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The algebraic theory of recombination spaces
, 2000
"... A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the s ..."
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Cited by 29 (15 self)
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A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourierdecomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for Pstructures. For binary string recombination the elementary landscapes are exactly the pspin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...
Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Large Cayley Graphs and Digraphs with Small Degree and Diameter
, 1995
"... We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter. ..."
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Cited by 11 (0 self)
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We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.
Large graphs with small degree and diameter: A voltage assignment approach
"... Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general up ..."
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Cited by 5 (2 self)
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Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general upper bound on the diameter of a lift in terms of the properties of the base voltage graph, and prove some results on vertextransitive lifts. The potential of the new method is highlighted by showing that all currently known largest Cayley graphs (of given degree and diameter) for semidirect products of cyclic groups can be described by means of a voltage assignment construction, using simpler groups. This research started when J. Plesn'ik and J. Sir'an were visiting the Department of Computer Science of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 Introduction In the past few decades there has been growing interest in the design of interconnection ...
A note on constructing large Cayley graphs of given degree and diameter by voltage assignments
, 1997
"... Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diame ..."
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Cited by 4 (1 self)
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Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertextransitive graphs of given degree and diameter....
Matrix Techniques for Strongly Regular Graphs and Related Geometries
"... this paper gives a survey of the recent results on strongly regular graphs. It is a sequel to Hubaut [27] earlier survey of constructions. Seidel [35] gives a good treatment of the theory. 2 Association schemes ..."
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Cited by 3 (0 self)
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this paper gives a survey of the recent results on strongly regular graphs. It is a sequel to Hubaut [27] earlier survey of constructions. Seidel [35] gives a good treatment of the theory. 2 Association schemes
Moore Graphs, Latin Squares and Outer Automorphisms
, 2003
"... A connection between Moore graphs, outer automorphisms of symmetric groups, and families of almost orthogonal latin squares is exhibited. ..."
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Cited by 2 (0 self)
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A connection between Moore graphs, outer automorphisms of symmetric groups, and families of almost orthogonal latin squares is exhibited.
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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Cited by 1 (1 self)
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
High Performance Interconnection Networks
, 2002
"... The thesis is concerned with the design of high performance interconnection networks for use predominantly in parallel computing systems and wide area networks. The most important indicating a combined measure of hardware complexity and worstcast message routing complexity. Furthermore, a high perf ..."
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The thesis is concerned with the design of high performance interconnection networks for use predominantly in parallel computing systems and wide area networks. The most important indicating a combined measure of hardware complexity and worstcast message routing complexity. Furthermore, a high performance network should also have the properties of regular and planar topology, high bisection width and routing simplicity. Specifically, the following problems are studied: (i) constructing the largest possible networks that simultaneously exhibit a number of other properties including a small number of edges, high bisection width and planarity; and (ii) implementing high performance communication networks on a scale comparable to that of the Internet. With respect to specific technology, the thesis addresses the following two questions: (i) exactly how can optical internetworking be achieved on a world wide scale so as to maximize performance and (ii) just how big can an optical internetwork be, given the present/future technological limits and performance constraints.