Cayley graphs and interconnection networks

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

[1]  Pradip K. Srimani,et al.  A New Family of Cayley Graph Interconnection Networks of Constant Degree Four , 1996, IEEE Trans. Parallel Distributed Syst..

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

[3]  Brian Alspach Cayley Graphs with Optimal Fault Tolerance , 1992, IEEE Trans. Computers.

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

[5]  Ginette Gauyacq Routages uniformes dans les graphes sommet-transitifs , 1995 .

[6]  Chris Godsil Connectivity of minimal Cayley graphs , 1981 .

[7]  Joy Morris,et al.  Connectivity of Cayley Graphs : A Special Family , 1996 .

[8]  D. Frank Hsu,et al.  Combinatorial Properties of Generalized Hypercube Graphs , 1996, Inf. Process. Lett..

[9]  Sheldon B. Akers,et al.  On Group Graphs and Their Fault Tolerance , 1987, IEEE Transactions on Computers.

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

[11]  M. Krishnamoorthy,et al.  Fault diameter of interconnection networks , 1987 .

[12]  W. Mader,et al.  Minimalen-fach kantenzusammenhängende Graphen , 1971 .

[13]  Paz Morillo,et al.  Diameter vulnerability of iterated line digraphs , 1996, Discret. Math..

[14]  Saïd Bettayeb,et al.  On the Multiply-Twisted Hypercube , 1994, Canada-France Conference on Parallel and Distributed Computing.

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

[16]  Filippo Mignosi,et al.  Garden of Eden Configurations for Cellular Automata on Cayley Graphs of Groups , 1993, SIAM J. Discret. Math..

[17]  Ming-Yang Kao,et al.  Optimal Broadcast in All-Port Wormhole-Routed Hypercubes , 1994, ICPP.

[18]  Shahram Latifi On the Fault-Diameter of the Star Graph , 1993, Inf. Process. Lett..

[19]  Patrick Solé,et al.  The edge-forwarding index of orbital regular graphs , 1994, Discret. Math..

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

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

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

[23]  Miguel Angel Fiol,et al.  Vertex-symmetric Digraphs with Small Diameter , 1995, Discret. Appl. Math..

[24]  Marc Baumslag,et al.  On the Fault-Tolerance of Quasi-Minimal Cayley Networks , 1991, ICCI.

[25]  Yahya Ould Hamidoune,et al.  Vosperian and superconnected Abelian Cayley digraphs , 1991, Graphs Comb..

[26]  Christos H. Papadimitriou,et al.  Bounds for sorting by prefix reversal , 1979, Discret. Math..

[27]  Max H. Garzon Cayley Automata , 1993, Theor. Comput. Sci..

[28]  John J. Cannon Construction of defining relators for finite groups , 1973, Discret. Math..

[29]  D.Frank Hsu,et al.  On Container Width and Length in Graphs, Groups,and Networks--Dedicated to Professor Paul Erdös on the occasion of his 80th birthday-- , 1994 .

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

[31]  Lynne L. Doty,et al.  Cayley Graphs with Neighbor Connectivity One , 1996, SIAM J. Discret. Math..

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

[33]  Vassilios V. Dimakopoulos,et al.  Tota; Exchange in Cayley Networks , 1996, Euro-Par, Vol. I.

[34]  Farhad Shahrokhi Approximation Algorithms for the Maximum Concurrent Flow Problem , 1989, INFORMS J. Comput..

[35]  Zsuzsanna Róka,et al.  One-Way Cellular Automata on Cayley Graphs , 1993, Theor. Comput. Sci..

[36]  Jiuqiang Liu Hamiltonian Decompositions of Cayley Graphs on Abelian Groups of Odd Order , 1996, J. Comb. Theory, Ser. B.

[37]  Jaroslav Opatrny,et al.  Forwarding indices of consistent routings and their complexity , 1994, Networks.

[38]  Dominique Barth,et al.  Emulating Networks by Bus Networks: Application to Trees and Hypermeshes , 1996, SIROCCO.

[39]  Cheryl E. Praeger,et al.  On orbital regular graphs and frobenius graphs , 1998, Discret. Math..

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

[41]  Ralph Tindell,et al.  Circulants and their connectivities , 1984, J. Graph Theory.

[42]  Gene Cooperman,et al.  New Methods for Using Cayley Graphs in Interconnection Networks , 1992, Discret. Appl. Math..

[43]  Olivier Delmas Communications par commutation de circuits dans les réseaux d'interconnexion. (Circuit-switched communications in interconnection networks) , 1997 .

[44]  Claude Berge Principes de combinatoire , 1970 .

[45]  M. H. Schultz,et al.  Topological properties of hypercubes , 1988, IEEE Trans. Computers.

[46]  Yahya Ould Hamidoune,et al.  Quelques problèmes de connexité dans les graphes orientés , 1981, J. Comb. Theory, Ser. B.

[47]  Ivan Hal Sudborough,et al.  On Sorting by Prefix Reversals and the Diameter of Pancake Networks , 1992, Heinz Nixdorf Symposium.

[48]  Richard M. Stafford,et al.  Processor Interconnection Networks from Cayley Graphs , 1992, Discret. Appl. Math..

[49]  Gert Sabidussi,et al.  The lexicographic product of graphs , 1961 .

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

[51]  Erik A. van Doorn,et al.  Connectivity of circulant digraphs , 1986, J. Graph Theory.

[52]  William Y. C. Chen,et al.  Cycle prefix digraphs for symmetric interconnection networks , 1993, Networks.

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

[54]  Chandra Gowri Sankaran Broadcasting on recursively decomposable Cayley graphs , 1994, Discret. Appl. Math..

[55]  Ginette Gauyacq,et al.  On Quasi-Cayley Graphs , 1997, Discret. Appl. Math..

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

[57]  Joseph A. Gallian,et al.  Hamiltonian cycles and paths in Cayley graphs and digraphs - A survey , 1996, Discret. Math..

[58]  Danny Dolev,et al.  A new look at fault tolerant network routing , 1984, STOC '84.

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

[60]  Yahya Ould Hamidoune,et al.  The Connectivity of Hierarchical Cayley Digraphs , 1992, Discret. Appl. Math..

[61]  Yahya Ould Hamidoune,et al.  An Isoperimetric Problem in Cayley Graphs , 1999, Theory of Computing Systems.

[62]  Gregory Butler,et al.  Fundamental Algorithms for Permutation Groups , 1991, Lecture Notes in Computer Science.

[63]  Zsolt Tuza,et al.  The Forwarding Index of Directed Networks , 1996, Discret. Appl. Math..

[64]  Willibald Dörfler,et al.  Every regular graph is a quasigroup graph , 1974, Discret. Math..

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

[66]  Bernard Elspas,et al.  Topological constraints on interconnection-limited logic , 1964, SWCT.

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

[68]  Polina Strogova,et al.  Finding a Finite Group Presentation Using Rewriting , 1998 .

[69]  Selim G. Akl,et al.  Optimal Communication Primitives on the Generalized Hypercube Network , 1996, J. Parallel Distributed Comput..

[70]  Complexity of the Forwarding Index Problem , 1993, SIAM J. Discret. Math..

[71]  Frank Harary,et al.  Graph Theory , 2016 .

[72]  Rainer Feldmann,et al.  The Cube-Connected Cycles Network Is a Subgraph of the Butterfly Network , 1992, Parallel Process. Lett..

[73]  Khaled Day,et al.  Arrangement Graphs: A Class of Generalized Star Graphs , 1992, Inf. Process. Lett..

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

[75]  Shahram Latifi,et al.  Conditional Fault Diameter of Star Graph Networks , 1996, J. Parallel Distributed Comput..

[76]  Farhad Shahrokhi,et al.  On Canonical Concurrent Flows, Crossing Number and Graph Expansion , 1994, Combinatorics, Probability and Computing.

[77]  Shietung Peng,et al.  Fault Tolerant Routing in Hypercubes and Star Graphs , 1996, Parallel Process. Lett..

[78]  Zsuzsanna Roka,et al.  Automates cellulaires sur graphes de Cayley , 1994 .

[79]  W. Mader Minimale n-fach kantenzusammenh angende graphen , 1971 .

[80]  Khaled Day,et al.  A Comparative Study of Topological Properties of Hypercubes and Star Graphs , 1994, IEEE Trans. Parallel Distributed Syst..

[81]  Marie-Claude Heydemann,et al.  On forwarding indices of networks , 1989, Discret. Appl. Math..

[82]  R. M. Damerell On Moore graphs , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[83]  Pradip K. Srimani,et al.  Fault Diameter of Star Graphs , 1993, Inf. Process. Lett..

[84]  Pradip K. Srimani,et al.  Trivalent Cayley Graphs for Interconnection Networks , 1995, Inf. Process. Lett..

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

[86]  Gene Cooperman,et al.  Applications of Cayley Graphs , 1990, AAECC.

[87]  Miguel Angel Fiol,et al.  On Cayley line digraphs , 1995, Discret. Math..

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

[89]  N. S. Barnett,et al.  Private communication , 1969 .

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

[91]  Alan J. Hoffman,et al.  On Moore Graphs with Diameters 2 and 3 , 1960, IBM J. Res. Dev..

[92]  Yahya Ould Hamidoune,et al.  Small cutsets in quasiminimal Cayley graphs , 1996, Discret. Math..

[93]  Fan Chung Graham,et al.  The forwarding index of communication networks , 1987, IEEE Trans. Inf. Theory.

[94]  Christian Lavault,et al.  Interconnection Networks: Graph- and Group Theoretic Modelling , 1999 .

[95]  Charles Delorme,et al.  Table of Large (Delta, D)-Graphs , 1992, Discret. Appl. Math..

[96]  Jean-Claude Bermond,et al.  Bus Interconnection Networks , 1996, Discret. Appl. Math..

[97]  Yannis Manoussakis,et al.  Computation of the forwarding index via flows: A note , 1994, Networks.

[98]  Farhad Shahrokhi,et al.  The maximum concurrent flow problem , 1990, JACM.

[99]  Ginette Gauyacq Edge-forwarding Index of Star Graphs and Other Cayley Graphs , 1997, Discret. Appl. Math..

[100]  Harold Neville Vazeille Temperley,et al.  Graph theory and applications , 1981 .

[101]  Afonso Ferreira,et al.  Optimal information dissemination in Star and Pancake networks , 1993, Proceedings of 1993 5th IEEE Symposium on Parallel and Distributed Processing.

[102]  Jaroslav Opatrny,et al.  Embeddings of complete binary trees into star graphs with congestion 1 , 1995, Proceedings of the Twenty-Eighth Annual Hawaii International Conference on System Sciences.

[103]  E T. Leighton,et al.  Introduction to parallel algorithms and architectures , 1991 .

[104]  F. Comellas,et al.  Broadcasting in Cycle Preex Digraphs ? , 1996 .

[105]  Marc Baumslag Cayley networks: a group-theoretic approach to the design and analysis of parallel networks , 1992 .

[106]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[107]  Selim G. Akl,et al.  Optimal Communication algorithms on Star Graphs Using Spanning Tree Constructions , 1995, J. Parallel Distributed Comput..

[108]  László Babai,et al.  Bounds on the diameter of Cayley graphs of the symmetric group , 1988, Journal of Algebraic Combinatorics.

[109]  Dharma P. Agrawal,et al.  Generalized Hypercube and Hyperbus Structures for a Computer Network , 1984, IEEE Transactions on Computers.

[110]  Franco P. Preparata,et al.  The cube-connected-cycles: A versatile network for parallel computation , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[111]  C. Berge Graphes et hypergraphes , 1970 .

[112]  Oded Goldreich,et al.  The Minimum-Length Generator Sequence Problem is NP-Hard , 1981, J. Algorithms.

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

[114]  Patrick Solé,et al.  Expanding and Forwarding , 1995, Discret. Appl. Math..