Graph Theory is the branch of mathematics that studies graphs, structures consisting of vertices connected by edges, used to model pairwise relationships between objects. It examines properties such as connectivity, paths, cycles, colourings, matchings and flows, and classifies graphs by structure (for example trees, bipartite and planar graphs). Originating with Euler’s 1736 solution of the Seven Bridges of Konigsberg problem, it now underpins network analysis, optimisation and computer science. Graph algorithms are fundamental to routing, scheduling, social network analysis and the representation of knowledge.
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- A graph abstracts a system into vertices and the edges that connect them, which may be directed or undirected and may carry weights. This simple model captures a wide variety of situations, from transport networks and circuit layouts to dependencies between tasks and relationships between people or concepts.
- Core problems include finding shortest paths, determining connectivity, computing spanning trees, colouring vertices so that neighbours differ, and identifying maximum flows or matchings. Many of these have efficient algorithms, while others, such as finding a Hamiltonian cycle or the chromatic number, are computationally hard and motivate the study of complexity.
- Graph theory is foundational to computer science and data analysis. Knowledge graphs, social networks, recommendation systems and routing protocols all rest on graph models, and the spectral study of graphs connects the field to linear algebra and machine learning.