Introduction to Graph Theory (Dover Books on Advanced Mathematics) [Paperback]

Although it's an "introduction", this gem of a book ends up in some quite deep territory. Trudeau starts off with some basic definitions of set theory concepts and then moves forward to define graphs in those terms.

Concepts such as planarity, connectedness, polygonality and colourings are quickly and smoothly reached, and the back end of the book covers genuses (which I thought was pretty incongruous for an "introduction"). Proofs of the Five Colour Theorem and the Heawood Colouring Theorem are included, as well as demonstrations of Euler's Formulae and Kuratowski's Theorem.

Trudeau's style is completely non-indimidating and patient - almost conversational - and he conveys a real enjoyment of the subject. Non-mathematicians will be able to follow almost all of his arguments quite easily and, for this reason above all others, he deserves 5 stars.

P.S. I spotted quite a large howler towards the end of the book: the Four Colour Theorem is stated as having "just been proved" - it was proven in 1977, which goes to show how old this book is!

Having never encountered graph theory before, I decided to purchase this book. It is a delight to read, and progresses very gently through the subject. The author has targted this book at people who don't necessarily want to get bogged down with heavy math jargon, and any jargon delivered is introduced very nicely with great explanations.

The book is a small paperback so very transportable. A dedicated reader could probably swallow the contents of this book in a few days.

* Physical

This book's pages are in standard paperback paper and its graphs and text are in B&W. The binding is very good for a paperback and stands up to opening and closing the book due to reading the same pages repeatedly. And I do mean you will need to go over bits again and again.

* Topics

This PURE MATH book is a new topic to me, although I have read a good book before (See another of my reviews). So you can guess that I am not a expert in this domain of Pure Mathematics! The whole development in this book is to avoid a too steep increase in difficulty at any particular point in the development of the topic. It begins to explain from a educated level of a non too involved standpoint, that is geometry / functions and there potential isomorphism's. I was surprised that Graph Theory is nothing to do with graphing function's! (p 12) Its to do with collection of sets and inter - relationships of information. The topics worked into are as follows; Graphs, Planar Graphs, Euler's Formula, Platonic Graphs, Coloring, The Genus of a Graph, Euler Walks and Hamiltonian Walks, and some solutions and concluded with the famous 7 bridges of Konigsberg problem.

* The way to help understanding

The book is well filled with Definitions to help your quotations of information. You have to see the graphs as the text descriptions would be too clumsily copied by myself. To make you aware of the type of definitions that are peppered throughout this grand book, I have selected 2 out of many definitions to test your interests in this arena of pure Math.

(p 64, Definition 18) 'A graph is planar if it is isomorphic to the graph that has been drawn in a plane without edge-crossings. Otherwise it is a nonplanar.'

When learning this area, the need to check the graphs to carefully count of vertices and edges. Its helpful when learning to see the extraction of information from the graph. The fun is increased when two or more separate graphs are linked together to add connections to data, in a 'supergraphs'(part of a bigger graph) or 'subgraphs', (a smaller group within a graph). Although as the author explains, Euler avoided the requirement to draw what could be a VERY involved graph and then the need to count the vertices, the 'dots', and the 'edges', the lines between the dots, into a simple - to - handle equation. Its SO impressive when its realized the graphics could be amazingly grand and could have been very involved and to avoid drawing is a blessing!

* Development of Graph Theory, based on previous reading of this book

(Definition 21 'A 'walk' is a sequences A1 A2 A3...An, of not necessarily distinct vertices in which A1 is joined by an edge to A2 and is joined by an edge A3... and A(n-1) by an edge An'.

More fun involves a relationship of a isomorphic data in the form of a not necessarily distinct or obvious vertices and edges within a graph. The book later develops into 'Platonic' graphs solids that is greatly helped with the graphs showing the geometric figures, trust me you need the book to follow it! The great descriptive stuff is the 4 - color problems is described and broadly worked out using the Appel - Haken proof. (see page 127 - 139)

* Summary

This book is stiffer than some at this level, but the connectivity between concepts is denser. So the incline in difficulty is partially camouflaged. You have to stick with it and have some fun. You do not need to crank your mind through every bit, but its a fine book to learn from. In fact I am going to read it again and better my grasp of this topic!