Commutativity of association schemes of prime square order having non-trivial thin closed subsets

Through a study of the structure of the modular adjacency algebra over a field of positive characteristic p for a scheme of prime order p and utilizing the fact that every scheme of prime order is commutative, we show that every association scheme of prime square order having a non-trivial thin closed subset is commutative. The second author was supported by Korea Research Foundation Grant (KRF-2006-003-00008).     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

Clifford theory for commutative association schemes

In this paper, Clifford's well-known theorem on irreducible characters of finite groups is generalized to finite commutative association schemes. Our theorem relates irreducible characters of finite commutative schemes to those of their strongly normal closed subsets.     

On association schemes with multiplicities 1 or a prime p

The main result of this paper gives a classification of commutative association schemes, all irreducible characters of which have multiplicity 1 or a prime p.     

Nilpotent schemes and group-like schemes

We give a definition of nilpotent association schemes as a generalization of nilpotent groups and investigate their basic properties. Moreover, for a group-like scheme, we characterize the nilpotency by its character products.     

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.     

Characters of association schemes containing a strongly normal closed subset of prime index

We consider a relation between characters of an association scheme and its strongly normal closed subsets with prime index. As an application of our result, we show that an association scheme of prime square order with a proper strongly normal closed subset is commutative.

     

Eigenvalues of association schemes of quadratic forms

The duality and primitivity of the association scheme Qua(n,q) of quadratic forms in n variables and the association scheme Sym(n,q) of symmetric bilinear forms in n variables over the finite field are discussed by Wang et al. [Association schemes of quadratic forms and symmetric bilinear forms, J. Algebraic Combin. 17 (2003) 149–161]. In this paper, eigenvalues of Qua(n,q) are computed, where q is a power of 2. As an application, two fusion schemes of Qua(n,q) are discussed and their dual schemes are constructed.     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

A modular absolute bound condition for primitive association schemes

The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field. Partially supported by RFBR grants 07-01-00485, 08-01-00379 and 08-01-00640. The paper was done during the stay of the author at the Faculty of Science of Shinshu University.     

Investigations on association schemes with elementary abelian thin residue

In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of order $p^2$. We then study the automorphism groups of these schemes and determine whether these schemes are schurian.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

New families of Q-polynomial association schemes

In this paper, we construct the first known infinite family of primitive Q-polynomial schemes which are not generated by distance-regular graphs. To construct these examples, we introduce the notion of a relative hemisystem of a generalized quadrangle with respect to a subquadrangle.     

New parameters of subsets in polynomial association schemes

We define new parameters, a zero interval and a dual zero interval, of subsets in P- or Q-polynomial association schemes. A zero interval of a subset in a P-polynomial association scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a Q-polynomial association scheme is a successive interval index for which the dual inner distribution vanishes. We derive bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a P-polynomial association scheme (resp. a Q-polynomial association scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a Q-polynomial association scheme). Moreover, we consider the spherical analogue of a dual zero interval.     

Convergence analysis of some first order and second order time accurate gradient schemes for semilinear second order hyperbolic equations

This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so‐called Sine‐Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in the discrete seminorm of and in the norm of . The convergence rate is proved to be for the first scheme and for the second scheme, where (resp. k) is the mesh size of the spatial (resp. time) discretization. The existence, uniqueness, and convergence results stated above do not require any relation between k and . The analysis presented in this work is also applicable in the gradient schemes framework. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 5–33, 2017     

On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s satisfies n_s2 and n_s^*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//t, where G is a finite group and t an involution of G weakly closed in C_G(t).

Groups G with an involution t weakly closed in C_G(t) have been described in Glauberman's Z^*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.