Complete Rotations in Cayley Graphs

As it is introduced by Bermond, Perennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a complete rotation, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to investigate such Cayley graphs. We relate some symmetries of a graph with potential algebraic symmetries appearing in its definition as a Cayley graph on a group.

[1]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[2]  G. Sabidussi.  On a class of fixed-point-free graphs , 1958 .

[3]  Gert Sabidussi,et al.  Graph multiplication , 1959 .

[4]  M. Watkins.  Connectivity of transitive graphs , 1970 .

[5]  M. Watkins,et al.  On the action of non-Abelian groups on graphs , 1971 .

[6]  A. White.  Graphs, Groups and Surfaces , 1973 .

[7]  Walter Knödel,et al.  New gossips and telephones , 1975, Discret. Math..

[8]  Arthur T. White,et al.  Permutation Groups and Combinatorial Structures , 1979 .

[9]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[10]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

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

[12]  Stéphane Pérennes,et al.  Gossiping in Cayley Graphs by Packets , 1995, Combinatorics and Computer Science.

[13]  Selim G. Akl,et al.  Communication and fault tolerance algorithms on a class of interconnection networks , 1995 .

[14]  D. Frank Hsu,et al.  Distributed Loop Computer Networks: A Survey , 1995, J. Parallel Distributed Comput..

[15]  Marie-Claude Heydemann,et al.  Cayley graphs and interconnection networks , 1997 .

[16]  Selim G. Akl,et al.  Spanning subgraphs with applications to communication on a subclass of the Cayley-graph-based networks , 1998, Discret. Appl. Math..

[17]  S. Pérennes,et al.  Cayley Graphs with Complete Rotations , 1999 .

[18]  Dominique Désérable.  A Family of Cayley Graphs on the Hexavalent Grid , 1999, Discret. Appl. Math..

[19]  Pierre Fraigniaud,et al.  Minimum linear gossip graphs and maximal linear (Delta, k)-gossip graphs , 2001, Networks.

[20]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.