Dr. Mikhail Klin
Born: 1946, Ukraine
Ph.D. 1975, Kiew University
Researcher 1992.
 Department of Mathematics - Research Grade A

Research Interests:
Algebraic combinatorics. Finite permutation groups. Graph theory. Computer algebra. Mathematical chemistry. Discrete geometry.
Research Projects:
Algebraic graph theory and its applications in mathematical chemistry (1995-1998, jointly with R.Poeschel, G.Tinhofer and G. Adelson-Velskii). Investigation of primitive graphs (1995-1998, jointly with Y. Segev).
Abstracts of Current Research:
• Directed strongly regular graphs.: We start from the classical definition of a strongly regular graph (srg). One of its formulations: a regular undirected graph is an srg if and only if it has exactly three distinct eigenvalues. A more general notion of a directed strongly regular graph (dsrg) was introduced in 1988 by A.M.Duval. These graphs also have 3 distinct eigenvalues. A new approach to the construction of dsrgs via use of coherent (cellular) algebras was elaborated.In particular, new infinite classes of dsrgs were discovered inside of flag algebras of incidence structures (such as Steiner 2-designs, generalised quadrangles, symmetric designs, etc). New sporadic examples of dsrgs were constructed with the aid of a computer. These results are obtained jointly with F.Fiedler (Dresden-Newark), A.Munemasa (Fukuoka), M.Muzychuk, P.-H.Zieschang (Kiel) and Ch.Pech (Dresden).
• Circulant graphs via Schur ring theory.: Circulant graphs via Schur ring theory. circulant graph with vertices is a Cayley graph over a cyclic group oforder . Our approach to circulant graphs is based on the following methodology: to describe all Schur rings over , to find their automorphism groups and then to use the results for the identification of circulant graphs. All automorphism groups of circulantgraphs with an odd prime-power number of vertices are described with the aidof a new operation over permutation groups, we call it a subwreath product. The knowledge of necessary and sufficientconditions for the isomorphism ofn-vertex circulant graphs allows to solvethe problem of their analytical enumeration. We obtained closedcounting formulae and generating functions (by valency) for directed and undirected non-isomorphic circulant graphs on vertices.Similar problems for arbitrary values of are considered.This is a joint project with V.Liskovets (Minsk), M.Muzychuk (Ramat-Gan),R. P\oschel (Dresden), G.Tinhofer (Munich) and A.Woldar (Villanova).
• On transitive permutation groups without semi-regular subgroups: The following question is, to our knowledge, still open: find an example of afinite graph $\Gamma$ with the vertex set $V$ such that the automorphism group$\mbox{Aut}(\Gamma)$ acts transitively on $V$ and does not have a semi-regularsubgroup. In our attempts to construct such an example we started from a fewseries of transitive permutation groups $(H,V)$ about which it was known(W.M. Kantor) that they do not contain a semi-regular subgroup. We found 2-orbits of $(H,V)$ and computed the 2-closure $(H^{(2)},V)$ (in the sense ofH. Wielandt). It turns out that the 2-closure does not have the desiredproperty. We intend to attack this problem more systematically, exemining wider classes of finite permutation groups with similar properties.This is a joint work with G. Jones (Southampton).\end{document}
Publications:
• Brouwer, A. E., Koolen, J. H., Klin M. H.,. A root graph that is locally the line graph of the Petersen graph. Discrete Mathematics : (2001)
• Jones, G., Klin, M., Moshe Y.. Primitivity of permutation groups, coherent algebras and matrices. Journal of Combinatorial Theory A : (2001)
• Cameron, P. J., Giudici M. , Jones, G. A., Kantor, W. M, Klin, M. H., Marusic, D. L., Nowitz A.. Transitive permutation groups without semiregular subgroups. The London Mathematical Society : (2001)
• Fieldler, F., Klin, M. Muzychuk, M.. Small vertex-transitive directed strongly regular graphs. Discrete Mathematics : (2001)
• Muzychuk, M., Klin, M., Poschel R.. The isomorphism problem for circulant graphs via Schur ring theory. DIMACS Series in Discrete Mathematics 56: 241-264 (2001)
Keywords:Cellular Algebra, Schur Ring, Strongly Regular Graph, Automorhism Group, Galois Correspondence.
Phones:
1. Fax: 972-8-6472910
 Email: klin@math.bgu.ac.il klin@cs.bgu.ac.il