Pseudocyclic association schemes and strongly regular graphs

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}).

corrected a typo

Cited by in corpus (9)

• Commutative association schemes
• Constructions of Strongly Regular Cayley Graphs and Skew Hadamard Difference Sets from Cyclotomic Classes
• Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes
• Constructions of Strongly Regular Cayley Graphs using Even Index Gauss Sums
• Three-class association schemes from cyclotomy
• New constructions of strongly regular Cayley graphs on abelian groups
• Strongly Regular Graphs From Unions of Cyclotomic Classes
• Constructions of Strongly Regular Cayley Graphs Using Index Four Gauss Sums
• Cyclotomic Constructions of Skew Hadamard Difference Sets