Divisible Design Graph

• Divisible design graphs are k-regular graphs on mn vertices, partitioned into m classes with fixed common neighbor counts within and between classes.
• They are constructed using group-theoretic (Cayley) and design-theoretic methods, incorporating affine designs, symplectic graphs, and difference sets.
• These graphs generalize strongly regular graphs and association schemes, revealing insights into spectral properties, equitable partitions, and combinatorial parameter relations.

A divisible design graph (DDG) is a $k$-regular simple graph on $v=mn$ vertices, whose vertex set can be partitioned into $m$ classes of size $n$ such that any two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and any two vertices from different classes have exactly $\lambda_2$ common neighbors. This structure generalizes strongly regular graphs and group-divisible designs, and links combinatorial design theory, algebraic graph theory, and association scheme theory.

1. Definition and Basic Properties

A graph $\Gamma=(V,E)$ is a divisible design graph with parameters $(v, k, \lambda_1, \lambda_2, m, n)$ if:

• $|V| = v = m n$ and $\Gamma$ is $k$-regular,
• The vertex set has a canonical partition $V = X_1 \cup \ldots \cup X_m$, $|X_i| = n$,
• For any two distinct vertices $x, y \in V$:
  • If $x, y \in X_i$, then $|N(x) \cap N(y)| = \lambda_1$,
  • If $x \in X_i, y \in X_j, i \neq j$, then $|N(x) \cap N(y)| = \lambda_2$.

Equivalently, the adjacency matrix $A$ satisfies

$$A^2 = k I_v + \lambda_1 (B - I_v) + \lambda_2 (J_v - B),$$

where $I_v$ is the $v \times v$ identity, $J_v$ is the $v \times v$ all-ones matrix, and $B = I_m \otimes J_n$ is the block-diagonal sum of $m$ all-ones $n \times n$ matrices. A DDG is called proper if $m,n > 1$ and $\lambda_1 \neq \lambda_2$; otherwise, it reduces to an improper case, such as a strongly regular graph or a complete multipartite graph (Kabanov et al., 2019, Crnković et al., 2021, Muzychuk et al., 15 Jan 2026).

Basic parameter relations include

$k = (n-1)\lambda_1 + (m-1)n \lambda_2,$

arising from counting neighbors for a fixed vertex, and eigenstructure constraints elaborated below.

2. Canonical Partition, Quotient Matrices, and Spectrum

The canonical partition in a DDG is always equitable. The corresponding quotient matrix $R$ of size $m\times m$ satisfies:

• Row sums: $\sum_{j=1}^m r_{ij} = k$ for all $i$,
• Quadratic: $R^2 = (k^2 - \lambda_2 v) I_m + \lambda_2 n J_m$,
• Trace: $\operatorname{tr} R^2 = m k^2 - (m-1)\lambda_2 v$.

Eigenvalue structure is determined by the combinatorial parameters. The eigenvalues of the adjacency matrix $A$ are: $$k,\quad \pm \sqrt{k-\lambda_1}\ (\text{multiplicity } m(n-1)),\quad \pm \sqrt{k^2-\lambda_2 v}\ (\text{multiplicity } m-1)$$ with associated multiplicities dictated by algebraic and combinatorial constraints. The canonical partition also ensures block-decomposition properties for both the adjacency matrix and $A^2$ (Gavrilyuk et al., 2023, Shalaginov, 2021, Muzychuk et al., 15 Jan 2026).

3. Constructions and Parameter Families

3.1. Group-Theoretic and Cayley Constructions

A central class of DDGs are Cayley divisible design graphs, equivalent to divisible difference sets relative to subgroups. Let $G$ be a finite group, $N \le G$ a subgroup, and $S \subseteq G \setminus \{e\}$ inverse-closed. If the group algebra identity $$SS^{-1} = a (N \setminus \{e\}) + b (G \setminus N) + k\{e\}$$ holds and $A=N\setminus\{e\}$, then $\operatorname{Cay}(G,S)$ is a DDG with canonical partition induced by $N$ and parameters $(|G|, |S|, a, b, [G:N], |N|)$ (Kabanov et al., 2019, Crnković et al., 2021).

Notable infinite Cayley DDG families arise from constructions over semidirect products of affine groups, vector-space difference sets, and strong or Kronecker products with classical graphs (Hadamard graphs, Paley graphs, SRGs) (Kabanov, 2021, Crnković et al., 2021).

3.2. Design-Theoretic and Geometric Constructions

Many DDG families are constructed via incidence structures with group-divisible design properties. Examples include:

• Graphs from affine (resolvable) designs and Latin squares (often using the Wallis–Fon-Der-Flaass modification) (Kabanov, 2021),
• DDGs from symplectic graphs and spreads in finite projective geometry—yielding both “special-spread” and “symplectic-spread” families, often not realizable as Cayley graphs for small parameters (Bruyn et al., 2024),
• Fusions of association schemes, such as those of Higmanian or Tatra type, where the union of certain basis relations defines the DDG edge set (Ryabov, 26 Jan 2026, Muzychuk et al., 15 Jan 2026).

Explicit parameter catalogues exist, with, for instance, $(4n, n+2, n-2, 2, 4, n)$ given by the line graph of $K_{4,n}$ and more general parameter sets for complex fusions over association schemes (Shalaginov, 2021, Muzychuk et al., 15 Jan 2026).

3.3. Exceptional Small Graphs and Classifications

Computational classification has determined all proper connected DDGs for $v \leq 39$ except for three parameter tuples, and every feasible small parameter set for $v \leq 27$ for Cayley DDGs is realized except five forbidden cases. These results demonstrate the unification and completion of sporadic construction records from group theory, design theory, and computer search (Panasenko et al., 2021, Crnković et al., 2021).

4. Connections to Strongly Regular Graphs and Association Schemes

A DDG with $m=1$ or $n=1$ or $\lambda_1 = \lambda_2$ is a strongly regular graph (SRG). DDGs thus interpolate between SRGs and group-divisible designs. Furthermore, decompositions of certain SRGs into a Hoffman coclique and an induced DDG component yield infinite families characterized by explicit algebraic relations and parameter constraints. The symplectic graph and its complement provide a canonical example where DDGs and SRGs are inextricably linked, and the classification of SRGs decomposable into a DDG and a Hoffman coclique is now explicit (Gavrilyuk et al., 2023, Bruyn et al., 2024).

Imprimitive symmetric association schemes (notably Higmanian and Tatra) provide natural settings for DDG constructions, where merging select basis relations yields DDGs with explicit algebraic and combinatorial parameters (Ryabov, 26 Jan 2026, Muzychuk et al., 15 Jan 2026).

5. Spectrum, Connectivity, and Structural Parameters

DDGs have tightly controlled spectral properties: at most five distinct eigenvalues, with multiplicities governed by the canonical partition and combinatorial equations. These properties ensure equitable partitions and often enable explicit determination of automorphism and symmetry groups.

Unlike primitive SRGs, whose vertex connectivity equals vertex degree, some DDGs exhibit reduced vertex connectivity; the difference $k - \kappa(G)$ can attain any power of two via Hadamard-family constructions (Panasenko, 2022). The spectrum also reveals that the presence of loops (in looped DDGs) or directed arcs (in divisible design digraphs) preserves close analogues of these algebraic constraints (Bhowmik et al., 6 May 2025, Muzychuk et al., 2024).

6. Infinite Families, Specialized Subclasses, and Open Directions

Infinite families are constructed from symplectic graphs over fields and certain special rings, Cayley graphs for affine groups, fusions of association schemes, and via generalized difference sets (e.g., Singer complements, Paley sets) (Bhowmik et al., 2024, Muzychuk et al., 15 Jan 2026). Specialized subclasses include “thin” DDGs ($n=2$), which connect directly to weighing matrices, fixed-point-free involutions, and orthogonal signings, and have recursive constructions for regular Hadamard matrices and symmetric weighing matrices (Goryainov et al., 18 Dec 2025).

A substantial open problem is the complete enumeration and classification of DDGs for mid-size parameter ranges, particularly in the context of their automorphism groups and fusion types in higher-rank schemes. Another is the classification and enumeration of special spreads in symplectic geometry that yield distinct DDGs, especially for larger projective spaces (Bruyn et al., 2024, Ryabov, 26 Jan 2026).

7. Summary Table: Canonical Parameter Relations in DDGs

Below is a summary table of the principal parameter identities appearing in DDGs, as synthesized from the referenced literature.

Parameter	Identity / Equation	Context
$v$	$v = m n$	size, classes, class size
$k$	$k = (n-1)\lambda_1 + (m-1) n \lambda_2$	vertex degree
$A^2$	$$A^2 = k I_v + \lambda_1 (B - I_v) + \lambda_2 (J_v - B)$$	adjacency matrix equation
$R$ (quotient)	$R^2 = (k^2 - \lambda_2 v) I_m + \lambda_2 n J_m$	equitable partition quotient matrix
Spectrum	$k$, $\pm\sqrt{k - \lambda_1}$, $\pm\sqrt{k^2 - \lambda_2 v}$	eigenvalues [multiplicities: $1, m(n-1), m-1$]
Properness	$m, n > 1$, $\lambda_1 \neq \lambda_2$	proper DDG
Regularity	$k(k-1) = (n-1)\lambda_1 + (m-1)n \lambda_2$	(Bhowmik et al., 6 May 2025)

These relations are the fundamental combinatorial and algebraic underpinnings for the construction, classification, and analysis of divisible design graphs (Gavrilyuk et al., 2023, Bruyn et al., 2024, Muzychuk et al., 15 Jan 2026).