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

TL;DR

This paper constructs an infinite sequence of strongly regular digraphs with explicit formulas for their adjacency matrices, using block circulant matrices and polynomial arithmetic, and verifies their properties with computational tools.

Contribution

It introduces a novel construction method for strongly regular digraphs using block circulant matrices and polynomial operations, solving previously open existence questions.

Findings

Explicit formulas for adjacency matrices of the digraphs.

Examples with previously unknown parameters, such as (63, 21, 8, 5, 8) and (81, 27, 10, 7, 10).

Hypothesis on automorphism group structures of the digraphs.

Abstract

The paper constructs an infinite sequence of strongly regular directed graphs. The construction is based on representing adjacency matrices as block matrices composed of circulant blocks, together with the use of a compactification operation consistent with polynomial arithmetic modulo $x^{2n+3}-1$. Using computer search with the pychoco library and subsequent analysis of automorphism groups in the GAP system, a stable structural pattern was identified, which made it possible to formulate and prove an explicit formula for the adjacency matrices of the infinite sequence of directed graphs. Among the obtained digraphs, there are examples with parameters $(63, 21, 8, 5, 8)$ and $(81, 27, 10, 7, 10)$, for which the question of existence had previously remained open. A hypothesis on the structure of the automorphism groups of the digraphs in the constructed sequence is also formulated.