Graphs are fundamental data structures modeling relationships between entities — consisting of vertices (nodes) connected by edges. They appear throughout computer science, from social networks and routing algorithms to dependency analysis and circuit design. Graphs can be directed or undirected, weighted or unweighted, cyclic or acyclic — each variant enabling different algorithmic approaches. A key insight: the choice between adjacency list and adjacency matrix representation profoundly impacts both memory usage and traversal speed — sparse graphs favor lists ($O(V + E)$ space), while dense graphs benefit from matrices' $O(1)$ edge lookups at the cost of $O(V^2)$ space.

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