Pseudocyclic association schemes and strongly regular graphs

TL;DR

This paper investigates pseudocyclic association schemes where all nontrivial relations are strongly regular graphs with identical eigenvalues, revealing their eigenstructure and providing new examples and classifications.

Contribution

It characterizes the eigenstructure of such schemes, connects them to symmetric designs, and presents new non-amorphous and amorphous examples, expanding the understanding of their structure.

Findings

Principal eigenmatrix is a linear combination of an incidence matrix and all-ones matrix.

Several new non-amorphous association schemes are constructed, including classes 4, 5, and 7.

Connections between symmetric designs and association schemes are established.

Abstract

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on GF(2^{12}).