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On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$

Viktor A. Byzov, Igor A. Pushkarev

Abstract

The paper constructs an infinite sequence of strongly regular directed graphs. The construction relies on representing adjacency matrices as block matrices of circulant blocks and using a compactification operation compatible with polynomial arithmetic modulo $x^{2n+3}-1$. Computer search with the pychoco library, followed by automorphism group analysis in GAP, revealed a consistent structural pattern, enabling the formulation and proof of an explicit formula for the adjacency matrices of the sought digraphs. It is proved that the obtained digraphs satisfy the defining equations for strongly regular digraphs. A conjecture on the structure of their automorphism groups is formulated.

On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$

Abstract

The paper constructs an infinite sequence of strongly regular directed graphs. The construction relies on representing adjacency matrices as block matrices of circulant blocks and using a compactification operation compatible with polynomial arithmetic modulo . Computer search with the pychoco library, followed by automorphism group analysis in GAP, revealed a consistent structural pattern, enabling the formulation and proof of an explicit formula for the adjacency matrices of the sought digraphs. It is proved that the obtained digraphs satisfy the defining equations for strongly regular digraphs. A conjecture on the structure of their automorphism groups is formulated.
Paper Structure (5 sections, 3 theorems, 19 equations, 1 table)

This paper contains 5 sections, 3 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A computer experiment to find strongly regular digraphs
  4. An infinite sequence of strongly regular digraphs
  5. Conclusion

Key Result

Lemma 3

The sequence of matrices has the following properties:

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Conjecture 8