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33
An optimal lower bound on the number of variables for graph identification
 In Prac. 30th IEEE Syrup. Foundations of Computer Science
, 1989
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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 35 (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 11 (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...
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 10 (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.
Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices
 ELECTRON. J. COMBIN., 10(1):RESEARCH PAPER
, 2003
"... We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,2 ..."
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Cited by 8 (1 self)
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We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a byproduct, a few new amorphic association schemes with 3 classes on 100 points are discovered.
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 8 (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
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
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 8 (3 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.
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...