Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes

Abstract

We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov's conjecture by Ikuta and Munemasa [15].1 |X| J, E 1 , . . . , E d be the primitive idempotents of the Bose-Mesner algebra of the scheme (X, {R i } 0≤i≤d ), where J is the all-one matrix of size |X| × |X|. The basis transition matrix from {E 0 , E 1 , . . . , E d } to {A 0 , A 1 , . . . , A d } is denoted by P = (p j (i)) 0≤i,j≤d , and usually called the first eigenmatrix (or character table) of the scheme. Explicitly P is the (d + 1) × (d + 1) matrix with rows and columns indexed by 0, 1, 2, . . . , d such thatLet k i = p i (0) and m i = rank(E i ). The k i 's and m i 's are called valencies and multiplicities of the scheme, respectively. We say that the scheme (X, {R i } 0≤i≤d ) is pseudocyclic if there exists an integer t such that m i = t for all i ∈ {1, . . . , d}. A classical example of pseudocyclic association schemes is the cyclotomic association scheme over a finite field, which we define below.

Strongly regular graphs from unions of cyclotomic classes

“…Note that D 0 is the same as D in It is natural to ask whether (F q , {R ′ k } 0≤k≤p 1 ) is an association scheme. We give an affirmative answer to this question in a subsequent paper [7]. Also included in [7] are some interesting properties of this fusion scheme in relation to A.V.…”

Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes

“…However, there had been known only a few counterexamples in the primitive case. Recently, in [11], the authors generalized the counterexamples of Van Dam and Ikuta-Munemasa into infinite series using strongly regular Cayley graphs based on index 2 Gauss sums of type N = p m Then, one can similarly prove that (F q , {R k } 0≤k≤p 1 p 2 ) is a pseudocyclic and non-amorphic association scheme in which every nontrivial relation is a strongly regular graph. Table 2 yields three new infinite series of pseudocyclic and non-amorphic association schemes, where each of the nontrivial relations is strongly regular.…”

New strongly regular decompositions of the complete graphs with prime power vertices

“…In the case where (p, p 1 , f ) = (3,11,5), it satisfies the condition of Theorem 10 for d = 1 but not for d ≥ 2. Therefore, we obtain a sporadic example of a strongly regular decomposition of the complete graph on F 3 10 .…”

“…(p, m, f ) = (3,11,5), (5,19,9), (3,35,12), (7,37,9), (11,43,7), (17,67,33) (3, 107, 53), (5,133,18), (41, 163, 81), (3,323,144), (5,499,249).…”

“…Recently, there was a progress on the existence of strongly regular decompositions by Feng et al [10,11]. They found infinite families of strongly regular decompositions, which do not form amorphous association schemes.…”