Sergei Evdokimov

Permutation group approach to association schemes

We survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed.

On a New High Dimensional Weisfeiler-Lehman Algorithm

We investigate the following problem: how different can a cellular algebra be from its Schurian closure, i.e., the centralizer algebra of its automorphism group? For this purpose we introduce the notion of a Schurian polynomial approximation scheme measuring this difference. Some natural examples of such schemes arise from high dimensional generalizations of the Weisfeiler-Lehman algorithm which constructs the cellular closure of a set of matrices. We prove that all of these schemes are dominated by a new Schurian polynomial approximation scheme defined by the m-closure operators. A sufficient condition for the m-closure of a cellular algebra to coincide with its Schurian closure is given.

Isomorphism of Coloured Graphs with Slowly Increasing Multiplicity of Jordan Blocks

n-vertex edge coloured graphs with multiplicity of Jordan blocks bounded by k can be done in time .

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group that is similar to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study, for cellular algebras, the notion of a base, which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally, we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential. Bibliography: 16 titles.

Forestal algebras and algebraic forests (on a new class of weakly compact graphs)

In this paper we introduce and investigate a new class of graphs called algebraic forests for which isomorphism testing can be done in time O(n3 log n). The class of algebraic forests admits a membership test of the same complexity, it includes cographs, trees and interval graphs, and even a joint superclass of the latter two, namely, rooted directed path graphs. In fact, our class is much larger than these classes, since every graph is an induced subgraph of some algebraic forest. The key point of our approach is the study of the class of forestal cellular algebras de ned inductively from one point algebras by taking direct sums and wreath products. In fact, algebraic forests are exactly the graphs the cellular algebras of which are forestal. We prove that each weak isomorphism of two forestal algebras is induced by a strong isomorphism. This implies that all forestal algebras are compact cellular algebras and so all algebraic forests are weakly compact graphs. We also present a complete characterization of cellular algebras of disconnected graphs.

C-algebras and algebras in plancherel duality

For an arbitrary (possibly noncommutative) C-algebra, a positivity condition generalizing the Krein condition for a commutative case is defined. We show that the class of positive C-algebras includes those arising in algebraic combinatorics from association schemes (possibly noncommutative). It is proved that the category of positive C-algebras is equivalent to the category of pairs of algebras in Plancherel duality, one of which is commutative.

Two-closure of odd permutation group in polynomial time

We present a polynomial-time algorithm which constructs the 2-closure of a permutation group of odd order.