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Configurations from a Graphical Viewpoint
Tomaž Pisanski and Brigitte Servatius
Publisher: Birkhäuser (2013)
Details: 279 pages, Hardcover
Series: Birkhäuser Advanced Texts
Topics: Graph Theory, Combinatorics, Combinatorial Geometry
MAA Review[Reviewed by Miklós Bóna, on 01/02/2013]
This is a very reader-friendly and informative book, which looks at various areas of mathematics through the lense of graphs. First, the authors grab our attention by discussing a classical theorem of the 16-year-old Blaise Pascal. This theorem states that if the vertices of a hexagon are on the boundary of a conic, then the three points in which pairs of opposite sides meet will be on the same line. This introductory example leads to very entertaining exercises, such as “Plant nine trees in ten rows so that each row has three trees.”
Because the whole book is about graphs, the authors continue with an introduction to graph theory. Even this part contains material that will be new to most readers, such as generalized Petersen graphs.
The main part of the book is divided into four chapters. The first one, devoted to occurrences of graphs in algebra, introduces the notion of groups through automorphisms of graphs. Cayley graphs, permutation groups, and various notions of transitive group action are discussed.
The next chapter focuses on graphs in topology. Perhaps this is the part of the book that is furthest away from the usual perspectives on graphs, namely combinatorics. Still, even here a combinatorialist will find applications of some familiar theorems, such as Euler’s theorem on planar graphs.
The last two chapters are devoted to various configurations, many of which are classical, such as the Fano plane. In Chapter 5, they are treated in a purely combinatorial way, while in Chapter 6, Geometry is also involved.
The book is full of beautiful theorems, and I kept wishing that I knew all of them. There are plenty of exercises, some of which are quite amusing, as I mentioned above. Too bad that none have their solutions included in the book. Based on the diverse areas that the authors cover, this reviewer believes that there will be a broad range of readers who will enjoy at least one chapter of the book.
Miklós Bóna is Professor of Mathematics at the University of Florida.