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A connected graph must have a path between every pair of vertices. The minimum structure achieving this is a tree, which by definition has exactly N-1 edges for N vertices. Adding any fewer edges would disconnect the graph, and any additional edges would create cycles (redundant for connectivity).

Step-by-step Derivation:
For a graph with N vertices to be connected, every vertex must be reachable from every other vertex. The minimal connected structure is a tree. A tree on N vertices has exactly N-1 edges by the fundamental tree property: edges = vertices - 1. For example: N=3 vertices need minimum 2 edges (forming a path or triangle without redundancy); N=4 vertices need minimum 3 edges. This is both necessary (fewer edges leave the graph disconnected) and sufficient (a tree is connected).