A First Look at Graph Theory
This book is intended to be an introductory text for mathematics and computer science students at the second and third year levels in universities. It gives an introduction to the subject with sufficient theory for students at those levels, with emphasis on algorithms and applications.
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I have taught undergraduate graph theory from this book many times, and I very much like its level, exposition, and problem selection. It is neither the most basic nor the most advanced introduction I've seen, but probably somewhere on the low side of middle. I supplement it with excellent texts (Bondy & Murty, Diestel) that are available for free online.