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The second edition of this well-received textbook is devoted to Combinatorics and Graph Theory, which are cornerstones of Discrete Mathematics. Every section begins with simple model problems. Following their detailed analysis, the reader is led through the derivation of definitions, concepts, and methods for solving typical problems. Theorems then are formulated, proved, and illustrated by more problems of increasing difficulty.
Preface to the second edition
Preface to the first edition
Introductory combinatorics and graph theory
Basic counting
Combinatorics of finite sets
The sum and product rules
Arrangements and permutations
Combinations
Permutations with identified elements
Probability theory on finite sets
Basic graph theory
Vocabulary
Connectivity in graphs
Trees
Eulerian graphs
Planarity
Graph coloring
Hierarchical clustering and dendrogram graphs
Introduction
Model example
Hubert’s single-link algorithm
Hubert’s complete-link algorithm
Case study
Combinatorial analysis
Enumerative combinatorics
The inclusion–exclusion principle
Inversion formulas
Generating functions I. Introduction
Generating functions II. Applications
Enumeration of equivalence classes
Existence theorems in combinatorics
Ramsey’s theorem
Systems of distinct representatives
Block designs
Systems of triples
Secondary structures of the RNA
RNAs, graphs, and the Cauchy–Hadamard formula
Counting the primary structures
Diagrams
Secondary structures
Asymptotic enumeration of the secondary structures. Examples
Answers/solutions to selected problems
Bibliography
Index