Brown J.I. Discrete Structures and Their Interactions

Springer, 2013. — 214 p.Imagine walking along a beach, looking for seashells. Each seashell you unearth is precious and exquisite, with its own unique beauty. Discovering the connections between different areas of mathematics is much the same. There are treasures to be found in applying one field to another in unexpected ways.
This text is intended for graduate and upper level undergraduate students in mathematics who have taken an initial course in discrete mathematics or graph theory. This 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 indeed, some researchers in the field) seldom experience. These discrete structures have important applications to many areas, both inside and outside combinatorics, and unless you aware of the structures, you will miss valuable connections that could be forged within your research.
Discrete structures can, from the right viewpoint, interact with one another, affording one the ability to use ideas and techniques that are “natural” in the second area to the first, often with striking results. On the other hand, we illustrate that discrete structures can be used sometimes even if one is interested in another seemingly unrelated area of mathematics or computer science, merely to represent the salient features and discover the underlying combinatorial principles. Finally, other areas of mathematics, such as linear and multilinear algebra, commutative algebra, topology, probability theory and logic, to name but a few, can have startling applications to problems exclusively within combinatorics, and this text highlights a few of such jewels.
This book does not attempt to delve deeply into any one area of discrete structures or to give the most recent or most involved applications in the research literature; the emphasis is on the connections between different structures and fields, and I hope that the examples chosen are accessible to most. This book is a leisurely stroll along the beach, looking for interesting approaches, ones that you might not find otherwise. I hope that after reading through this book, you will find some of your own.Discrete Structures - A Common Framework
Graphs and Directed Graphs
Preorders and Partial Orders
Hypergraphs
Complexes and Multicomplexes
Research Problems
A: Set Theory
B: Matrix Theory and Linear Algebra
C: Abstract Algebra
D: Probability
E: Topology
F: Logic