On the diameter and bisector size of Cayley graphs

We present bounds on two combinatorial properties of Cayley graphs in terms relating to the structure of their underlying group, Included in this work is a presentation of lower bounds on the diameter of Cayley graphs of groups with nilpotent subgroups and upper bounds on the size of node bisectors of Cayley graphs of groups with solvable subgroups.Cayley graphs, being endowed with algebraic structure, have been increasingly recognized as a source of interconnection networks underlying parallel computers. Their structure has been shown to endow parallel architectures with advantages, for example, in terms of algorithmic efficiency. Our results demonstrate limits on the communication power of certain classes of well-structured interconnection networks.

[1]  László Babai,et al.  Small-diameter Cayley Graphs for Finite Simple Groups , 1989, Eur. J. Comb..

[2]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[3]  Gunnar E. Carlsson,et al.  Interconnection Networks Based on a Generalization of Cube-Connected Cycles , 1985, IEEE Transactions on Computers.

[4]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[5]  Bruce M. Maggs,et al.  Expanders might be practical: fast algorithms for routing around faults on multibutterflies , 1989, 30th Annual Symposium on Foundations of Computer Science.

[6]  Michael R. Fellows,et al.  Dense symmetric networks from linear groups , 1988, Proceedings., 2nd Symposium on the Frontiers of Massively Parallel Computation.

[7]  E. Szemerédi,et al.  Sorting inc logn parallel steps , 1983 .

[8]  János Komlós,et al.  Sorting in c log n parallel sets , 1983, Comb..

[9]  Paul Feldman,et al.  Wide-Sense Nonblocking Networks , 1988, SIAM J. Discret. Math..

[10]  Hyman Bass,et al.  The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups , 1972 .

[11]  M. Murty.  Ramanujan Graphs , 1965 .

[12]  Richard N. Draper,et al.  An overview of supertoroidal networks , 1991, SPAA '91.

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

[14]  Gábor Hetyei,et al.  On the diameter of finite groups , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[15]  Maria M. Klawe,et al.  Limitations on Explicit Constructions of Expanding Graphs , 1984, SIAM J. Comput..

[16]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[17]  Greenleaf,et al.  Invariant Means on Topological Groups , 1969 .

[18]  László Babai,et al.  Local expansion of vertex-transitive graphs and random generation in finite groups , 1991, STOC '91.

[19]  A. L. Rosenberg,et al.  Processor-Time Tradeoffs for Cayley Graph Interconnection Networks , 1991, The Sixth Distributed Memory Computing Conference, 1991. Proceedings.

[20]  Arnold L. Rosenberg,et al.  Group Action Graphs and Parallel Architectures , 1990, SIAM J. Comput..