Complete Graph
A Graph in which each pair of Vertices is connected by anEdge. The complete graph with 
Vertices is denoted 
. In olderliterature, complete Graphs are called Universal Graphs.
is the Tetrahedral Graph and is therefore a Planar Graph. 
is nonplanar. Conway and Gordon (1983)proved that every embedding of 
is Intrinsically Linked with at least one pair of linked triangles. They alsoshowed that any embedding of 
contains a knotted Hamiltonian Cycle.
The number of Edges in 
is 
, and the Genus is
for 
. The number of distinct variations for (Graphs whichcannot be transformed into each other without passing nodes through an Edge or another node)for 
, 2, ... are 1, 1, 1, 1, 1, 1, 6, 3, 411, 37, .... The Adjacency Matrix of the complete graph takesthe particularly simple form of all 1s with 0s on the diagonal.
It is not known in general if a set of Trees with 1, 2, ..., 
Edges can alwaysbe packed into . However, if the choice of Trees is restricted to either the path or star from eachfamily, then the packing can always be done (Zaks and Liu 1977, Honsberger 1985).
References
Chartrand, G. Introductory Graph Theory. New York: Dover, pp. 29-30, 1985. Conway, J. H. and Gordon, C. M. ``Knots and Links in Spatial Graphs.'' J. Graph Th. 7, 445-453, 1983. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 60-63, 1985. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986. Zaks, S. and Liu, C. L. ``Decomposition of Graphs into Trees.'' Proc. Eighth Southeastern Conference on Combinatorics, Graph Theory, and Computing. pp. 643-654, 1977.
- Wilcoxon Signed Rank Test
- Wild Knot
- Wild Point
- Wilf-Zeilberger Pair
- Wilkie's Theorem
- Williams p+1 Factorization Method
- Wilson Plug
- Wilson Prime
- Wilson Quotient
- Douady's Rabbit Fractal
- Double Bubble
- Double Bubble Conjecture
- Double Contraction Relation
- Double Cusp
- Double Exponential Distribution
- Double Exponential Integration
- Double Factorial
- Double Folium
- Double-Free Set
- Double Gamma Function
- Double Point
- Double Sixes
- Double Sum
- Doublet Function
- Doubly Even Number