Properties
- Given a bipartite graph, finding its complete bipartite subgraph Km,n with maximal number of edges mn is an NP-complete problem.
- A planar graph cannot contain K3,3 as a minor; an outerplanar graph cannot contain K3,2 as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary).
- A complete bipartite graph. Kn,n is a Moore graph and a (n,4)-cage.
- A complete bipartite graph Kn,n or Kn,n+1 is a Turán graph.
- A complete bipartite graph Km,n has a vertex covering number of min{m,n} and an edge covering number of max{m,n}.
- A complete bipartite graph Km,n has a maximum independent set of size max{m,n}.
- The adjacency matrix of a complete bipartite graph Km,n has eigenvalues √(nm), −√(nm) and 0; with multiplicity 1, 1 and n+m−2 respectively.
- The laplacian matrix of a complete bipartite graph Km,n has eigenvalues n+m, n, m, and 0; with multiplicity 1, m−1, n−1 and 1 respectively.
- A complete bipartite graph Km,n has mn−1 nm−1 spanning trees.
- A complete bipartite graph Km,n has a maximum matching of size min{m,n}.
- A complete bipartite graph Kn,n has a proper n-edge-coloring corresponding to a Latin square.
- The last two results are corollaries of the Marriage Theorem as applied to a k-regular bipartite graph.
Read more about this topic: Complete Bipartite Graph
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)