In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k − 1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices. 1

Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.

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The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W (A) which includes A. In case when A = A(\Gamma) is the adjacency matrix of a graph \Gamma the algorithm examines all configurations in \Gamma having three vertices and, according to this information, partitions vertices and ordered pairs of vertices into equivalence classes. The resulting construction allows to associate to each graph \Gamma a matrix algebra W (\Gamma) := W (A(\Gamma)) which is an invariant of the graph \Gamma. For many classes of graphs, in particular for most of the molecular graphs, the algebra W (\Gamma) coincides with the centralizer algebra of the automorphism group Aut(\Gamma). In such a case the partition returned by the stabilization algorithm is equal to the partition into orbits of Aut(\Gamma). We give algebraic and combinatorial descriptions of the Weisfeiler--Leman algorithm and present an efficient com...

INTRODUCTION Evolutionary change is caused by the spontaneously generated genetic variation and its subsequent fixation by drift and/or selection. Consequently, the main focus of evolutionary theory has been to understand the genetic structure and dynamics of populations, see e.g. [101]. In recent years, however, alternative approaches have gained increasing prominence in evolutionary theory. This development has been stimulated to some extent by the application of evolutionary models to designing evolutionary algorithms such as Genetic Al- Evolutionary Dynamics edited by J.P. Crutchfield and P. Schuster, 1999 2 Spectral Landscape Theory gorithms, Evolution Strategies, and Genetic Programming, as well as by the theory of Complex Adaptive Systems [69, 79, 38]. The generic structure of an evolutionary model is x 0 = S (x; w) ffi T (x;<F

To each coherent configuration (scheme) C and positive integer m we associate a natural scheme � C (m) on the m-fold Cartesian product of the point set of C having the same automorphism group as C. Using this construction we define and study two positive integers: the separability number s(C) and the Schurity number t(C) ofC. It turns out that s(C) ≤ m iff C is uniquely determined up to isomorphism by the intersection numbers of the scheme � C (m). Similarly, t(C) ≤ m iff the diagonal subscheme of � C (m) is an orbital one. In particular, if C is the scheme of a distance-regular graph Γ, then s(C) = 1 iff Γ is uniquely determined by its parameters whereas t(C)=1iffΓis distance-transitive. We show that if C is a Johnson, Hamming or Grassmann scheme, then s(C) ≤ 2andt(C) = 1. Moreover, we find the exact values of s(C) andt(C) for the scheme C associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, s(C)=t(C)=2ifC is the scheme of a Doob graph. In addition, we prove that s(C) ≤ 2andt(C) ≤ 2 for any imprimitive 3/2-homogeneous scheme. Finally, we show that s(C) ≤ 4, whenever C is a cyclotomic scheme on a prime number of points. 1

We define and study m-closed cellular algebras (coherent configurations) and m-isomorphisms of cellular algebras which can be regarded as mth approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If m = 1 we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are m-closed for all m 1 are exactly Schurian ones whereas the weak isomorphisms which are m-isomorphisms for all m 1 are exactly ones induced by strong isomorphisms. We show that for any m there exist m-closed algebras on O(m) points which are not Schurian and m-isomorphisms of cellular algebras on O(m) points which are not induced by strong isomorphisms. This enables us to find for any m an edge colored graph with O(m) vertices satisfying the m-vertex condition ...

Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstruction a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a so-called Robinson graph by means of the nearest neighborhood interchange move. The coherent algebra and spectra of Robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure. We consider simple model landscapes as well as landscapes arising from the maximum parsimony problem, focusing on two complementary measures of ruggedness: the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points.

All graphs considered in the paper are directed. Let be a graph on n vertices which we identify with the elements of the additive cyclic group Z n f0; 1;...;n 1g. The graph is called circulant if it has a cyclic symmetry, that is, if the permutation ð0; 1; 2;...;n 1Þ is anautomorphism of the graph.

We show that the symmetry groups of the cut cone Cutn and the metric cone Metn both consist of the isometries induced by the permutations on {1,..., n}; that is, Is(Cutn) = Is(Metn) � Sym(n) for n ≥ 5. For n = 4 we have Is(Cut4) = Is(Met4) � Sym(3) × Sym(4). This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp n) � Sym(n) for n ≥ 5, where Hyp n denotes the hypermetric cone.