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Register a new userA characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers
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Recent classification of $frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${mathrm{PSL}}(2,q)$ and ${mathrm{PGamma L}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.
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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. Ivanovs conjecture by Ikuta and Munemasa [15].
Every Hadamard matrix $H$ of order $n > 1$ induces a graph with $4n$ vertices, called the Hadamard graph $Gamma(H)$ of $H$. Since $Gamma(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme $(Omega, S)$ of order $4n$. In this article we deal with fission schemes of $(Omega, S)$ under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme.
{Let ${Cal X}$ be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matrices $T$ such that $(PT)^3=I$. Moreover, all of the solutions for the classical infinite families of such schemes (including the Hamming scheme) are classified.
An association scheme is called quasi-thin if the valency of each its basic relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of it forms a Klein group with respect to the relation product. It is proved that any Kleinian scheme arises from near-pencil on~$3$ points, or affine or projective plane of order~$2$. The main result is that any non-Kleinian quasi-thin scheme a) is the two-orbit scheme of a suitable permutation group, and b) is characterized up to isomorphism by its intersection number array. An infinite family of Kleinian quasi-thin schemes for which neither a) nor b) holds is also constructed.
In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.
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