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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.
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In this paper we characterize large regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that large association schemes become $P$-polynomial association schemes. Our results are summarized as follows. Let $G=(V,E)$ be a connected $k$-regular graph with $d+1$ distinct eigenvalues $k=theta_0>theta_1>cdots>theta_d$. Since the diameter of $G$ is at most $d$, we have the Moore bound [ |V| leq M(k,d)=1+k sum_{i=0}^{d-1}(k-1)^i. ] Note that if $|V|> M(k,d-1)$ holds, the diameter of $G$ is equal to $d$. Let $E_i$ be the orthogonal projection matrix onto the eigenspace corresponding to $theta_i$. Let $partial(u,v)$ be the path distance of $u,v in V$. Theorem. Assume $|V|> M(k,d-1)$ holds. Then for $x,y in V$ with $partial(x,y)=d$, the $(x,y)$-entry of $E_i$ is equal to [ -frac{1}{|V|}prod_{j=1,2,ldots,d, j e i} frac{theta_0-theta_j}{theta_i-theta_j}. ] If a symmetric association scheme $mathfrak{X}=(X,{R_i}_{i=0}^d)$ has a relation $R_i$ such that the graph $(X,R_i)$ satisfies the above condition, then $mathfrak{X}$ is $P$-polynomial. Moreover we show the dual version of this theorem for spherical sets and $Q$-polynomial association schemes.
{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.
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.
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].
We utilize association schemes to analyze the quality of semidefinite programming (SDP) based convex relaxations of integral packing and covering polyhedra determined by matchings in hypergraphs. As a by-product of our approach, we obtain bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine exactly or provide bounds on the performance of Lov{a}sz-Schrijver SDP hierarchy, and illustrate the usefulness of obtaining commutative subschemes from non-commutative schemes via contraction in this context.
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