A complete graph Kₙ: build it edge by edge, count the edges via degrees, contrast its density against a sparse tree, or explore it with the n slider in free play.

Complete · every pair joined vertex active joined tree
Edges drawn so far
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Press Play. A complete graph Kn joins every single pair of vertices — the densest simple graph possible.

n6 Speed
n6
Drag the n slider to relayout the circle. Click a vertex then another to toggle an edge; click Show all to complete the graph.
graph
K6
edges
0
n(n−1)/2
0
degree (n−1)
0