Discrete Structures and Their Interactions by Brown, Jason I.;

Short description:

With exercises and research problems, this text highlights the connections among various discrete structures, including graphs, directed graphs, hypergraphs, partial orders, finite topologies, and simplicial complexes. It also explores their relationships to classical areas of mathematics, such as linear and multilinear algebra, analysis, probability, logic, and topology. The author explains how discrete structures have important applications in many areas inside and outside of combinatorics. He also discusses how to recognize valuable research connections through the structures.

Long description:

Discover the Connections between Different Structures and Fields

Discrete Structures and Their Interactions highlights the connections among various discrete structures, including graphs, directed graphs, hypergraphs, partial orders, finite topologies, and simplicial complexes. It also explores their relationships to classical areas of mathematics, such as linear and multilinear algebra, analysis, probability, logic, and topology.

The text introduces a number of discrete structures, such as hypergraphs, finite topologies, preorders, simplicial complexes, and order ideals of monomials, that most graduate students in combinatorics, and even some researchers in the field, seldom experience. The author explains how these structures have important applications in many areas inside and outside of combinatorics. He also discusses how to recognize valuable research connections through the structures.

Intended for graduate and upper-level undergraduate students in mathematics who have taken an initial course in discrete mathematics or graph theory, this book shows how discrete structures offer new insights into the classical fields of mathematics. It illustrates how to use discrete structures to represent the salient features and discover the underlying combinatorial principles of seemingly unrelated areas of mathematics.

"The book is a collection of examples, each of which shows either how discrete structures interact with each other or how discrete structures interact with other parts of mathematics. … I am certain I will use some examples I found in this book when I teach combinatorics in the upcoming semester."
—Miklós Bóna, MAA Reviews, December 2013

Table of Contents:

Introduction. Discrete Structures—A Common Framework. Graphs and Directed Graphs. Preorders and Partial Orders. Hypergraphs. Complexes and Multicomplexes. Research Problems. Bibliography. Selected Solutions. Appendices. Index.