Mikhail Klin

Transitive Permutation Groups Without Semiregular Subgroups

A transitive nite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and ane group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups. Part of the motivation for studying this class of groups was a conjecture due to Maru si c, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

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.

Amorphic Cellular Rings

The enumeration problem for cellular subrings of a given cellular ring has found numerous applications in combinatorics and graph theory (see for example [17], [22]). Usually one uses a computer to solve this problem. A general idea underlying searching algorithms for the enumeration of subrings was proposed in [19] and consists of the following.

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.

Some new strongly regular graphs

We show that three pairwise 4-regular graphs constructed by the second author are members of infinite families.

A root graph that is locally the line graph of the Petersen graph

We construct a root graph on 192 vertices that is locally the line graph of the Petersen graph, a new distance-regular graph on 96 vertices (with intersection array {15,10,1;1,2,15} and automorphism group 2^4.Sym(6)), and several new strongly regular graphs (with parameters (v,k,@l,@m)=(96,20,4,4) and (96,19,2,4)) and square 2-(96,20,4) designs.

A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices

New constructions of regular distance regular antipodal covers (in the sense of Godsil-Hensel) of complete graphs K n are presented. The main source of these constructions are skew generalized Hadamard matrices. It is described how to produce such a matrix of order n 2 over a group T from an arbitrary given generalized Hadamard matrix of order n over the same group T . Further, a new regular cover of K 45 on 135 vertices is produced with the aid of a decoration of the alternating group A 6 .

Graphical t -designs via polynomial Kramer-Mesner matrices

Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for fixed values of k and t in which the entries are polynomials in n the number of vertices of the underlying graph. From this we obtain an elementary proof that with a few exceptions Sn[2] is a maximal subgroup of Sn2 or An2. We also show that there are only finitely many graphical incomplete t-(v,k,λ) designs for fixed values of 2 ⩽ t and k at least in the cases k = t + 1, t = 2, and 2 ⩽ t < k ⩽ 6. All graphical t-designs are determined by the program DISCRETA3 for various small parameters. Most parameter sets are new for graphical designs, some also for general simple t-designs. The largest value of t for which graphical designs were found is t = 5. Some of the smaller designs which are block transitive are drawn as graphs.

Examples of computer experimentation in algebraic combinatorics

We introduce certain paradigms for procuring computer-free explanations from data acquired via computer algebra experimentation. Our established context is the field of algebraic combinatorics, with special focus on coherent configurations and association schemes. All results presented here were obtained by the authors with the aid of computer algebra systems, especially COCO and GAP. A number of examples are explored, in particular of objects on 28, 50, 63, and 210 points. In a few cases, initial experimental data pointed to appropriate theoretical generalizations that yielded an infinite class of related combinatorial structures. Special attention is paid to algebraic automorphisms (of a coherent algebra), a fairly new concept that has already proved to have far-reaching consequences. Finally, we focus on the Doyle-Holt graph on 27 vertices, and some of its related structures.

Algorithmic Algebraic Combinatorics and Grbner Bases

Tutorials.- Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics.- Siamese Combinatorial Objects via Computer Algebra Experimentation.- Using Grobner Bases to Investigate Flag Algebras and Association Scheme Fusion.- Enumerating Set Orbits.- The 2-dimensional Jacobian Conjecture: A Computational Approach.- Research Papers.- Some Meeting Points of Grobner Bases and Combinatorics.- A Construction of Isomorphism Classes of Oriented Matroids.- Algorithmic Approach to Non-symmetric 3-class Association Schemes.- Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings.- A Construction of Designs from PSL(2,q) and PGL(2,q), q=1 mod 6, on q+2 Points.- Approaching Some Problems in Finite Geometry Through Algebraic Geometry.- Computer Aided Investigation of Total Graph Coherent Configurations for Two Infinite Families of Classical Strongly Regular Graphs.