Delta$, and use classical S-ring arguments plus the action of a Singer cycle's normalizer to link automorphisms to $\Gamma L(1, q)$. The overall argument reduces the automorphism group to the standard affine group even for large classes of generalized Paley graphs (Ponomarenko, 23 Nov 2025).

3. Connectivity, Component Structure, and Bipartiteness

Connectivity depends on the additive generation properties of the subgroup $S$:

• $\mathrm{GPaley}(q, \frac{q-1}{k})$ is connected if and only if $S$ additively generates $\mathbb{F}_q$.
• Disconnectivity is characterized by subfield containment conditions: the graph splits into isomorphic copies of a smaller generalized Paley graph over some proper subfield $\mathbb{F}_{p^a}\subset \mathbb{F}_q$, specifically when $(q-1)/(p^a-1)|k$ (Podestá et al., 30 Sep 2024, Bonini et al., 5 Sep 2024).

With rare exceptions, these graphs are non-bipartite; the unique bipartite case occurs when $q=2^m$, $k=2^{m-1}$, yielding the perfect matching $K_2 \sqcup \cdots \sqcup K_2$ (Podestá et al., 30 Sep 2024). A concise summary appears in the table:

Case	Connected?	Bipartite?
$k=2^{m-1}$, $q=2^m$	Yes	Yes
else	Varies	No

4. Spectral Theory and Strong Regularity

The eigenvalues of $\mathrm{GPaley}(q,\frac{q-1}{k})$ are given by the Gaussian periods $\eta_i^{(k,q)}$, arising from the $k$-th cyclotomic decomposition of $\mathbb{F}_q^*$ and traced through the additive characters of the field: $$\eta_i^{(k,q)} = \sum_{x \in C_i} \zeta_p^{\operatorname{Tr}(x)}$$ where $C_i$ are the cyclotomic classes, $0 \leq i < k$, and $\zeta_p$ a primitive $p$-th root of unity.

Key spectral properties:

• For small $k$ ($k = 2,3,4$), explicit closed formulas for the Gaussian periods and spectra exist (Podestá et al., 2023, Podestá et al., 2019).
• Integrality: The graph is integral (all eigenvalues in $\mathbb{Z}$) if and only if $k | (q-1)/(p-1)$.
• Semiprimitive case: When $k | (p^{m/2}+1)$ and $m$ even, the spectrum collapses to three eigenvalues, and the graph is strongly regular (srg) (Podestá et al., 2023, Podestá et al., 2019, Podestá et al., 2018). The parameters can be expressed in terms of $q, k$, and the Gaussian periods.

A substantial number of generalized Paley graphs are strongly regular or distance-regular (particularly the Van Lint–Schrijver graphs obtained for special $k$) and are of (pseudo-)Latin-square type or negative Latin-square type depending on field-theoretic parameters (Podestá et al., 2023, Podestá et al., 2018).

5. Extremal and Ramsey-Theoretical Properties

The clique number problem in generalized Paley graphs is one of the major open questions in extremal combinatorics. Principal results include:

• Square-root bound: For $q$ odd, $\omega(\mathrm{GPaley}(q,d)) \leq \sqrt{q}$, with equality if and only if $d | (\sqrt{q}+1)$; in this case, the cliques come from subfields (Yip, 2021).
• Construction of maximal cliques: For $q \equiv 1 \pmod{d}$, there are maximal cliques of size $\sim q/m$ for each integer $m$ with $\operatorname{rad}(m)|\operatorname{rad}(d)$, using intersections of shifted subfields and character sum methods (Martin et al., 7 Mar 2024).
• Upper bounds and improvements: For cubic Paley graphs and related cases, constant factor improvements on the clique number (from the folklore $\sqrt{q}$) have been achieved—e.g., $\omega < 0.769\sqrt{q} + 1$ for generic cubic Paley graphs (Yip, 2020).
• Triangle/quadrilateral counts: Exact formulas for the number of $K_3, K_4$ subgraphs have been developed in terms of Jacobi sums and finite-field hypergeometric functions, yielding nontrivial lower bounds on multicolor Ramsey numbers (Dawsey et al., 2020).

These extremal results interact with algebraic number theory (Gauss sum evaluations, Stickelberger's theorem) and combinatorial geometry (direction lemmas, subfield clique constructions).

6. Algorithmic Identification: Weisfeiler–Leman Dimension

For $q \gg k$, the Weisfeiler–Leman (WL) dimension for identification via the color refinement algorithm is sharply bounded:

• $$\mathrm{dim}_{\mathrm{WL}}\,\mathrm{GPaley}(q,(q-1)/k) \leq 5$$ for sufficiently large $q$ (Ponomarenko, 23 Nov 2025).
• For Van Lint–Schrijver graphs, $3 \leq \mathrm{dim}_{\mathrm{WL}} \leq 5$, reflecting that 2-dimensional WL cannot distinguish these from certain non-isomorphic SRGs on the same parameters.

The bounds are achieved using combinatorial properties of normal circulant schemes and base-size arguments, leveraging the connection to the automorphism group.

7. Applications and Broader Contexts

Generalized Paley graphs are central to multiple research themes:

• Spectral graph theory: Their spectra are tightly related to coding theory (irreducible cyclic codes and two-weight codes) (Podestá et al., 2019), Ramanujan graph construction, and energy/integrality phenomena (Podestá et al., 2018).
• Permutation group theory: Exact coincidence of clique and chromatic number (synchronization) in these graphs directly links to non-synchronizing primitive affine groups (Schneider et al., 2013).
• Algebraic combinatorics: These graphs provide infinite towers of explicit Ramanujan graphs in all characteristics, serve as test beds for character sum bounds, and sit at the interface between extremal combinatorics, number theory, and the theory of pseudorandom graphs.
• Metric geometry: Condensed Ricci curvature can be calculated explicitly for classes of generalized Paley graphs satisfying the global matching condition (Bonini et al., 5 Sep 2024).

Their paper continues to illuminate both the structure of finite fields and fundamental limits in graph theory, with ongoing open questions regarding clique numbers, chromatic numbers, automorphism groups for small $q$, and spectral sharpness.