Discrete Structures and Their Interactions
Chapman and Hall/CRC – 2013 – 224 pages
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
Discrete Structures—A Common Framework
Properties, Parameters and Operations
Representations and Models
Graphs and Directed Graphs
Graphs and Directed Graphs as Models
Graphs and Other Branches of Mathematics
Preorders and Partial Orders
Finite Topologies and Preorders
Representing Preorders and Partial Orders
Complexes and Multicomplexes
Representations of Complexes and Multicomplexes
Applications of Complexes and Multicomplexes
Appendix A Set Theory
Appendix B Matrix Theory and Linear Algebra
Appendix C Abstract Algebra
Appendix D Probability
Appendix E Topology
Appendix F Logic
Jason I. Brown is a professor of mathematics at Dalhousie University. He received a Ph.D. from the University of Toronto and has written over 70 refereed articles. His research interests include graphs, hypergraphs, partial order, finite topologies, and simplicial complexes, with a focus on the applications of other fields of mathematics to discrete problems. His mathematical research that uncovered how the Beatles played the opening chord of "A Hard Day’s Night" was featured in various media, including NPR and BBC radio, Guitar Player Magazine, and the Wall Street Journal website.