Difference Graph of a Finite Group

• The Difference Graph is a structure with vertices as nontrivial proper subgroups where an edge exists if their join equals the group but their product does not.
• It uncovers links between subgroup generation and graph parameters, illustrating how properties like simplicity, nilpotency, and solvability are reflected in graph invariants.
• The reduced graph D*(G) filters out isolated vertices from normal subgroups, offering practical insights into group characteristics such as supersolvability and cyclic behavior.

The difference graph of a finite group captures intricate relationships between group-theoretic structure and graph-theoretic properties by encoding certain 'difference phenomena' in subgroup generation, element orders, or quotient sets. This article focuses on the Difference Subgroup Graph $D(G)$ of a finite group $G$, as developed in (Das et al., 6 Nov 2025), and situates this graph within the broader landscape of difference graphs stemming from combinatorial group theory. The vertices of $D(G)$ are the nontrivial proper subgroups of $G$; distinct subgroups $H$ and $K$ are adjacent exactly when $\langle H, K \rangle = G$ but $HK \neq G$. This construction isolates those pairs whose join covers $G$ without having the subgroup product fill the group, thereby distinguishing between generating and multiplying subgroups. The investigation of $D(G)$ yields strong connections between fundamental graph invariants and core structural properties of $G$, including simplicity, nilpotency, solvability, and supersolvability.

1. Definition and Foundational Aspects

Let $G$ be a finite group and $S = \{ H < G : H \neq 1,\ H \neq G \}$ the set of all nontrivial proper subgroups of $G$. The difference subgroup graph is the simple undirected graph

$V(D(G)) = S,$

with edges $E(D(G))$ specified by

$$E(D(G)) = \left\{ \{H, K\} : \langle H, K \rangle = G\ \text{and}\ HK \neq G \right\}.$$

This definition sets $D(G)$ as the setwise difference between the join graph $\Delta(G)$ (edges for pairs generating $G$) and the comaximal subgroup graph $\Gamma(G)$ (edges for pairs multiplying to $G$). Thus,

$D(G) = \Delta(G) \setminus \Gamma(G).$

$D(G)$ is sensitive to global generation: it filters subgroup pairs by stringent generation versus multiplication criteria.

Many vertices of $D(G)$ are isolated. For group-theoretic focus, the reduced graph $D^*(G)$ is defined by deleting all isolated vertices from $D(G)$; in particular, every nontrivial normal subgroup is isolated in $D(G)$.

2. Structural Properties and Symmetries

Basic properties

• Nontrivial normal subgroups are isolated vertices.
• Conjugation acts as a graph automorphism: for $g \in G$, $H \sim K$ implies $gHg^{-1} \sim gKg^{-1}$.
• All non-isolated vertices have degree at least $2$; there are no leaves.
• Degrees among non-isolated vertices are variable.
• If $G \cong H \rtimes K$, then $D(K)$ embeds as an induced subgraph of $D(G)$.
• If $N \triangleleft G$, then $D(G/N)$ embeds as an induced subgraph.

Lower bounds

If $H \sim K$ is an edge in $D(G)$:

• At least $3$ edges emanate from $H$ and $K$ if conjugate.
• At least $4$ edges if not conjugate.

3. Connectivity, Forbidden Subgraphs, and Girth

Connectivity

• Theorem: $D(G)$ is connected if and only if $G$ is simple.
• A nontrivial normal subgroup yields an isolated vertex, so only simple groups have fully connected difference subgroup graphs.

Triangle-freeness and Bipartiteness

• Theorem: If $D(G)$ is triangle-free (hence bipartite), then $G$ is nilpotent.
• Corollary: If $D(G)$ has any edge, then its girth is either $3$ or $4$.
  • Non-nilpotent $G$ yields triangles via non-normal maximal subgroups.
  • For nilpotent $G$, edges induce $4$-cycles whenever present.

Universal vertices

• $D(G)$ never has a universal vertex and is never complete. This is a consequence of subgroup order constraints in finite simple groups.

4. Reduced Graph $D^*(G)$: Classification Results

• If $D^*(G)$ has a universal vertex, then $$G \cong \mathbb{Z}_q^\beta \rtimes \mathbb{Z}_{p^\alpha}$$, primes $p \ne q$.
• $D^*(G)$ is complete if and only if $G \cong \mathbb{Z}_q \rtimes \mathbb{Z}_{p^\alpha}$, with $|V(D^*(G))| = n_p = 1 + p l = q$.
• If $D^*(G)$ forms a cycle, it is necessarily $C_3$ or $C_4$.

5. Graph Parameters and Their Group-Theoretic Consequences

Independence number $\alpha(D(G))$:

• $\alpha(D(G)) \leq 5 \implies G$ non-nilpotent.
• $\alpha(D(G)) \leq 13 \implies G$ is a $p$-group or non-nilpotent.
• $\alpha(D(G)) \leq 3 \implies G$ supersolvable.
• $\alpha(D(G)) \leq 14 \implies G$ solvable.
• These bounds are tight, demonstrated by appropriate group families.

Clique number $\omega(D(G))$:

• $\omega(D(G)) \leq 4$ forces supersolvability.
• $\omega(D(G)) \leq 7$ forces solvability.

6. Forbidden Subgraph Characterizations

• Claw-free: $D(G)$ is claw-free iff $G$ is supersolvable.
• Cograph ($P_4$-free): $D(G)$ is a cograph iff $G$ is solvable.
• Chordality combined with cograph structure also forces solvability.

7. Examples

• Abelian (Dedekind, Iwasawa) groups: $D(G)$ is edgeless; for cyclic $G$, $D(C_n)$ is empty.
• Dihedral group $D_4$: $D(D_4) \cong C_4$.
• Symmetric group $S_3$: $D^*(S_3) \cong C_3$.
• Alternating group $A_4$: $\alpha(D(A_4))=4$, $\omega(D(A_4))=5$.
• Simple group $A_5$ ($57$ nontrivial subgroups): $D(A_5)$ is connected; $\alpha(D(A_5))=15$.

8. Analytical Significance and Open Problems

The study of $D(G)$ yields precise characterizations linking graph-theoretic invariants to deep group-theoretic properties. The systematic correspondence between forbidden graph substructures and algebraic properties (solvability, nilpotency, simplicity) enhances the toolkit available for both group theorists and combinatorialists.

Main open questions:

• When is $D^*(G)$ connected? The conjecture states that for non-nilpotent $G$, $D^*(G)$ is always connected.
• For non-abelian $p$-groups with $p$ odd and $D(G)$ nonempty, must the girth equal $3$?
• Does $D(G) \cong D(H)$ (both connected) imply $G \cong H$? For nilpotent $G$, does $D(G) \cong D(H)$ force $H$ nilpotent?
• If $D(G)$ is perfect, is $G$ necessarily solvable?
• It is known that $\omega(D(G)) \leq 7$ implies solvability; the optimal bound may be $\omega(D(G)) \leq 15$.

9. Connections and Future Directions

$D(G)$ synthesizes aspects of the classical join and comaximal subgroup graphs. It detects subtle deviations between generation and product phenomena, reflecting the asymmetry in group structure. This graph also interfaces with combinatorial parameters (clique, independence numbers) that serve as proxies for algebraic properties not readily accessible via traditional group-theoretic invariants.

The investigation of $D(G)$ is poised for further exploration in the field of computational group theory, classification problems, and the study of automorphism groups of combinatorial structures defined by subgroups. The robustness of the difference graph paradigm suggests possible extensions in infinite group settings, algebraic semigroups, and the analysis of spectral graph properties with algebraic significance (Das et al., 6 Nov 2025).