Complete graph

A complete graph is a type of graph where every pair of vertices has a direct edge connecting them. In other words, in a complete graph, you can get from any vertex to any other vertex in just one step.

Complete graph: a graph where every pair of vertices is directly connected by an edge.

Notation

A complete graph with n vertices is often written as K_n. For example:

• K_3 is a complete graph with 3 vertices, where each vertex is connected to the other two vertices.
• K_4 is a complete graph with 4 vertices, where each vertex is connected to the other three vertices.

Number of edges

In a complete graph with n vertices:

\text{Number of edges} = \frac{n(n-1)}{2}

This assumes the graph is undirected.

Number of hamiltonian cycles

For a complete graph with n vertices:

\text{Number of Hamiltonian cycles} = \frac{(n-1)!}{2}