Mikhail E. Muzychuk

A´da´m's conjecture is true in the square-free case

Abstract Adams conjecture [1] formulates necessary and sufficient conditions for cyclic (circulant) graphs to be isomorphic. It is known to be true if the number n of vertices is either prime ([4]), a product of two primes ([12]) or satisfies the condition n, φ(n)) = 1, where φ is Eulers function ([15]). On the other hand, it is also known that the conjecture fails if n is divisible by 8 or by an odd square. It was newly conjectured in [15] that Adams conjecture is true for all other values of n. We prove that the conjecture is valid whenever n is a square-free number.

On graphs with three eigenvalues

Abstract We consider undirected non-regular connected graphs without loops and multiple edges (other than complete bipartite graphs) which have exactly three distinct eigenvalues (such graphs are called non-standard graphs). The interest in these graphs is motivated by the questions posed by W. Haemers during the 15th British Combinatorial Conference (Stirling, July 1995); the main question concerned the existence of such graphs. A brief review of two papers by Bridges and Mena (1979, 1981) is followed by the presentation of our new results and examples concerning, in particular, the construction of some non-standard graphs. This answers problems posed by Haemers. Other open problems are suggested and discussed in the final section.

On Ádám's conjecture for circulant graphs

Abstract Adams (1967) conjecture formulates necessary and sufficient conditions for cyclic (circulant) graphs to be isomorphic. It is known that the conjecture fails if n is divisible by either 8 or by an odd square. On the other hand, it was shown in [?] that the conjecture is true for circulant graphs with square-free number of vertices. In this paper we prove that Adams conjecture remains also true if the number of vertices of a graph is twice square-free.

On the isomorphism problem for cyclic combinatorial objects

We prove that the number of cyclic combinatorial objects on n elements isomorphic to a given one is less than or equal to ϕ(n). We also show that if any two prime divisors p ≠ q of n satisfy the property p∤(q − 1), q∤(p − 1), then the isomorphism problem for cyclic combinatorial objects on n elements may be reduced to the one on prime power number of elements.

Difference Sets with n = 2p ^m

Let D be a (v,k,λ) difference set over an abelian group G with even n = k - λ. Assume that t ∈ N satisfies the congruences t ≡ qifi (mod exp(G)) for each prime divisor qi of n/2 and some integer fi. In [4] it was shown that t is a multiplier of D provided that n > λ, (n/2, λ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > λ may be removed. As a corollary we obtain that in the case of n= 2pa when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].

The Structure of Schur Rings Over Cyclic Groups of Square-Free Order

We give the complete classification of Schur rings over cyclic groups of square-free order. In this classification we use the topological language originally suggested by Ya. Yu. Gol’fand. Each Schur ring over Zn is uniquely determined by a finite topology L on the set of prime divisors of n and by a family {GP}P ∈ L of finite groups satisfying an additional condition.

Subschemes of the Johnson scheme

Abstract Let X = ( X, {R i } i =0 d ) and X ′ = ( X′, {R′ i } i =0 d ′ ) be two association schemes defined on the same set X . We say thatX′is a subscheme ofXif each relation R′ i is a union of some R i . The subscheme lattice of the Johnson scheme J(n, m) is studied. We prove that it is trivial for m ≥ 3n + 4 ≥ 13 .

Subschemes of Hamming association schemes H(n, q), q≥4

The subschemes of the Hamming schemes H(n, q) for q greater than or equal to 4 are studied. We prove that there are no nontrivial subschemes when q > 4 and there exists only one nontrivial subscheme when q = 4.

On the mathematical model of triangulanes

The mathematical model of spirocondensed cyclopropanes, suggested by S.S. Tratch, is considered in the paper. According to the model every such compound is represented by a set of congruent equilateral triangles connected to each other by specific rules. One can associate an abstract graph with such a configuration of triangles. We investigate under what conditions a given graph can be realized as a graph of some spatial triangle configuration.