Graph theory is the mathematical study of graphs — structures consisting of vertices (nodes) connected by edges (links) — originating in the 18th century with Euler's solution to the Seven Bridges of Königsberg problem. It provides essential foundations for modeling relationships in computer science, social networks, transportation systems, and biological networks. The field's power lies in its ability to represent complex interconnected systems as simple mathematical objects, enabling analysis through traversal algorithms, shortest path computations, and connectivity measures. A key distinction to internalize: edges can be directed (one-way arrows) or undirected (bidirectional connections), and graphs can be weighted (edges have costs) or unweighted — these properties fundamentally change which algorithms apply and what problems can be solved efficiently.

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