Graph Edges Calculator (Complete Graph)

Formula

How this is calculated

A complete graph K_n connects every pair of distinct vertices. Enter n, the number of vertices (a whole number, n ≥ 0), and choose the graph type. In an undirected complete graph each unordered pair {a, b} contributes a single edge, so the total is the binomial coefficient C(n, 2) = n(n−1)/2 — the classic "handshake" number, since it also counts how many handshakes occur when n people each shake hands once with everyone else.

A directed (oriented) complete graph instead counts every ordered pair (a, b) with a ≠ b, allowing an arrow in each direction. There are n choices for the first vertex and (n−1) for the second, giving n(n−1) edges — exactly twice the undirected count. Both formulas grow quadratically with n, which is why the edges-versus-vertices curve curves upward sharply.

The calculator floors any decimal input to an integer and rejects negative values. Edge cases behave naturally: n = 0 and n = 1 both yield 0 edges (no pairs exist), and n = 2 yields 1 undirected or 2 directed edges. Self-loops and multi-edges are not counted, matching the definition of a simple complete graph.