Large Networks with Small Diameter

The construction of large networks with small diameter D for a given maximal degree Δ is a major goal in combinatorial network theory. Using genetic algorithms, together with Cayley graph techniques, new results for this degree/diameter problem can be obtained. A modification of the Todd-Coxeter algorithm yields further results and allows, with Sabidussi's representation theorem, a uniform representation of vertex-symmetric graphs. The paper contains an updated table of the best known (Δ, D)-graphs and a table with the largest known graphs for a given Δ and maximum average distance µ between the nodes.

[1]  Gérard Memmi,et al.  Some New Results About the (d, k) Graph Problem , 1982, IEEE Transactions on Computers.

[2]  Steve Gregory,et al.  Constructing Minimum Path Configurations for Multiprocessor Systems , 1993, Parallel Comput..

[3]  N. Biggs.  Algebraic Graph Theory: COLOURING PROBLEMS , 1974 .

[4]  Michael J. Dinneen,et al.  New results for the degree/diameter problem , 1994, Networks.

[5]  Shahid H. Bokhari,et al.  Reducing the Diameters of Computer Networks , 1986, IEEE Transactions on Computers.

[6]  Denis Trystram,et al.  Parallel algorithms and architectures , 1995 .

[7]  Charles Delorme,et al.  Strategies for Interconnection Networks: Some Methods from Graph Theory , 1986, J. Parallel Distributed Comput..

[8]  J.W. Moore,et al.  Small Diameter Symmetric Networks from Linear Groups , 1992, IEEE Trans. Computers.

[9]  K. Wendy Tang,et al.  Representations and routing for Cayley graphs [computer networks] , 1991, IEEE Trans. Commun..

[10]  S. Lakshmivarahan,et al.  Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey , 1993, Parallel Comput..

[11]  G. Farhi,et al.  Large Graphs with Given Degree and Diameter III , 1982 .

[12]  Charles Delorme,et al.  Examples of Products Giving Large Graphs with Given Degree and Diameter , 1992, Discret. Appl. Math..

[13]  H. Felsch,et al.  Programmierung der Restklassenabzählung einer Gruppe nach Untergruppen , 1961 .

[14]  Christian Scheideler,et al.  Universal continuous routing strategies , 1996, SPAA '96.

[15]  J. A. Todd,et al.  A practical method for enumerating cosets of a finite abstract group , 1936 .

[16]  Charles Delorme,et al.  Large graphs with given degree and diameter. II , 1984, J. Comb. Theory, Ser. B.

[17]  Michael Sampels.  Massively parallel architectures and systolic communication , 1997, PDP.

[18]  H. Yap.  Some Topics in Graph Theory , 1986 .

[19]  G. Sabidussi.  Vertex-transitive graphs , 1964 .

[20]  D. Frank Hsu,et al.  Graph Theory, Combinatorics, Algorithms, and Applications , 1991 .

[21]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[22]  Béla Bollobás,et al.  The Diameter of Graphs: a survey , 1981 .

[23]  Friedhelm Meyer auf der Heide,et al.  Universal algorithms for store-and-forward and wormhole routing , 1996, STOC '96.

[24]  Friedhelm Meyer auf der Heide,et al.  A Packet Routing Protocol for Arbitrary Networks , 1995, STACS.

[25]  Paul R. Hafner.  Large Cayley Graphs and Digraphs with Small Degree and Diameter , 1995 .

[26]  F. Comellas,et al.  New large graphs with given degree and diameter , 1994, math/9411218.

[27]  R. M. Storwick,et al.  Improved Construction Techniques for (d, k) Graphs , 1970, IEEE Transactions on Computers.

[28]  Michael Sampels.  Cayley Graphs as Interconnection Networks: A Case Study , 1996, Parcella.

[29]  Charles Delorme,et al.  Large Graphs with Given Degree and Diameter - Part I , 1984, IEEE Trans. Computers.