De Gruyter Series in Discrete Mathematics and Applications
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Edited by:
Maria Chudnovsky
This series is devoted to the publication of high-level monographs which cover the whole spectrum of current discrete mathematics and its applications in various fields. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of discrete mathematics. Contributions which are on the borderline of discrete mathematics and related fields and which stimulate further research at the crossroads of these areas are particularly welcome.
Author / Editor information
Topics
Algebraic combinatorics is the study of combinatorial objects as an extension of the study of finite permutation groups, or, in other words, group theory without groups. In the spirit of Delsarte's theory, this book studies combinatorial objects such as graphs, codes, designs, etc. in the general framework of association schemes, providing a comprehensive overview of the theory as well as pointing out to extensions.
Written for mathematicians working with the theory of graph spectra, this (primarily theoretical) book presents relevant results considering the spectral properties of regular graphs. The book begins with a short introduction including necessary terminology and notation. The author then proceeds with basic properties, specific subclasses of regular graphs (like distance-regular graphs, strongly regular graphs, various designs or expanders) and determining particular regular graphs. Each chapter contains detailed proofs, discussions, comparisons, examples, exercises and also indicates possible applications. Finally, the author also includes some conjectures and open problems to promote further research.
Contents
Spectral properties
Particular types of regular graph
Determinations of regular graphs
Expanders
Distance matrix of regular graphs
This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art.
The monograph presents a well-balanced combination of
- a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis),
- clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals),
- ready-to-use computer packages.
A result is the solution in practice of a large general class of Diophantine equations.
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.
