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26
An Optimal Lower Bound on the Number of Variables for Graph Identification
 Combinatorica
, 1992
"... In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open q ..."
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Cited by 135 (9 self)
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In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding 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
Combinatorial Landscapes
 SIAM REVIEW
, 2002
"... 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, ne ..."
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Cited by 33 (2 self)
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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.
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. I. Permutation Groups and Coherent (Cellular) Algebras.
, 1997
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Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the WeisfeilerLeman Algorithm
, 1997
"... 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 hav ..."
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Cited by 10 (4 self)
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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 WeisfeilerLeman algorithm and present an efficient com...
Separability number and Schurity number of coherent configurations
 Electronic J. Combinatorics
"... To each coherent configuration (scheme) C and positive integer m we associate a natural scheme � C (m) on the mfold 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 th ..."
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Cited by 9 (3 self)
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To each coherent configuration (scheme) C and positive integer m we associate a natural scheme � C (m) on the mfold 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 distanceregular graph Γ, then s(C) = 1 iff Γ is uniquely determined by its parameters whereas t(C)=1iffΓis distancetransitive. 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 distanceregular 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 3/2homogeneous scheme. Finally, we show that s(C) ≤ 4, whenever C is a cyclotomic scheme on a prime number of points. 1
A solution of the isomorphism problem for circulant graphs
 Proc. London Math. Soc
, 2004
"... 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. ..."
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Cited by 9 (0 self)
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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.
Spectral Landscape Theory
 Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality, Accident, and Function
, 1999
"... 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 ..."
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Cited by 8 (3 self)
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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
On Highly Closed Cellular Algebras And Highly Closed Isomorphisms
, 1998
"... We define and study mclosed cellular algebras (coherent configurations) and misomorphisms 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 ..."
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Cited by 7 (4 self)
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We define and study mclosed cellular algebras (coherent configurations) and misomorphisms 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 mclosed for all m 1 are exactly Schurian ones whereas the weak isomorphisms which are misomorphisms for all m 1 are exactly ones induced by strong isomorphisms. We show that for any m there exist mclosed algebras on O(m) points which are not Schurian and misomorphisms 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 mvertex condition ...
Coherent Configurations, Association Schemes and Permutation Groups
 Groups, Combinatorics and Geometry
, 2003
"... Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a nontrivial coherent configuration. However, symmetric coherent configurations have a much longer history, having b ..."
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Cited by 7 (0 self)
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Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a nontrivial coherent configuration. However, symmetric coherent configurations have a much longer history, having been used in statistics under the name of association schemes. The relationship between permutation groups and association schemes is quite subtle; there are groups which preserve no nontrivial association scheme, and other groups for which there is not a unique minimal association scheme. This paper gives a brief outline of the theory of coherent configurations and association schemes, and reports on some recent work on the connection between association schemes and permutation groups. 1 Coherent configurations This section contains the definitions of coherent configurations and of various specialisations (including association schemes), and their connection with finite permutation...
Landscapes on Spaces of Trees
, 2001
"... 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 socalled ..."
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Cited by 6 (1 self)
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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 socalled 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.