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Handbook of discrete and combinatorial mathematics. 2nd revised and updated edition. (English) Zbl 1421.05001

Discrete Mathematics and Its Applications. Boca Raton, FL: CRC Press (ISBN 978-1-58488-780-5/hbk; 978-1-58488-781-2/ebook). xxiv, 1590 p. (2018).
This book, consisting of twenty chapters, provides a comprehensive reference textbook on discrete and combinatorial mathematics. The book begins with an overview of the fundamental concepts of discrete mathematics including set theory, logic and proof techniques. This is followed by a chapter on counting problems, permutations, combinations, partitions and Young tableaux, before moving on to sequences of numbers, generating functions and their use in recurrence relations, finite sums and limits of sequences. The fourth and fifth chapters then consider the fundamentals of number theory, including prime numbers, conguences, factorization and diophantine equations as well as the underlying general algebraic structures, groups, rings, fields and boolean algebras.
The next chapter consists of a detailed overview of linear algebra and includes the properties of vector spaces, matrix algebra, eigenvectors, linear systems and matrix decomposition. The seventh chapter introduces discrete probability theory, random variables, Markov chains, random walks and queueing theory. The next two chapters move on to graph theory and trees, where the concepts of graph isomorphiosm, colorings, planar graphs, topological and analytic graph theory and spanning trees are studied in depth.
The tenth chapter considers network flow models where techniques for their solution are presented and their links to combinatorial mathematics are demonstrated. This is followed by three slightly more advanced chapters which consider, respectively, partially ordered sets, combinatorial designs and discrete and combinatorial geometry.
The next three chapters study more practical problems in combinatorial mathematics, namely coding theory, cryptography and discrete optimization, where a large number of examples help the reader easily understand complex concepts. The last four chapters consider more recent developments in discrete and combinatorial mathematics, those around computational complexity, data structures, data mining and applications in discrete bioinformatics.
Throughout this very interesting book, a comprehensive exposition of the fundamental properties is presented as well as the major theorems with proofs which help educate the interested researcher on the importance and applicability of each topic. The structure of each chapter follows the same presentation and includes several facts and examples which make the book very easy to read. The book concludes with an interesting biography chapter and an index.

MSC:

05-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to combinatorics
06-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to ordered structures
11-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to number theory
68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to computer science
90-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to operations research and mathematical programming
94-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to information and communication theory
00A20 Dictionaries and other general reference works

Citations:

Zbl 1044.00002
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