Mikhail Muzychuk

A Solution of the Isomorphism Problem for Circulant Graphs

The isomorphism problem for circulant graphs (Cayley graphs over the cyclic group) which has been open since 1967 is completely solved in this paper. The main result of the paper gives an efficient isomorphism criterion for circulant graphs of arbitrary order. This result also solves an isomorphism problem for colored circulant graphs and some classes of cyclic codes.

On Sylow Subgraphs of Vertex-Transitive Self-Complementary Graphs

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive selfcomplementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congruence pm3 1(mod4), where pm is the highest power of p which divides the order of the graph.

An Elementary Abelian Group of Rank 4 Is a CI-Group

In this paper we prove that Z4p is a CI-group; i.e., two Cayley graphs over the elementary abelian group Z4p are isomorphic if and only if their connecting sets are conjugate by an automorphism of the group Z4p.

Schur rings

In this paper we survey the recent developments in the theory of Schur rings and its applications to different problems that appear in theory of association schemes, Cayley graphs and other parts of algebraic combinatorics.

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.

Symmetric Bush-type Hadamard matrices of order 4m^4 exist for all odd m

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order 4m 4 for all odd integers m.

Association schemes generated by a non-symmetric relation of valency 2

We give some observations on association schemes with a relation of valency 2 from the representation theory of Bose-Mesner algebras and the basic structure theory. One of the applications of these observations is the classification of the association schemes with a nonsymmetric relation of valency 2 if the cardinality of the point set is a product of two primes, and another is the proof of Lis conjecture that any finite simple group is a connected 2-DCI-group.

The Structure of Rational Schur Rings over Cyclic Groups

Rational Schur rings over cyclic groups are studied in the paper. We prove (see Main Theorem) that there exists a one-to-one correspondence between rational Schur rings over Cn and sublattices of the divisor lattice of n.

On pseudocyclic association schemes

The notion of a pseudocyclic association scheme is generalized to the non-commutative case. It is proved that any pseudocyclic scheme the rank of which is much more than the valency is the scheme of a Frobenius group and is uniquely determined up to isomorphism by its intersection number array. An immediate corollary of this result is that any scheme of prime degree, valency k and rank at least k 4 is schurian.

Small vertex-transitive directed strongly regular graphs

We consider directed strongly regular graphs defined in 1988 by Duval. All such graphs with n vertices, n ≤ 20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is presented. This, together with a recent result by Jorgensen, gives a complete answer on Duvals question about the existence of directed strongly regular graphs with n ≤ 20. The paper includes catalogues of all generated graphs and certain theoretical generalizations based on some known and new graphs.