Graph theory is unusual for sitting squarely between pure mathematics and computer science: the same object — dots joined by lines — is both a rich source of elegant theorems and the backbone of practical algorithms for networks, scheduling, and optimization. That dual nature is why the reading can feel scattered; a proof-heavy text and an algorithms text can seem to be about different fields entirely.
The order that works establishes a shared foundation, then lets you lean toward either the algorithmic or the theoretical side, and finally reaches the advanced structures that unify them. Each book is chosen to build on a common core rather than leaving you to reconcile disconnected treatments.
The foundations
Start with Introduction to Graph Theory by West, a clear and comprehensive first course that balances intuition and rigor, or Graph theory by Diestel, the elegant modern standard that many consider the definitive graduate text. Graph theory by Harary is the classic older reference, terse but historically central. Working through one of the first two gives you the vocabulary — trees, connectivity, matchings, coloring — that every later book assumes.
Algorithms and optimization
Next, turn toward computation. Algorithm design by Kleinberg is a superb bridge, teaching graph algorithms in the context of general algorithmic thinking, and Graph algorithms is the focused classic on the algorithmic side. Network Flow is the definitive treatment of flow problems — a topic with enormous practical reach — and Combinatorial optimization extends the ideas into the optimization problems that graphs so often model. This stage connects the theory to problems you can actually solve on a computer.
Advanced theory
The final arc is where graph theory meets deeper mathematics. Algebraic graph theory studies graphs through linear algebra and group theory, and Spectral graph theory develops the powerful connection between a graph's structure and the eigenvalues of its matrices — ideas now central to machine learning and network science. Random graphs introduces the probabilistic theory of graphs, a beautiful and heavily applied area. These reward the foundation the earlier books built.
Read in this order and graph theory's two personalities become one flexible toolkit. Follow the full path to go from basic definitions to spectral and probabilistic methods.
