scholarly journals Cubic Cayley graphs with small diameter.

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Eugene Curtin
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Permutation Groups ◽  
Large Graphs ◽  
Polya's Theorem ◽  
International Audience ◽  
Pólya’S Theorem

International audience In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.

scholarly journals Equivalence classes of functions on finite sets

1982 ◽  
Vol 5 (4) ◽  
pp. 745-762
Author(s):  
Chong-Yun Chao ◽  
Caroline I. Deisher
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Permutation Groups ◽  
Weak Equivalence ◽  
Finite Sets ◽  
The Family ◽  
Polya's Theorem ◽  
Finite Set ◽  
Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

scholarly journals Borel Cayley Graph-Based Topology Control for Consensus Protocol in Wireless Sensor Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Junghun Ryu ◽  
Jaewook Yu ◽  
Eric Noel ◽  
K. Wendy Tang
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Interconnection Networks ◽  
Topology Control ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Wireless Sensor ◽  
Control Algorithms ◽  
Transmission Range ◽  
Consensus Protocol

Borel Cayley graphs have been shown to be an efficient candidate topology in interconnection networks due to their small diameter, short path length, and low degree. In this paper, we propose topology control algorithms based on Borel Cayley graphs. In particular, we propose two methods to assign node IDs of Borel Cayley graphs as logical topologies in wireless sensor networks. The first one aims at minimizing communication distance between nodes, while the entire graph is imposed as a logical topology; while the second one aims at maximizing the number of edges of the graph to be used, while the network nodes are constrained with a finite radio transmission range. In the latter case, due to the finite transmission range, the resultant topology is an “incomplete” version of the original BCG. In both cases, we apply our algorithms in consensus protocol and compare its performance with that of the random node ID assignment and other existing topology control algorithms. Our simulation indicates that the proposed ID assignments have better performance when consensus protocols are used as a benchmark application.

scholarly journals Some remarks concerning harmonic functions on homogeneous graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Cayley Graphs ◽  
Group Cohomology ◽  
Radial Variation ◽  
International Audience

International audience We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

On a generalization of Pólya’s theorem

2007 ◽  
Vol 81 (1-2) ◽  
pp. 247-259 ◽  
Author(s):  
I. P. Rochev
Keyword(s):  
Polya's Theorem ◽  
Pólya’S Theorem

On Pólya's Theorem

1967 ◽  
Vol 19 ◽  
pp. 792-799 ◽  
Author(s):  
J. Sheehan
Keyword(s):  
Finite Groups ◽  
Scalar Product ◽  
Combinatorial Analysis ◽  
Simple Extension ◽  
Group Characters ◽  
Polya's Theorem ◽  
Representations Of Finite Groups ◽  
Pólya’S Theorem ◽  
Special Case ◽  
Superposition Theorem

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

2001 ◽  
Vol 33 (6) ◽  
pp. 653-661 ◽  
Author(s):  
CAI HENG LI ◽  
CHERYL E. PRAEGER
Keyword(s):  
Wreath Product ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Permutation Groups ◽  
General Study ◽  
Product Action ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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