Prime Graph Characterization

• Prime graphs are graphs that lack nontrivial modules, making them indecomposable with respect to substitution or modular decomposition.
• The framework encompasses module-based constructions, Gruenberg–Kegel graphs, and prime distance graphs, linking structural graph theory with group theory.
• Forbidden subgraph characterizations and vertex-minor operations play a crucial role in relating prime graphs to arithmetic constraints and group invariants.

A prime graph is a central object in the intersection of algebraic graph theory and combinatorial group theory, encoding fundamental structural information about groups, graphs, and number-theoretic constructs via adjacency relations grounded in “primality” or “irreducibility.” The term encompasses several distinct, but interrelated, graph-theoretic frameworks united by the theme of obstructing certain decompositional structures, realizing group-theoretic invariants, or encoding prime-based adjacency. This article presents a rigorous characterization of prime graphs, including module-based prime graphs in combinatorics, Gruenberg–Kegel (prime) graphs of finite groups, prime distance graphs, and forbidden-subgraph characterizations.

1. Prime Graphs in Structural Graph Theory

A (combinatorial) graph $G=(V,E)$ is prime if it possesses no nontrivial module. A module is a subset $M \subseteq V$ such that every $v \in V \setminus M$ is either adjacent to all of $M$ or to none of $M$. The trivial modules are $V$, $\emptyset$, and any singleton $\{v\}$.

Prime graphs can equivalently be viewed as indecomposable objects with respect to substitution or modular decomposition: $G$ is prime if the only modules are trivial and $|V(G)| \geq 4$. This notion plays a crucial role in graph decomposition, structure theorems, and algorithmic reductions (Boussaïri et al., 2013).

Prime Bound Function

Given a (not necessarily prime) graph $G$, the prime bound $p(G)$ is the smallest integer $p$ such that there exists a prime supergraph $H$ with $V(H) \supseteq V(G)$, $H[V(G)] = G$, and $|V(H) \setminus V(G)| = p$. The function $p(G)$ admits a complete characterization in terms of the largest modular clique $\omega_M(G)$ and modular independent set $\alpha_M(G)$:

• If $m(G) = \max(\omega_M(G), \alpha_M(G)) \geq 2$ and $\log_2 m(G) \notin \mathbb{Z}$, then $p(G)=\lceil\log_2 m(G)\rceil$.
• If $m(G) = 2^k$, then $p(G)=k$ except when $G$ or $\overline{G}$ has $2^k$ isolated vertices, in which case $p(G)=k+1$.
• If $G$ is non-prime with $\alpha_M(G)=\omega_M(G)=1$, $p(G)=1$ (Boussaïri et al., 2013).

2. Gruenberg–Kegel (Prime) Graphs of Finite Groups

The prime graph (or Gruenberg–Kegel graph) $\Gamma(G)$ of a finite group $G$ is defined as follows:

• Vertices: $\pi(G)$, the set of prime divisors of $|G|$.
• Edges: $\{p,q\}$ whenever $G$ contains an element of order $pq$.

Prime graphs are essential for distinguishing finite simple and almost-simple groups, elucidating subgroup structure, and encoding arithmetic constraints on element orders. Key group-theoretic phenomena, like the presence of elements of composite order, normal Hall subgroups, and Frobenius actions, are mirrored by adjacency patterns in $\Gamma(G)$ (Burness et al., 2014, Khramova et al., 2021, Chen et al., 20 Apr 2025).

Graph-Theoretic Characterization for Solvable Groups

A fundamental result is that a finite simple graph $\Gamma$ occurs as the prime graph of a finite solvable group if and only if its complement $\overline{\Gamma}$ is triangle-free and 3-colorable (Gruber et al., 2013, Huang et al., 2022). This yields a purely graph-theoretic recognition principle: $$\Gamma \cong \Gamma(G) \text{ for solvable } G \iff \overline{\Gamma} \text{ triangle-free and } \chi(\overline{\Gamma}) \le 3.$$ Minimal prime graphs correspond to maximal Frobenius action configurations among Sylow subgroups, and can be generated by vertex-duplication and clique-generation from core “base graphs” (e.g., complements of certain circulant graphs or $C_5$).

3. Prime Distance and Prime-Order Graphs

Prime Distance Graphs

A graph $G$ is a prime distance graph if its vertices admit an injective labeling $L: V(G) \to \mathbb{Z}$ such that for each edge $uv$, $|L(u)-L(v)|$ is prime (Laison et al., 2021). Key results include:

• Every bipartite graph and every cycle $C_n$ admits a prime distance labeling.
• The existence of prime distance labelings on certain windmill or paper mill graphs is equivalent to classical conjectures such as the Twin Prime and de Polignac’s conjectures.

Prime Order Element Graphs

Given a finite group $G$, the prime-order element graph $\Gamma(G)$ has vertex set $G$ with $x \sim y$ if $o(xy)$ is prime. Forbidden-subgraph characterizations yield tight correspondence between group structure (e.g., exponent and Sylow subgroup properties) and classical graph classes (e.g., perfection, cographs, chordal graphs, claw-freeness). For example, $\Gamma(G)$ is perfect if and only if $G$ has no subgroup isomorphic to $\mathbb{Z}_{pq}$ ($p,q$ distinct odd primes) or $\mathbb{Z}_{2p} \times \mathbb{Z}_2$ \; (Manna et al., 2024).

4. Characterization by Vertex-Minor and Split Structure

A graph is prime with respect to vertex-minor operations if it has no split—i.e., no partition $(A,B)$ with $|A|,|B| \ge 2$ and the adjacency submatrix $A_G[A,B]$ of rank at most $1$. This tightly links prime graphs to local and pivot-minor equivalence, and to strong 2-connectivity in the Tutte minor sense (Kim et al., 2022). The exceptional families locally equivalent to cycles or multi-path-graphs precisely delimit those with fewer than two or three non-essential vertices.

5. Classification, Uniqueness, and Forbidden Subgraph Phenomena

Minimal Prime Graphs

A minimal prime graph is a connected graph on at least two vertices whose complement is triangle-free, 3-colorable, and edge-maximal with respect to these properties (Huang et al., 2022, Alvarez et al., 30 Nov 2025). Every graph whose minimal prime graph complement contains a vertex of degree $2$ is exactly a $C_5$-reseminant—i.e., a graph generated by finite duplications of three specific vertices in a $5$-cycle, two of which are adjacent (Alvarez et al., 30 Nov 2025).

Recognizability

Specific families of finite simple groups, including $2E_6(2)$, $E_8(q)$ ($q$ in various small fields), and sporadic simple groups, are recognizable by isomorphism type of their prime graphs—i.e., any group sharing the same prime graph is isomorphic to the given simple group (Chen et al., 20 Apr 2025, Wang et al., 2020). For sporadic groups, disconnectedness of the prime graph together with matching group order suffices for group recognition.

Forbidden Subgraphs and Group Structure

Classification of prime graphs using forbidden subgraphs (e.g., odd cycles, $P_4$, $C_k$, $K_{1,3}$) naturally encodes deep group-theoretic invariants: perfection, chordality, cograph structure, and claw-freeness correspond to precise restrictions on group exponents, subgroup composition, and commutator structure (Manna et al., 2024).

The table below summarizes graph types and the corresponding group-theoretic conditions leading to forbidden subgraph characterizations in the prime order element graph:

Graph Property	Minimal Forbidden Subgraph	Group-Theoretic Condition
Perfection	Odd hole $C_{2k+1}$, odd antihole	Exclusion of $\mathbb{Z}_{pq}$, $\mathbb{Z}_{2p}\times\mathbb{Z}_2$
Cograph	$P_4$	EPPO group, restricted exponents
Chordal	$C_k$, $k\geq 4$	Only very small specific groups
Claw-free	$K_{1,3}$	Strong restrictions on direct products, exponents

6. Interplay with Group Theory, Colorability, and Decomposition

Prime graph characterization exposes a deep synergy between group theory and combinatorial graph properties. Triangle-freeness and bounded chromatic number in complements reflect the solvable group setting. Vertex-duplication, strong and Cartesian products, and clique-generation give rise to infinite families with structured automorphism groups. The realization of certain prime graphs by groups is governed by number-theoretic and group-theoretic obstructions such as the existence of elements of prescribed order, normal Hall subgroups, and module structure (Huang et al., 2022, DeGroot et al., 2021, Laubacher et al., 2020).

The prime graph landscape thus forms a unifying language for expressing maximal decompositional rigidity, recognizing finite simple and solvable group structures, and elucidating profound connections between forbidden subgraph theory and arithmetic invariants. Open directions include the full classification of minimal prime graphs with higher minimum degree, extension to infinite families and character-theoretic graphs, and further exploration of cross-links between chromatic, minor-, and modular-decomposition hierarchies (Huang et al., 2022, Alvarez et al., 30 Nov 2025, Manna et al., 2024).