complete graph

in graph theory, a complete graph is a graph in which every possible pair of graph vertices is connected by one edge.

as seen in the handshake problem, if there are $n$ vertices that must handshake with every other one, each possible handshake represents an edge in the graph

all the possible handshakes can be found by using the combination function with $C(n,2)$, therefore, a complete graph with $n$ vertices has $C(n,2)$ edges.

for reference, the combination functions page says $C(n,2)=\frac{n(n-1)}{2}$

complete graphs in real life situations include round robin tournaments, where each team plays all the other teams once