A Treatise on Graph Theory

A Treatise on Graph Theory: Classical Foundations and Artificial Intelligence presents graph theory as a living mathematical discipline rather than a catalogue of isolated theorems.

Beginning with graphs, digraphs, degree, connectivity, trees, cycles, colouring and planarity, the book develops the classical foundations carefully before advancing through perfect graphs, extremal methods, random graphs, spectral theory, symmetry, algorithms and computational complexity. The later chapters extend this framework into graph embeddings, graph neural networks, learning-augmented algorithms, generative models, link prediction, graph limits and large-scale intelligent systems.

What distinguishes this treatise is the way artificial intelligence is integrated into the mathematics. Learning systems are introduced only where the underlying graph-theoretic structure has already been established, allowing modern methods to be understood as extensions, approximations or computational counterparts of classical ideas rather than replacements for them.

Definitions are developed before use, equations are placed within their derivations, major results are connected to their historical and mathematical context, and exercises are accompanied by worked solutions. The final chapters examine networks in science, technology and society, before turning to open problems at the frontier between exact mathematical structure and learned representation.

Written for advanced students, researchers, engineers and readers interested in the mathematical foundations of contemporary artificial intelligence, A Treatise on Graph Theory offers both a rigorous introduction to the classical subject and a modern account of why graph theory has become one of the central languages of intelligent systems.