Kombinatorik og grafteori

This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own.Studies in spectral graph theory seek to determine properties of a graph through matrices associated with it. It turns out that eigenvalues and eigenvectors have surprisingly many connections with the structure of a graph. This book approaches this subject under the perspective of eigenvalue location algorithms. These are algorithms that, given a symmetric graph matrix M and a real interval I, return the number of eigenvalues of M that lie in I. Since the algorithms described here are typically very fast, they allow one to quickly approximate the value of any eigenvalue, which is a basic step in most applications of spectral graph theory. Moreover, these algorithms are convenient theoretical tools for proving bounds on eigenvalues and their multiplicities, which was quite useful to solve longstanding open problems in the area. This book brings these algorithms together, revealing how similar they are in spirit, and presents some of their main applications.This work can be of special interest to graduate students and researchers in spectral graph theory, and to any mathematician who wishes to know more about eigenvalues associated with graphs. It can also serve as a compact textbook for short courses on the topic.

"e;Can one hear the shape of a drum?"e; This striking question, made famous by Mark Kac, conceals a precise mathematical problem, whose study led to sophisticated mathematics. This textbook presents the theory underlying the problem, for the first time in a form accessible to students.Specifically, this book provides a detailed presentation of Sunada's method and the construction of non-isometric yet isospectral drum membranes, as first discovered by Gordon-Webb-Wolpert. The book begins with an introductory chapter on Spectral Geometry, emphasizing isospectrality and providing a panoramic view (without proofs) of the Sunada-Berard-Buser strategy. The rest of the book consists of three chapters. Chapter 2 gives an elementary treatment of flat surfaces and describes Buser's combinatorial method to construct a flat surface with a given group of isometries (a Buser surface). Chapter 3 proves the main isospectrality theorems and describes the transplantation technique on Buser surfaces. Chapter 4 builds Gordon-Webb-Wolpert domains from Buser surfaces and establishes their isospectrality.Richly illustrated and supported by four substantial appendices, this book is suitable for lecture courses to students having completed introductory graduate courses in algebra, analysis, differential geometry and topology. It also offers researchers an elegant, self-contained reference on the topic of isospectrality.

The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its applications. The research work of each team was led by two experts in the corresponding area, who prior to the workshop, carefully selected relevant and meaningful open problems that would yield high-quality research and results of strong impact. As a result, all six teams have made significant contributions to several open problems in their respective areas. The workshop led to the creation of the Women in Graph Theory and Applications Research Collaboration Network, which provided the framework to continue collaborating and to produce this volume.This book contains six chapters, each of them on one of the different areas of research at the Workshop for Women in Graph Theory and Applications, and written by participants of each team.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras. This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

You don't need to buy expensive statistical software like SPSS. This book teaches you R (R can be downloaded for free), People Analytics, Social Media Analytics, Text Mining and Sentiment Analysis. It is written for people with absolutely NO knowledge of R programming, with step-by-step print-screen instructions. The sample R codes are kept simple, short & straightforward so that you are not overwhelmed with too much unnecessary information, and focuses on the R codes relevant to people analytics, so that you'll be up-and-running in no time. If you are new to R programming, this is the book for you. As R is developed specially for statistical analysis, you can run complicated statistical number crunching (Correlation, Multiple & Logistic Regression, etc.) by simply entering a few commands. This book covers a wide People Analytics scope (Benefits, Compensation, Culture, Diversity & Inclusion, Engagement, Leadership, Learning & Development, Personality Traits, Performance Management, Recruitment, Sales Incentives) with numerous real-world examples, and shows how People Analytics with R can help you: 1) Run Social Media Analytics, Text mining & Sentiment Analysis with R. 2) Predict employees' flight-risk using R's Correlation & Logistic Regression function. 3) Identify the personality traits of top performing Customer Service staff and Sales staff using R's correlation function.4) Predict impact of Employee Engagement on Customer Satisfaction, Revenue and Shareholder Returns, etc. using R's Correlation & Multiple Regression function. 5) Predict impact of Learning & Development on Sales, using R's Multiple Regression function.6) Predict Diversity & Inclusion's impact on Revenue and EBIT using R's Multiple Regression function. You'll be taught how to convert your company's ethnicity diversity mix (70% Chinese: 20% Malay: 10% Indian) to an index number (e.g. diversity index of 2.7), then use R's Multiple Regression to predict your company sales, if your company's "diversity Index" is 2.7 and if you spend $90 on advertising.

Many applications in different domains need to calculate the shortest-path between two points in a graph. In this paper we describe this shortest path problem in detail, starting with the classic Dijkstra's algorithm and moving to more advanced solutions that are currently applied to road network routing, including the use of heuristics and precomputation techniques. Since several of these improvements involve subtle changes to the search space, it may be difficult to appreciate their benefits in terms of time or space requirements. To make methods more comprehensive and to facilitate their comparison, this book presents a single case study that serves as a common benchmark. The paper also compares the search spaces explored by the methods described, both from a quantitative and qualitative point of view, and including an analysis of the number of reached and settled nodes by different methods for a particular topology.Table of Contents: List of Figures / List of Tables / Acknowledgments / Introduction / Graph Theory Basics / Classical Algorithms / Hierarchical Preprocessing-Dependent Approaches / Non-Hierarchical Preprocessing-Dependent Approaches / Analysis and Comparison of Approaches / Conclusions / Bibliography / Authors' Biographies

Graph theory, being a rigorously investigated field of combinatorial mathematics, is adopted by a wide variety of disciplines addressing a plethora of real-world applications. Advances in graph algorithms and software implementations have made graph theory accessible to a larger community of interest. Ever-increasing interest in machine learning and model deployments for network data demands a coherent selection of topics rewarding a fresh, up-to-date summary of the theory and fruitful applications to probe further. This volume is a small yet unique contribution to graph theory applications and modeling with graphs. The subjects discussed include information hiding using graphs, dynamic graph-based systems to model and control cyber-physical systems, graph reconstruction, average distance neighborhood graphs, and pure and mixed-integer linear programming formulations to cluster networks.

A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant and the connection between combinatorics and topology often arises as an unexpected surprise.The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry. The text contains a large number of figures that support the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given, and each chapter ends with a series of exercises. The extensive appendix makes the book completely self-contained.The textbook is well suited for advanced undergraduate or beginning graduate mathematics students. Previous knowledge in topology or graph theory is helpful but not necessary. The text may be used as a basis for a one- or two-semester course as well as a supplementary text for a topology or combinatorics class.

This SpringerBrief provides the first systematic review of the existing works of cohesive subgraph search (CSS) over large heterogeneous information networks (HINs). It also covers the research breakthroughs of this area, including models, algorithms and comparison studies in recent years. This SpringerBrief offers a list of promising future research directions of performing CSS over large HINs.The authors first classify the existing works of CSS over HINs according to the classic cohesiveness metrics such as core, truss, clique, connectivity, density, etc., and then extensively review the specific models and their corresponding search solutions in each group. Note that since the bipartite network is a special case of HINs, all the models developed for general HINs can be directly applied to bipartite networks, but the models customized for bipartite networks may not be easily extended for other general HINs due to their restricted settings. The authors also analyze and compare these cohesive subgraph models (CSMs) and solutions systematically. Specifically, the authors compare different groups of CSMs and analyze both their similarities and differences, from multiple perspectives such as cohesiveness constraints, shared properties, and computational efficiency. Then, for the CSMs in each group, the authors further analyze and compare their model properties and high-level algorithm ideas.This SpringerBrief targets researchers, professors, engineers and graduate students, who are working in the areas of graph data management and graph mining. Undergraduate students who are majoring in computer science, databases, data and knowledge engineering, and data science will also want to read this SpringerBrief.

Combinatorial optimization is a fascinating topic. Combinatorial optimization problems arise in a wide variety of important fields such as transportation, telecommunications, computer networking, location, planning, distribution problems, etc. Important and significant results have been obtained on the theory, algorithms and applications over the last few decades. In combinatorial optimization, many network design problems can be generalized in a natural way by considering a related problem on a clustered graph, where the original problem's feasibility constraints are expressed in terms of the clusters, i.e., node sets instead of individual nodes. This class of problems is usually referred to as generalized network design problems (GNDPs) or generalized combinatorial optimization problems. The express purpose of this monograph is to describe a series of mathematical models, methods, propositions, algorithms developed in the last years on generalized network design problems in a unified manner. The book consists of seven chapters, where in addition to an introductory chapter, the following generalized network design problems are formulated and examined: the generalized minimum spanning tree problem, the generalized traveling salesman problem, the railway traveling salesman problem, the generalized vehicle routing problem, the generalized fixed-charge network design problem and the generalized minimum vertex-biconnected network problem. The book will be useful for researchers, practitioners, and graduate students in operations research, optimization, applied mathematics and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will find this book useful, too.

Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Concisely written, gentle introduction to graph theory suitable as a textbook or for self-studyGraph-theoretic applications from diverse fields (computer science, engineering, chemistry, management science)2nd ed.