This book covers both theoretical and practical results for graph polynomials. Graph polynomials have been developed for measuring combinatorial graph invariants and for characterizing graphs. Various problems in pure and applied graph theory or discrete mathematics can be treated and solved efficiently by using graph polynomials. Graph polynomials have been proven useful areas such as discrete mathematics, engineering, information sciences, mathematical chemistry and related disciplines.

Graphs on Surfaces: Dualities, Polynomials, and Knots offers an accessible and comprehensive treatment of recent developments on generalized duals of graphs on surfaces, and their applications. The authors illustrate the interdependency between duality, medial graphs and knots; how this interdependency is reflected in algebraic invariants of graphs and knots; and how it can be exploited to solve problems in graph and knot theory. Taking a constructive approach, the authors emphasize how generalized duals and related ideas arise by localizing classical constructions, such as geometric duals and Tait graphs, and then removing artificial restrictions in these constructions to obtain full extensions of them to embedded graphs. The authors demonstrate the benefits of these generalizations to embedded graphs in chapters describing their applications to graph polynomials and knots. Graphs on Surfaces: Dualities, Polynomials, and Knots also provides a self-contained introduction to graphs on surfaces, generalized duals, topological graph polynomials, and knot polynomials that is accessible both to graph theorists and to knot theorists. Directed at those with some familiarity with basic graph theory and knot theory, this book is appropriate for graduate students and researchers in either area. Because the area is advancing so rapidly, the authors give a comprehensive overview of the topic and include a robust bibliography, aiming to provide the reader with the necessary foundations to stay abreast of the field. The reader will come away from the text convinced of advantages of considering these higher genus analogues of constructions of plane and abstract graphs, and with a good understanding of how they arise.

Graph Theory (as a recognized discipline) is a relative newcomer to Mathematics. The first formal paper is found in the work of Leonhard Euler in 1736. In recent years the subject has grown so rapidly that in today's literature, graph theory papers abound with new mathematical developments and significant applications.As with any academic field, it is good to step back occasionally and ask Where is all this activity taking us?, What are the outstanding fundamental problems?, What are the next important steps to take?. In short, Quo Vadis, Graph Theory?. The contributors to this volume have together provided a comprehensive reference source for future directions and open questions in the field.

The 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2006) was held on the island of Sotra close to the city of Bergen on the west coast of Norway. The workshop was organized by the Algorithms Research Group at the Department of Informatics, University of Bergen, and it took place from June 22 to June 24. The 78 participants of WG 2006 came from the universities and research institutes of 17 di?erent countries. The WG 2006 workshop continues the series of 31 previous WG workshops. Since 1975, WG has taken place 20 times in Germany, four times in The Neth- lands, twice in Austria as well as oncein France, in Italy, in Slovakia, in Switz- land and in the Czech Republic, and has now been held for the ?rst time in Norway. The workshop aims at uniting theory and practice by demonstrating how graph-theoretic concepts can be applied to various areas in computer s- ence, or by extracting new problems from applications. The goal is to present recent research results and to identify and explore directions of future research. The talks showed how recent researchresults from algorithmic graph theory can be used in computer science and which graph-theoreticquestions arisefrom new developments in computer science. There were two fascinating invited lectures by Hans Bodlaender (Utrecht, The Netherlands) and Tandy Warnow (Austin, US

The 34th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2008) took place in Van Mildert College at Durham University, UK, 30 June – 2 July 2008. The approximately 80 participants came from va- ous countries all over the world, among them Australia, Brazil, Canada, Chile, Czech Republic, France, Greece, Hungary,Israel, Italy, Japan, The Netherlands, Norway, Poland, Spain, Switzerland, UK and the USA. WG 2008 continued the series of 33 previous WG conferences. Since 1975, the WG conference has taken place 21 times in Germany, four times in The Netherlands, twice in Austria as well as once in Italy, Slovakia, Switzerland, the Czech Republic, France, Norway and now in the UK. The WG conference traditionally aims at uniting theory and practice by demonstrating how graph-theoretic concepts can be applied to various areas in computer science, or by extracting new problems from applications. The goal is to present recent researchresults and to identify and exploredirections of future research. The continuing interest in the WG conferences was re?ected in the number and quality of submissions; 76 papers were submitted and in an evaluation p- cess with four reports per submission, 30 papers were accepted by the Program Committee for the conference. Due to the high number of submissions and the limited schedule of 3 days, various good papers could not be accepted. Therewereexcellent invited talks by Giuseppe Di Battista(UniversitàRoma Tre,Italy)onalgorithmicaspectsof(un)-stableroutingintheInternet,byLeszek G?sieniec (University of Liverpool, UK) on memory-e?cient graph exploration, andbyMartinGrohe(Humboldt-UniversitätzuBerlin,Germany)onalgorithmic meta theorems.

"Precalculus is intended for college-level precalculus students. Since precalculus courses vary from one institution to the next, we have attempted to meet the needs of as broad an audience as possible, including all of the content that might be covered in any particular course. The result is a comprehensive book that covers more ground than an instructor could likely cover in a typical one- or two-semester course; but instructors should find, almost without fail, that the topics they wish to include in their syllabus are covered in the text. Many chapters of OpenStax College Precalculus are suitable for other freshman and sophomore math courses such as College Algebra and Trigonometry; however, instructors of those courses might need to supplement or adjust the material. OpenStax will also be releasing College Algebra and Algebra and trigonometry titles tailored to the particular scope, sequence, and pedagogy of those courses."--Preface.

The 35th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2009) took place at Montpellier (France), June 24–26 2009. About 80 computer scientists from all over the world (Australia, Belgium, Canada, China, Czech Republic, France, Germany, Greece, Israel, Japan, Korea, The Netherlands, Norway, Spain, UK, USA) attended the conference. Since1975,ithastakenplace20timesinGermany,fourtimesinTheNeth- lands, twice in Austria, as well as once in Italy, Slovakia, Switzerland, the Czech Republic, France, Norway, and the UK. The conference aims at uniting theory and practice by demonstrating how graph-theoretic concepts can be applied to various areas in computer science, or by extracting new problems from appli- tions. The goal is to present recent research results and to identify and explore directions of future research. The conference is well-balanced with respect to established researchers and young scientists. There were 69 submissions. Each submission was reviewed by at least three, and on average four, Program Committee members. The Committee decided to accept 28 papers. Due to the competition and the limited schedule, some good papers could not be accepted. Theprogramalsoincludedexcellentinvitedtalks:onegivenbyDanielKràlon “AlgorithmsforClassesofGraphswithBoundedExpansion,” the otherbyDavid Eppsteinon“Graph-TheoreticSolutionstoComputationalGeometryProblems.” The proceedings contains two survey papers on these topics.

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of “Graph Theory”. Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view.

The Tutte Polynomial touches on nearly every area of combinatorics as well as many other fields, including statistical mechanics, coding theory, and DNA sequencing. It is one of the most studied graph polynomials. Handbook of the Tutte Polynomial and Related Topics is the first handbook published on the Tutte Polynomial. It consists of thirty-four chapters written by experts in the field, which collectively offer a concise overview of the polynomial’s many properties and applications. Each chapter covers a different aspect of the Tutte polynomial and contains the central results and references for its topic. The chapters are organized into six parts. Part I describes the fundamental properties of the Tutte polynomial, providing an overview of the Tutte polynomial and the necessary background for the rest of the handbook. Part II is concerned with questions of computation, complexity, and approximation for the Tutte polynomial; Part III covers a selection of related graph polynomials; Part IV discusses a range of applications of the Tutte polynomial to mathematics, physics, and biology; Part V includes various extensions and generalizations of the Tutte polynomial; and Part VI provides a history of the development of the Tutte polynomial. Features Written in an accessible style for non-experts, yet extensive enough for experts Serves as a comprehensive and accessible introduction to the theory of graph polynomials for researchers in mathematics, physics, and computer science Provides an extensive reference volume for the evaluations, theorems, and properties of the Tutte polynomial and related graph, matroid, and knot invariants Offers broad coverage, touching on the wide range of applications of the Tutte polynomial and its various specializations

This book constitutes the refereed proceedings of the Second International Conference on Computability in Europe, CiE 2006, held in Swansea, UK, June/July 2006. The book presents 31 revised full papers together with 30 invited papers, including papers corresponding to 8 plenary talks and 6 special sessions on proofs and computation, computable analysis, challenges in complexity, foundations of programming, mathematical models of computers and hypercomputers, and Gödel centenary: Gödel's legacy for computability.