As the title suggests, this book addresses the mathematics and theory of hypergraphs. The target audience includes graduate students and researchers with an interest in math and computer science (CS).

Graph theory has served as the foundation of chemistry and CS for decades. While graph theory advances the theory of sciences, researchers find that graphs can be generalized to hypergraphs. In contrast to a graph, where an edge connects two vertices only, a hypergraph contains a set of hyperedges where each hyperedge can connect a number of vertices. This unique feature gives hypergraphs different properties than graphs, and scientists expect the development of hypergraphs to further advance modern sciences such as chemistry, CS, and network science.

This book consists of seven chapters. Chapter 1 introduces the basic notions of hypergraphs. Chapter 2 contrasts hypergraphs with graphs and teaches some basic properties of hypergraphs, such as subtrees, the König property, and linear spaces. Chapter 3 addresses the coloring problem in hypergraphs.

Chapter 4 introduces variations of hypergraphs. The author describes various types of hypergraphs, such as interval hypergraphs, unimodular hypergraphs, balanced hypergraphs, planar hypergraphs, and normal hypergraphs. Chapter 5 explores generic algorithms and spanning tree algorithms. One important feature of graph theory is in the development of algorithms. Novel algorithms can assist investigations of chemical compounds and facilitate the design of computational methods. Similarly, we can develop algorithms on hypergraphs. Although there are plenty of algorithms in graph theory, not many have been developed for hypergraphs because hypergraph theory is still quite new.

Chapter 6 explores the quite recent topic of directed hypergraphs (dirhypergraphs). This field has not yet stabilized. However, its importance is surging because many of the theorems in graph theory can be extended to dirhypergraphs.

Chapter 7 concludes the book with examples of hypergraphs. Hypergraphs have not been applied to solving real-world problems. Nevertheless, the author introduces how to use the concepts of hypergraphs to describe problems. Examples include telecommunications, parallel data structure, databases, and image processing. Although this chapter is short, the author provides motivation for future research on hypergraphs.

Hypergraph theory is a hard science and a topic in pure mathematics. Fortunately, the author introduces the theory step by step, so the reader does not get lost in the middle of reading. I expect readers of this book will be motivated to advance this field, which in turn can advance other sciences.