Finite geometry and combinatorics is the art of counting any phenomena that can be described by a diagram. Everyday life is full of applications; from telephones to compact disc players, from the transmission of confidential information to the codes on any item on supermarket shelves. This is a collection of thirty-five articles on covering topics such as finite projective spaces, generalized polygons, strongly regular graphs, diagram geometries and polar spaces. Included here are articles from many of the leading practitioners in the field including, for the first time, several distinguished Russian mathematicians. Many of the papers contain important new results and the growing use of computer algebra packages in this area is also demonstrated.

The aim of these notes is to link algorithmic problems arising in knot theory with statistical physics and classical combinatorics. Apart from the theory of computational complexity needed to deal with enumeration problems, introductions are given to several of the topics, such as combinatorial knot theory, randomized approximation models, percolation, and random cluster models.

This volume is comprised of the invited lectures given at the 14th British Combinatorial Conference. The lectures survey many topical areas of current research activity in combinatorics and its applications, and also provide a valuable overview of the subject, for both mathematicians and computer scientists.

This book is primarily aimed at graduate students and researchers in graph theory, combinatorics, or discrete mathematics in general. However, all the necessary graph theory is developed from scratch, so the only pre-requisite for reading it is a first course in linear algebra and a small amount of elementary group theory. It should be accessible to motivated upper-level undergraduates.

The Petersen graph occupies an important position in the development of several areas of modern graph theory because it often appears as a counter-example to important conjectures. In this account, the authors examine those areas, using the prominent role of the Petersen graph as a unifying feature. Topics covered include: vertex and edge colourability (including snarks), factors, flows, projective geometry, cages, hypohamiltonian graphs, and symmetry properties such as distance transitivity. The final chapter contains a mixture of other topics in which the Petersen graph has played its part. Undergraduate students should profit from reading this book as there are few prerequisite skills involved, and it could be used for a second course in graph theory. At the same time, the authors have also included a number of unsolved problems as well as topics of recent study. It should, therefore, also be useful as a reference for graph theorists.

First published in 1993, this thesis is concerned with the design of efficient algorithms for listing combinatorial structures. The research described here gives some answers to the following questions: which families of combinatorial structures have fast computer algorithms for listing their members? What general methods are useful for listing combinatorial structures? How can these be applied to those families which are of interest to theoretical computer scientists and combinatorialists? Amongst those families considered are unlabelled graphs, first order one properties, Hamiltonian graphs, graphs with cliques of specified order, and k-colourable graphs. Some related work is also included, which compares the listing problem with the difficulty of solving the existence problem, the construction problem, the random sampling problem, and the counting problem. In particular, the difficulty of evaluating Polya's cycle polynomial is demonstrated.

Combinatorics is a broad and important area of mathematics, and this textbook provides the beginner with the ideal introduction to many of the different aspects of the subject. By building up from the basics, and demonstrating the relationships between the various branches of combinatorics, Victor Bryant provides a readable text that presents its results in a straightforward way. Numerous examples and exercises, including hints and solutions, are included throughout and serve to lead the reader to some of the deeper results of the subject, many of which are usually excluded from elementary texts. This is an excellent textbook, by an experienced author, for introductory courses in combinatorics and graph theory.

The range of issues considered in graph drawing includes algorithms, graph theory, geometry, topology, order theory, graphic languages, perception, app- cations, and practical systems. Much research is motivated by applications to systems for viewing and interacting with graphs. The interaction between th- retical advances and implemented solutions is an important part of the graph drawing eld. The annually organized graph drawing symposium is a forum for researchers, practitioners, developers, and users working on all aspects of graph visualization and representations. The preceding symposia were held in M- treal (GD 98), Rome (GD 97), Berkeley (GD 96), Passau (GD 95), Princeton (GD 94), and Paris (GD 93). The Seventh International Symposium on Graph Drawing GD 99 was or- nized at Sti r n Castle, in the vicinity of Prague, Czech Republic. This baroque castle recently restored as a hotel and conference center provided a secluded place for the participants, who made good use of the working atmosphere of the conference. In total the symposium had 83 registered participants from 16 countries."

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations. The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics. In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too. Features The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems. An extensive bibliography and tables at the end make the book usable as a standard reference. Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.

Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems.

Enumerative Combinatorics presents elaborate and systematic coverage of the theory of enumeration. The first seven chapters provide the necessary background, including basic counting principles and techniques, elementary enumerative topics, and an extended presentation of generating functions and recurrence relations. The remaining seven chapters focus on more advanced topics, including, Stirling numbers, partitions of integers, partition polynomials, Eulerian numbers and Polya's counting theorem.

Extensively classroom tested, this text was designed for introductory- and intermediate-level courses in enumerative combinatorics, but the far-reaching applications of the subject also make the book useful to those in operational research, the physical and social science, and anyone who uses combinatorial methods. Remarks, discussions, tables, and numerous examples support the text, and a wealth of exercises-with hints and answers provided in an appendix--further illustrate the subject's concepts, theorems, and applications.

This book constitutes the thoroughly refereed post-proceedings of the 8th International Symposium on Graph Drawing, GD 2000, held in Colonial Williamsburg, VA, USA, in September 2000.The 36 revised full papers presented were carefully reviewed and selected from a total of 68 submissions. The book presents topical sections on empirical studies and standards, theory, application and systems, force-directed layout, k-level graph layout, orthogonal drawing, symmetry and incremental layout, and reports on a workshop on graph data formats and on the annual GD graph drawing contest.

This volume, the third in a sequence that began with The Theory of Matroids (1986) and Combinatorial Geometries (1987), concentrates on the applications of matroid theory to a variety of topics from geometry (rigidity and lattices), combinatorics (graphs, codes, and designs) and operations research (the greedy algorithm).

This volume contains the papers selected for presentation at IPCO VII, the Seventh Conference on Integer Programming and Combinatorial Optimization, Graz, Austria, June9{11,1999.Thismeetingisaforumforresearchersandpr- titioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, compu- tion, and applications of integer programming and combinatorial optimization. Topics include, but are not limited to: approximation algorithms, branch and bound algorithms, computational biology, computational complexity, compu- tional geometry, cutting plane algorithms, diophantine equations, geometry of numbers, graph and network algorithms, integer programming, matroids and submodular functions, on-line algorithms, polyhedral combinatorics, scheduling theory and algorithms, and semide nite programs. IPCO was established in 1988 when the rst IPCO program committee was formed. IPCO I took place in Waterloo (Canada) in 1990, IPCO II was held in Pittsburgh (USA) in 1992, IPCO III in Erice (Italy) 1993, IPCO IV in Cop- hagen (Denmark) 1995, IPCO V in Vancouver (Canada) 1996, and IPCO VI in Houston (USA) 1998. IPCO is held every year in which no MPS (Mathematical Programming Society) International Symposium takes place: 1990, 1992, 1993, 1995,1996,1998,1999,2001,2002,2004,2005,2007,2008: ::::: Since the MPS meeting is triennial, IPCO conferences are held twice in everythree-year period. As a rule, in even years IPCO is held somewhere in Northern America, and in odd years it is held somewhere in Europe. In response to the call for papers for IPCO 99, the program committee - ceived99submissions, indicatingastrongandgrowinginterestintheconfere

This text presents the salient features of the general theory of infinite electrical networks in a coherent exposition. Using the basic tools of functional analysis and graph theory, the author presents the fundamental developments of the past two decades and discusses applications to other areas of mathematics. The jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional to infinite-dimensional spaces. Many of the questions that are conventionally asked about finite networks are presently unanswerable for infinite networks, while questions that are meaningless for finite networks crop up for infinite ones and lead to surprising results, such as the occasional collapse of Kirchhoff's laws in infinite regimes. Some central concepts have no counterpart in the finite case, as for example the extremities of an infinite network, the perceptibility of infinity, and the connections at infinity.

Theareaofgraphtransformationoriginatedinthelate1960sunderthename "graph grammars" - the main motivation came from practical considerations concerning pattern recognition and compiler construction. Since then, the list of areas which have interacted with the development of graph transformation has grown impressively. The areas include: software speci?cation and development, VLSI layout schemes, database design, modeling of concurrent systems, m- sively parallel computer architectures, logic programming, computer animation, developmentalbiology, musiccomposition, distributedsystems, speci?cationl- guages, software and web engineering, and visual languages. As a matter of fact, graph transformation is now accepted as a fundamental computation paradigm where computation includes speci?cation, programming, and implementation. Over the last three decades the area of graph transfor- tion has developed at a steady pace into a theoretically attractive research ?eld, important for applications. Thisvolume consistsofpapersselectedfromcontributionsto the Sixth Int- national Workshop on Theory and Applications of Graph Transformation that took place in Paderborn, Germany, November 16-20, 1998. The papers und- went an additional refereeing process which yielded 33 papers presented here (out of 55 papers presented at the workshop). This collection of papers provides a very broad snapshot of the state of the art of the whole ?eld today. They are grouped into nine sections representing most active research areas. Theworkshopwasthe sixth in a seriesof internationalworkshopswhich take place every four years. Previous workshops were called "Graph Grammars and Their Application to Computer Science." The new name of the Sixth Workshop re?ectsmoreaccuratelythecurrentsituation, whereboththeoryandapplication play an equally central role.

This book stresses the connection between, and the applications of, design theory to graphs and codes. Beginning with a brief introduction to design theory and the necessary background, the book also provides relevant topics for discussion from the theory of graphs and codes.

Although graph theory, design theory, and coding theory had their origins in various areas of applied mathematics, today they are to be found under the umbrella of discrete mathematics. Here the authors have considerably reworked and expanded their earlier successful books on graphs, codes and designs, into an invaluable textbook. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. The discrete mathematics needed is developed in the text, making this book accessible to any student with a background of undergraduate algebra. Many exercises and useful hints are included througout, and a large number of references are given.

This volume contains nine invited papers that survey many areas of current research in combinatorics both on the theoretical and practical side. Several papers may be regarded as summarizing our present state of knowledge in a particular topic.

Graphdrawingaddressestheproblemofconstructingrepresentationsofabstract graphs, networks, and hypergraphs. The 6th Symposium on Graph Drawing (GD '98) was held August 13{15, 1998, atMcGillUniversity, Montr eal, Canada.ItimmediatelyfollowedtheTenth Canadian Conference on Computational Geometry (CCCG '98), held August 10{12 at McGill. The GD '98 conference attracted 100 paid registrants from academic and industrial institutions in thirteen countries. Roughly half the p- ticipantsalsoattendedCCCG'98.Asinthepast, interactionamongresearchers, practitioners, andstudents fromtheoreticalcomputer science, mathematics, and the application areas of graph drawing continued to be an important aspect of the graph drawing symposium. In response to the call for papers and system demonstrations, the program committee received 57 submissions, of which 10 were demos. Each submission was reviewed by at least 4 members of the program committee, and comments were returnedto the authors.Following extensive email discussions andmultiple rounds of voting, the program committee accepted 23 papers and 9 demos. GD '98 also held an unrefereed poster gallery. The poster gallery contained 16 posters, 14 of which have abstracts in this volume. The poster gallery served to encourageparticipationfromresearchersinrelatedareasandprovidedast- ulating environment for the breaks between the technical sessions. In keeping with the tradition of previous graph drawing conferences, GD '98 held a graph drawing contest. This contest, which is traditionally a conference highlight, servestomonitorandtochallengethestateoftheartingraphdrawing. A report on the 1998 contest appears in this volume.

Several geometric problems can be formulated in terms of the arrangement of a collection of curves in a plane, which has made this one of the most widely studied topics in computational geometry. This book, first published in 1991, presents a study of various problems related to arrangements of lines, segments, or curves in the plane. The first problem is a proof of almost tight bounds on the length of (n,s)-Davenport-Schinzel sequences, a technique for obtaining optimal bounds for numerous algorithmic problems. Then the intersection problem is treated. The final problem is improving the efficiency of partitioning algorithms, particularly those used to construct spanning trees with low stabbing numbers, a very versatile tool in solving geometric problems. A number of applications are also discussed. Researchers in computational and combinatorial geometry should find much to interest them in this book.

This book, suitable for graduate students and professional mathematicians alike, didactically introduces methodologies due to Furstenberg and others for attacking problems in chromatic and density Ramsey theory via recurrence in topological dynamics and ergodic theory, respectively. Many standard results are proved, including the classical theorems of van der Waerden, Hindman, and Szemer di. More importantly, the presentation strives to reflect the extent to which the field has been streamlined since breaking onto the scene around twenty years ago. Potential readers who were previously intrigued by the subject matter but found it daunting may want to give a second look.

The National Security Agency funded a conference on Coding theory, Cryp- tography, and Number Theory (nick-named Cryptoday) at the United States Naval Academy, on October 25-27, 1998. We were very fortunate to have been able to attract talented mathematicians and cryptographers to the meeting. Unfortunately, some people couldn't make it for either scheduling or funding reasons. Some of these have been invited to contribute a paper anyway. In addition, Prof. William Tutte and Frode Weierud have been kind enough to allow the inclusion of some very interesting unpublished papers of theirs. The papers basically fall into three catagories. Historical papers on cryp- tography done during World War II (Hatch, Hilton, Tutte, Ulfving, and Weierud), mathematical papers on more recent methods in cryptography (Cosgrave, Lomonoco, Wardlaw), and mathematical papers in coding theory (Gao, Joyner, Michael, Shokranian, Shokrollahi). A brief biography of the authors follows. - Peter Hilton is a Distinguished Professor of Mathematics Emeritus at the State University of New York at Binghamton. He worked from 1941 to 1945 in the British cryptanalytic headquarters at Bletchley Park. Profes- sor Hilton has done extensive research in algebraic topology and group theory. - William Tutte is a Distinguished Professor Emeritus and an Adjunct Pro- fessor in the Combinatorics and Optimization Department at the Univer- sity of Waterloo. He worked from 1941 to 1945 in the British cryptana- lytic headquarters at Bletchley Park. Professor Tutte has done extensive research in the field of combinatorics.

Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, from a summer school in Thurnau, aim to give an introduction to these topics, and to describe recent progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersection theory and algebraic geometry.

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."