Locally Finite, Planar, Edge-Transitive Graphs

Locally Finite, Planar, Edge-Transitive Graphs PDF

Author: Jack E. Graver

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 89

ISBN-13: 0821805568

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The nine finite, planar, 3-connected, edge-transitive graphs have been known and studied for many centuries. The infinite, locally finite, planar, 3-connected, edge-transitive graphs can be classified according to the number of their end. The 1-ended graphs in this class were identified by Grünbaum and Shephard; Watkins characterized the 2-ended members. Any remaining graphs in this class must have uncountably may ends. In this work, infinite-ended members of this class are shown to exist. A more detailed classification scheme in terms of the types of Petrie walks in the graphs in this class and the local structure of their automorphism groups is presented.

Topics in Topological Graph Theory

Topics in Topological Graph Theory PDF

Author: Lowell W. Beineke

Publisher: Cambridge University Press

Published: 2009-07-09

Total Pages: 387

ISBN-13: 1139643681

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The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.

Graph Symmetry

Graph Symmetry PDF

Author: Gena Hahn

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 434

ISBN-13: 9401589372

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The last decade has seen two parallel developments, one in computer science, the other in mathematics, both dealing with the same kind of combinatorial structures: networks with strong symmetry properties or, in graph-theoretical language, vertex-transitive graphs, in particular their prototypical examples, Cayley graphs. In the design of large interconnection networks it was realised that many of the most fre quently used models for such networks are Cayley graphs of various well-known groups. This has spawned a considerable amount of activity in the study of the combinatorial properties of such graphs. A number of symposia and congresses (such as the bi-annual IWIN, starting in 1991) bear witness to the interest of the computer science community in this subject. On the mathematical side, and independently of any interest in applications, progress in group theory has made it possible to make a realistic attempt at a complete description of vertex-transitive graphs. The classification of the finite simple groups has played an important role in this respect.

Topological Graph Theory

Topological Graph Theory PDF

Author: Jonathan L. Gross

Publisher: Courier Corporation

Published: 2001-01-01

Total Pages: 386

ISBN-13: 0486417417

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Iintroductory treatment emphasizes graph imbedding but also covers connections between topological graph theory and other areas of mathematics. Authors explore the role of voltage graphs in the derivation of genus formulas, explain the Ringel-Youngs theorem, and examine the genus of a group, including imbeddings of Cayley graphs. Many figures. 1987 edition.

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders PDF

Author: Richard Warren

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 183

ISBN-13: 082180622X

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The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called k-connected set transitivity (k-CS-transitivity), are analysed in some detail. Classification in many of the interesting cases is given. This work generlizes Droste's classification of the countable k-transitive trees (k>1). In a CFPO, the structure can be branch downwards as well as upwards, and can do so repeatedely (though it neverr returns to the starting point by a cycle). Mostly it is assumed that k>2 and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behaviour.

The Finite Irreducible Linear 2-Groups of Degree 4

The Finite Irreducible Linear 2-Groups of Degree 4 PDF

Author: Dane Laurence Flannery

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 93

ISBN-13: 0821806254

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This memoir contains a complete classification of the finite irreducible 2-subgroups of GL(4, C). Specifically, the author provides a parametrized list of representatives for the conjugacy classes of such groups, where each representative is defined by generating a set of monomial matrices. The problem is treated by a variety of techniques, including: elementary character theory; a method for describing Hasse diagrams of submodule lattices; and calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups and Schur indices of their defining characters are also considered

Directions in Infinite Graph Theory and Combinatorics

Directions in Infinite Graph Theory and Combinatorics PDF

Author: R. Diestel

Publisher: Elsevier

Published: 2016-06-06

Total Pages: 392

ISBN-13: 148329479X

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This book has arisen from a colloquium held at St. John's College, Cambridge, in July 1989, which brought together most of today's leading experts in the field of infinite graph theory and combinatorics. This was the first such meeting ever held, and its aim was to assess the state of the art in the discipline, to consider its links with other parts of mathematics, and to discuss possible directions for future development. This volume reflects the Cambridge meeting in both level and scope. It contains research papers as well as expository surveys of particular areas. Together they offer a comprehensive portrait of infinite graph theory and combinatorics, which should be particularly attractive to anyone new to the discipline.

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies PDF

Author: Wolfgang Bulla

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 97

ISBN-13: 0821808087

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In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived. Features: Simple and unified treatment of the topic. Self-contained development. Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.

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