Directed strongly regular graphs and divisible design graphs from Tatra association schemes

TL;DR

This work constructs directed strongly regular graphs and divisible design graphs by leveraging the structure of Tatra association schemes and their fusions. It provides a explicit finite-field construction of a noncommutative, rank-$2n$ scheme $\mathcal{X}_0$, analyzes its automorphism and isomorphism groups, and shows how to obtain infinite families of DSRGs and DDGs from these schemes. Two main results are established: (i) an infinite family of DSRGs with parameters $\left((q+1)p, q+\frac{p-1}{2}, q, \frac{q-1}{p}, \frac{q-1}{p}+1\right)$ under $p\equiv 3\mod 4$ and $q-1=p(p-3)/4$, with detailed automorphism structure; (ii) a general DDG construction from difference sets in cyclic groups, yielding parameters $\left(n(q+1),kq,\lambda q,\frac{k^2(q-1)}{n},q+1,n\right)$ and SRG specializations. The results connect association schemes, $S$-rings, and difference-set theory to produce new combinatorial graphs with rich symmetry and fusion properties, including schurian and automorphism-characterization outcomes. The work also discusses implications under the Bateman–Horn conjecture for infinitude and poses open questions about classifying nonisomorphic graphs from these constructions.

Abstract

In this paper, we construct directed strongly regular graphs and divisible design graphs with new parameters merging some basic relations of so-called Tatra associations schemes. We also study the above association schemes, their fusions and isomorphisms.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 3.1
• Corollary 3.2
• Corollary 3.3
• Lemma 3.4
• proof