Cayley Divisible Design Graphs

• Cayley Divisible Design Graphs are highly symmetric Cayley graphs whose vertices partition into classes with fixed common neighbor counts, generalizing strongly regular graphs.
• They utilize divisible difference sets and Schur rings to translate group-theoretic properties into combinatorial design parameters.
• Explicit constructions via affine groups, WL-rank analysis, and automorphism studies yield families with distinctive algebraic and spectral characteristics.

A Cayley divisible design graph (often abbreviated as Cayley DDG) is a highly symmetric combinatorial structure defined as a Cayley graph whose vertex set admits a canonical partition such that intra-class and inter-class pairs of vertices have precisely prescribed numbers of common neighbors. These graphs generalize the notion of strongly regular graphs and provide a rich interface between algebraic graph theory, combinatorial design theory, and group theory, with central roles played by concepts such as Deza graphs, Schur rings, and divisible difference sets. The study of Cayley DDGs encompasses undirected and directed settings and produces families exhibiting extremal algebraic properties, including graphs with maximal possible Weisfeiler–Leman rank and integral spectra.

1. Definitions and Algebraic Criteria

1.1 Structure and Parameters

A graph $\Gamma=(V,E)$ is a divisible design graph (DDG) with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ if it is simple, $k$-regular, has $v=mn$ vertices, and the vertex set admits a partition into $m$ classes (blocks) of size $n$ such that:

• Each pair within the same class shares exactly $\lambda_1$ common neighbors.
• Each pair from different classes shares exactly $\lambda_2$ common neighbors.

A Cayley DDG is a DDG that arises as a Cayley graph $\Cay(G,S)$ for some finite group $G$ and identity-free, inverse-closed connection set $(v,k,\lambda_1,\lambda_2,m,n)$0. The canonical partition then corresponds to the right cosets of a subgroup $(v,k,\lambda_1,\lambda_2,m,n)$1 of size $(v,k,\lambda_1,\lambda_2,m,n)$2, with index $(v,k,\lambda_1,\lambda_2,m,n)$3 in $(v,k,\lambda_1,\lambda_2,m,n)$4 (Kabanov et al., 2019, Crnković et al., 2021).

1.2 Divisible Difference Sets

A subset $(v,k,\lambda_1,\lambda_2,m,n)$5 is a divisible difference set (DDS) relative to a subgroup $(v,k,\lambda_1,\lambda_2,m,n)$6 if

$(v,k,\lambda_1,\lambda_2,m,n)$7

in the group ring $(v,k,\lambda_1,\lambda_2,m,n)$8, where $(v,k,\lambda_1,\lambda_2,m,n)$9. This criterion guarantees that $k$0 is a DDG with the subgroup-coset partition as its canonical partition (Kabanov et al., 2019).

1.3 Deza Graphs

A Deza graph with parameters $k$1 is a $k$2-regular graph such that any two distinct vertices have either $k$3 or $k$4 common neighbors. Every DDG is a Deza graph with $k$5, $k$6, but with the extra structure of a canonical partition (Churikov et al., 2021).

1.4 Directed Generalizations

In the directed setting, a divisible design digraph (DDD) has regular in-degree and out-degree, and the counts of shared in- or out-neighbors for pairs within the same or different classes are specified analogously (Crnkovic et al., 2019, Muzychuk et al., 2024).

2. Infinite Families and Explicit Constructions

Several infinite families of Cayley DDGs have been constructed using group-theoretic and combinatorial design methodologies.

2.1 Affine Group Construction

Let $k$7 with $k$8 a prime power, $k$9. Set $v=mn$0 (the affine group). For a suitable selection of subspaces $v=mn$1 of $v=mn$2 and a permutation $v=mn$3 satisfying symmetry conditions, the connection set

$v=mn$4

with $v=mn$5 a generator of $v=mn$6, yields a DDG with explicit parameters depending on $v=mn$7 and $v=mn$8, where the partition is given by cosets of $v=mn$9 (Kabanov et al., 2019).

2.2 Maximal WL-rank Family

Fix any odd integer $m$0. Define the group

$m$1

with $m$2 cyclic of order $m$3. The connection set

$m$4

where $m$5 and $m$6 generate involutory subgroups, produces a Cayley DDG with parameters $m$7. When $m$8 is odd, the Schur ring generated by $m$9 is the full group ring, so the Weisfeiler–Leman rank equals the number of vertices; the graphs are strictly Deza, divisible-design, and integral (Churikov et al., 2021).

2.3 Heisenberg DD Digraphs

For every odd prime power $n$0, let $n$1 be the Heisenberg group of order $n$2. Utilizing orbits under a cyclic subgroup of automorphisms, the Cayley digraphs $n$3, where $n$4 are specific DDSs, yield pairwise nonisomorphic, normal, arc-transitive Cayley DDDs with parameters $n$5. The family exhibits equality of neighborhood designs and uniform 2-dimensional WL type, but 3-dimensional WL distinguishes the members for $n$6 (Muzychuk et al., 2024).

2.4 Graph Product and Block Replacement Methods

Four main infinite constructions extend the framework using Kronecker and strong products or block-replacement with Hadamard matrices, generating Cayley DDGs with broad parametric ranges and realizations on various group types (Crnković et al., 2021).

3. Algebraic and Spectral Properties

3.1 Eigenvalues and Spectrum

Cayley DDGs universally admit highly structured spectra. For general parameter set $n$7, the spectrum is

$n$8

with multiplicities $n$9 respectively. For the maximal WL-rank family, the spectrum is explicitly integral: $\lambda_1$0 (Churikov et al., 2021, Kabanov et al., 2019).

3.2 Weisfeiler–Leman Rank and Dimension

The WL-rank of a Cayley graph is the rank of the Schur ring generated by the connection set. In the Churikov–Ryabov construction, this rank is maximal ($\lambda_1$1), while the underlying (2-dimensional) WL algorithm still identifies these graphs (WL-dimension 2). Conversely, the Heisenberg family is indistinguishable by 2-WL but differentiable for individual members by 3-WL (WL-dimension 3, for $\lambda_1$2) (Churikov et al., 2021, Muzychuk et al., 2024).

3.3 Automorphism Groups

Cayley DDGs possess regular automorphism groups by construction. Many are vertex-transitive, and some explicit families (notably the Heisenberg digraphs) are normal and arc-transitive, with automorphism group a semidirect product of the right regular group and a subgroup of automorphisms preserving the DDS structure (Muzychuk et al., 2024).

4. Classification, Finite Examples, and Open Constructions

Dedicated classification efforts, combining canonical algebraic characterizations and computational enumeration, have resolved the set of parameter-feasible Cayley DDGs for all $\lambda_1$3, identifying unique and non-isomorphic realizations for all but a finite number of cases (Crnković et al., 2021, Crnkovic et al., 2019). The tables below summarize sample small parameter sets and corresponding groups:

Parameters	Regular Group(s)	# Non-isomorphic
(8,3,0,1,4,2)	$\lambda_1$4	1
(12,5,1,2,4,3)	$\lambda_1$5	5
(27,10,9,1,3,9)	$\lambda_1$6	4

Nonexistence results confirm that not every parameter set under the DDG definition is Cayley-realizable, emphasizing the combinatorial and algebraic constraints specific to group and difference-set structure (Crnković et al., 2021).

5. Connections to Combinatorial and Algebraic Design Theory

Cayley DDGs occupy a central role linking symmetric group-divisible designs, Deza graphs, and difference-set theory. The canonical partition of a DDG aligns with the block structure of a group-divisible design. The construction and study of divisible difference sets directly translates group-theoretic properties into combinatorial regularities. Notably, the neighborhood designs of certain Cayley DDDs are symmetric group-divisible designs with isomorphic block structure over nonisomorphic underlying digraphs (Muzychuk et al., 2024).

The interaction with Schur rings (S-rings) provides a further classification tool by encoding the orbit partition of the group under the automorphism group, directly connecting graph isomorphism, distinction under the Weisfeiler–Leman algorithm, and spectral properties (Kabanov et al., 2019, Churikov et al., 2021).

6. Open Problems and Extensions

Outstanding problems focus on the precise enumeration of nonisomorphic Cayley DDGs within specific parameter classes, classification for larger graph orders, construction of new infinite families beyond Paley and affine-derived templates, and verification of conjectured connections to balanced generalized weighing matrices (BGW) for directed analogs (Crnkovic et al., 2019, Crnković et al., 2021). The behavior of the Weisfeiler–Leman dimension for newly-constructed families, the explicit isomorphism types of neighborhood designs, and the structural role of Schur ring invariants remain active avenues of research.

The ongoing development of Cayley divisible design graphs thus continues to shape the interconnections between algebraic combinatorics, group theory, and algorithmic graph theory, leveraging deep techniques from permutation groups, difference sets, and spectral analysis.