Definition:Complete Graph

Definition
Let $$G = \left({V, E}\right)$$ be a simple graph such that every vertices is adjacent to every other vertex.

Then $$G$$ is called complete.

A complete graph of order $$p$$ is $p-1$-regular and is denoted $$K_p$$.

Examples
The first five complete graphs are shown below:



Basic Properties

 * $$K_n$$ is Hamiltonian for all $$n \ge 3$$, from Ore's Theorem or trivially, by inspection.


 * $$K_1$$ is the edgeless graph $$N_1$$, and also the path graph $$P_1$$.


 * $$K_2$$ is the path graph $$P_2$$, and also the complete bipartite graph $$K_{1, 1}$$.


 * $$K_3$$ is the cycle graph $$C_3$$, and is also called a triangle.


 * $$K_4$$ is the graph of the tetrahedron.


 * The complement of $$K_n$$ is the edgeless graph $$N_n$$.