Undirected Graphs

A social network is a graph, a road map is a graph, an `import` graph in your codebase is a graph, and so is the dependency tree of a package manager. The reason graphs feel everywhere is that they are the most general relationship structure: just vertices and edges, with no required root, no acyclic constraint, and no fixed branching factor. **Graphs: Representation Basics** is where you stop drawing them on whiteboards and start storing them in code. This lesson defines the core vocabulary (vertex, edge, degree, path, cycle, directed and undirected, connected and disconnected) and walks through the three canonical representations: an adjacency list backed by a hash map of vertex to neighbors, an adjacency matrix for dense graphs and `O(1)` edge-existence queries, and an edge list for algorithms that just iterate over edges. You will analyze the space and time trade-offs of each so you can pick the right one before writing any traversal logic. In **Hash Map (Dictionary) Basics**, you saw that a hash map maps keys to values in expected `O(1)`; an adjacency list is exactly that, mapping each vertex to its list of neighbors. **Queue & Deque** gave you the FIFO machinery that BFS will run on top of these representations once traversal lessons begin. With a graph stored in memory, the algorithms track is where it comes alive: BFS, DFS, shortest paths, connected components, and the rest of the graph algorithm canon.