Divisible design graphs obtained by plugging a difference set into a construction for antipodal distance-regular graphs of diameter 3

Abstract: In this paper, we present a new construction of divisible design graphs with new parameters, obtained by plugging a difference set of a quotient group into a known construction of antipodal distance-regular graphs of diameter 3. Also, we show that in characteristic 2 the new divisible design graphs are Cayley graphs over an elementary abelian 2-group.

• The paper presents a novel construction of DDGs by inserting reversible difference sets into antipodal distance-regular graphs of diameter 3.
• It derives explicit parameter formulas and proves these DDGs realize as Cayley graphs in characteristic 2, revealing rich group and spectral properties.
• The construction offers significant implications for coding theory, cryptography, and combinatorial designs through infinite new families of DDGs.

Divisible Design Graphs via Difference Sets in Antipodal Distance-Regular Graphs of Diameter 3

Introduction

The paper "Divisible design graphs obtained by plugging a difference set into a construction for antipodal distance-regular graphs of diameter 3" (2607.05043) develops a novel algebraic combinatorial construction of divisible design graphs (DDGs), leveraging difference sets in quotient groups and classic antipodal distance-regular graphs with diameter three. The authors further elucidate structural properties of these graphs, including explicit parameterization and their realization as Cayley graphs in characteristic two. The study offers new infinite families of DDGs with previously unknown parameters, broadening the spectrum of DDGs arising from algebraic and geometric configurations, and connects the construction to bent functions, Hadamard difference sets, and polar graphs.

Preliminaries and Core Definitions

A divisible design graph (DDG) with parameters $(v, k, \lambda_1, \lambda_2, m, n)$ is a $k$-regular graph on $v$ vertices partitioned into $m$ classes of size $n$. Vertices in the same class share exactly $\lambda_1$ common neighbors; between distinct classes, there are exactly $\lambda_2$ common neighbors. DDGs generalize group divisible designs in design theory via adjacency matrices.

Distance-regular graphs, especially antipodal graphs of diameter $d=3$, serve as a base for the construction. Such graphs exhibit regularity in distance distributions among pairs of vertices. The paper references the classic family constructed from vector spaces $V$ of even dimension over $\mathbb{F}_q$ endowed with a nondegenerate symplectic form, and a subgroup $k$0 of the additive group with index $k$1. Vertices are tuples $k$2; adjacency is determined via the bilinear form and coset structure.

Difference sets $k$3 in an abelian group $k$4 play a critical role, particularly reversible and zero-free sets, including Hadamard difference sets and their geometric/combinatorial connections.

Construction of New Divisible Design Graphs

The central result modifies the adjacency rule of the antipodal distance-regular graphs by incorporating a difference set $k$5 of size $k$6 in the quotient group $k$7. The adjacency in the graph $k$8 depends on membership in $k$9, where $v$0 is the union of $v$1 cosets representing $v$2. Provided $v$3 is reversible and zero-free, this yields a $v$4-regular DDG. The authors derive explicit formulas for the DDG parameters:

• $v$5
• $v$6
• $v$7
• $v$8
• $v$9

These parameters depend tightly on the algebraic properties of the difference set and the underlying vector space. The construction generalizes complements of antipodal distance-regular graphs and, in characteristic two, yields numerous nontrivial DDGs with previously unattained parameters.

Cayley Graph Realization and Characteristic 2 Results

For $m$0 a power of 2, the constructed DDGs are isomorphic to Cayley graphs over elementary abelian $m$1-groups of order $m$2. The connection set in the Cayley graph is determined algebraically using the quadratic form $m$3 associated with the symplectic bilinear form. The explicit isomorphism $m$4 encodes each vertex $m$5 as $m$6, ensuring the adjacency structure matches the DDG construction.

This realization allows one to exploit group-theoretical and spectral properties of Cayley graphs for further analysis, including automorphism groups and possible symmetry properties. Notably, the construction does not extend straightforwardly to odd characteristic, confirming a sharp dichotomy in the algebraic context.

Numerical Results and Claims

The authors present strong enumeration results: the number of bent functions in even dimension variables is enormous and each support corresponds to a Hadamard difference set, thereby generating a vast number of distinct DDGs. Analytical expressions and explicit constructions for parameters demonstrate significant coverage of DDGs with prime power order. The claims regarding infinite families of new DDGs and their recursive construction extend the landscape of design theory.

In characteristic 2, all nontrivial difference sets are Hadamard, and thus the construction covers both trivial and Hadamard types. The connection sets of certain binary polar graphs and collinearity graphs of Desarguesian nets instantiate these constructions, further reinforcing the combinatorial richness of the framework.

Implications, Generalizations, and Future Directions

Practically, the construction enables systematic generation of DDGs from algebraic data—difference sets and vector spaces—which can be used in coding theory, cryptographic protocols, and combinatorial design applications. The theoretical implications are manifold: DDGs bridge design theory, association schemes, and spectral graph theory while the approach enables further generalizations.

The announcement of future work points toward recursive constructions of divisible difference sets, broader classes of antipodal distance-regular graphs, and extensions to non-symplectic and higher-diameter cases. The method is anticipated to yield infinitely many new DDGs and association schemes, providing potentially novel strongly regular graphs and new combinatorial configurations for use in discrete mathematics and theoretical computer science.

Conclusion

The paper establishes a rigorous and explicit construction of divisible design graphs by embedding reversible difference sets into antipodal distance-regular graphs of diameter 3. The construction yields new families of DDGs with explicit, algebraically derived parameters and facilitates their realization as Cayley graphs in characteristic 2. The approach unifies algebraic, combinatorial, and geometric perspectives, offers strong numerical results on the abundance of DDGs, and sets the stage for further recursive and infinite constructions in combinatorial design and algebraic graph theory.