Author: János Pach Publisher: American Mathematical Soc. ISBN: 0821834843 Category : Mathematics Languages : en Pages : 283
Book Description
The early development of graph theory was heavily motivated and influenced by topological and geometric themes, such as the Konigsberg Bridge Problem, Euler's Polyhedral Formula, or Kuratowski's characterization of planar graphs. In 1936, when Denes Konig published his classical ""Theory of Finite and Infinite Graphs"", the first book ever written on the subject, he stressed this connection by adding the subtitle Combinatorial Topology of Systems of Segments. He wanted to emphasize that the subject of his investigations was very concrete: planar figures consisting of points connected by straight-line segments. However, in the second half of the twentieth century, graph theoretical research took an interesting turn. In the most popular and most rapidly growing areas (the theory of random graphs, Ramsey theory, extremal graph theory, algebraic graph theory, etc.), graphs were considered as abstract binary relations rather than geometric objects.Many of the powerful techniques developed in these fields have been successfully applied in other areas of mathematics. However, the same methods were often incapable of providing satisfactory answers to questions arising in geometric applications. In the spirit of Konig, geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edges (or more generally, by edges represented by simple Jordan arcs). It is an emerging discipline that abounds in open problems, but it has already yielded some striking results which have proved instrumental in the solution of several basic problems in combinatorial and computational geometry. The present volume is a careful selection of 25 invited and thoroughly refereed papers, reporting about important recent discoveries on the way Towards a Theory of Geometric Graphs.
Author: László Lovász Publisher: American Mathematical Soc. ISBN: 1470450879 Category : Geometry Languages : en Pages : 444
Book Description
Graphs are usually represented as geometric objects drawn in the plane, consisting of nodes and curves connecting them. The main message of this book is that such a representation is not merely a way to visualize the graph, but an important mathematical tool. It is obvious that this geometry is crucial in engineering, for example, if you want to understand rigidity of frameworks and mobility of mechanisms. But even if there is no geometry directly connected to the graph-theoretic problem, a well-chosen geometric embedding has mathematical meaning and applications in proofs and algorithms. This book surveys a number of such connections between graph theory and geometry: among others, rubber band representations, coin representations, orthogonal representations, and discrete analytic functions. Applications are given in information theory, statistical physics, graph algorithms and quantum physics. The book is based on courses and lectures that the author has given over the last few decades and offers readers with some knowledge of graph theory, linear algebra, and probability a thorough introduction to this exciting new area with a large collection of illuminating examples and exercises.
Author: Stefan Felsner Publisher: Springer Science & Business Media ISBN: 3322803031 Category : Mathematics Languages : en Pages : 179
Book Description
Among the intuitively appealing aspects of graph theory is its close connection to drawings and geometry. The development of computer technology has become a source of motivation to reconsider these connections, in particular geometric graphs are emerging as a new subfield of graph theory. Arrangements of points and lines are the objects for many challenging problems and surprising solutions in combinatorial geometry. The book is a collection of beautiful and partly very recent results from the intersection of geometry, graph theory and combinatorics.
Author: János Pach Publisher: Springer Science & Business Media ISBN: 1461401100 Category : Mathematics Languages : en Pages : 610
Book Description
In many applications of graph theory, graphs are regarded as geometric objects drawn in the plane or in some other surface. The traditional methods of "abstract" graph theory are often incapable of providing satisfactory answers to questions arising in such applications. In the past couple of decades, many powerful new combinatorial and topological techniques have been developed to tackle these problems. Today geometric graph theory is a burgeoning field with many striking results and appealing open questions. This contributed volume contains thirty original survey and research papers on important recent developments in geometric graph theory. The contributions were thoroughly reviewed and written by excellent researchers in this field.
Author: Mathew Penrose Publisher: Oxford University Press ISBN: 0198506260 Category : Mathematics Languages : en Pages : 345
Book Description
This monograph provides and explains the mathematics behind geometric graph theory, which studies the properties of a graph that consists of nodes placed in Euclidean space so that edges can be added to connect points that are close to one another. For example, a collection of trees scattered in a forest and the disease that is passed between them, a set of nests of animals or birds on a region and the communication between them or communication between communications stations or nerve cells. Aimed at graduate students and researchers in probability, statistics, combinatorics and graph theory including computer scientists, it covers topics such as: technical tools, edge and component counts, vertex degrees, clique and chromatic number, and connectivity. Applications of this theory are used in the study of neural networks, spread of disease, astrophysics and spatial statistics.
Author: Matthias Keller Publisher: Cambridge University Press ISBN: 1108587380 Category : Mathematics Languages : en Pages : 493
Book Description
This book addresses the interplay between several rapidly expanding areas of mathematics. Suitable for graduate students as well as researchers, it provides surveys of topics linking geometry, spectral theory and stochastics.
Author: László Lovász Publisher: Springer Science & Business Media ISBN: 3642392865 Category : Mathematics Languages : en Pages : 720
Book Description
Paul Erdös was one of the most influential mathematicians of the twentieth century, whose work in number theory, combinatorics, set theory, analysis, and other branches of mathematics has determined the development of large areas of these fields. In 1999, a conference was organized to survey his work, his contributions to mathematics, and the far-reaching impact of his work on many branches of mathematics. On the 100th anniversary of his birth, this volume undertakes the almost impossible task to describe the ways in which problems raised by him and topics initiated by him (indeed, whole branches of mathematics) continue to flourish. Written by outstanding researchers in these areas, these papers include extensive surveys of classical results as well as of new developments.
Author: Peter Brass Publisher: Springer Science & Business Media ISBN: 0387238158 Category : Mathematics Languages : en Pages : 507
Book Description
This book is the result of a 25-year-old project and comprises a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.
Author: Clara Löh Publisher: Springer ISBN: 3319722549 Category : Mathematics Languages : en Pages : 389
Book Description
Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.
Author: Paul B. Larson Publisher: American Mathematical Soc. ISBN: 1470454629 Category : Education Languages : en Pages : 330
Book Description
This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo–Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.
Author: János Pach
Author: László Lovász
Author: Stefan Felsner
Author: János Pach
Author: Mathew Penrose
Author: Matthias Keller
Author: László Lovász
Author: Peter Brass
Author: Clara Löh
Author: Paul B. Larson