A complete graph is a term from graph theory and describes a simple graph in which every node is connected to every other node by an edge . The complete graph with nodes is uniquely determined (except for isomorphism ) and is denoted by.
If the set of nodes of the complete graph , then the set of edges is exactly the set of edges between pairs of different nodes .
A complete graph is also its maximal clique .
The complete graphs to are planar . All other complete graphs are not planar according to Kuratowski's theorem , since they contain them as a subgraph.
The number of edges of the complete graph corresponds to the triangle number
The complete graph is a - regular graph : each node has neighbors . Because of this, every node coloring of the graph has colors. Furthermore, it follows that the complete graphs are Eulerian for odd and not for even .
Complete graphs are for Hamiltonian graphs . The complete graph contains different Hamilton circles .
The idea of the complete graph can be transferred to -partite graphs . These are complete if every node of a partition is connected to all nodes of all other partitions. The complete multipartite graph with partition sets containing nodes is called .
If you provide a complete graph with an orientation, you get a tournament graph .
Using free Python - Library NetworkX to complete graphs can be generated. Example:
import networkx as nx import matplotlib.pyplot as plt G = nx.complete_graph(15) nx.draw_circular(G, with_labels=True, font_weight='bold') plt.show()
- Lutz Volkmann: Foundations of the graph theory. Springer, Vienna 1996, ISBN 3-211-82774-9 ; newer version: graphs at all corners and edges (PDF; 3.7 MB)
- Eric W. Weisstein : Complete Graph . In: MathWorld (English).