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In graph theory, the hypercube graph Q n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q 3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q n has 2 n vertices, 2 n – 1 n edges, and is a regular graph with n edges touching each vertex.
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract.It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.
In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces. It can be named by its Schläfli symbol {4,3 8}, being composed of 3 9-cubes around each ...
Let n be a positive integer, and let γ be a real number in the unit interval 0 ≤ γ ≤ 1.Suppose additionally that (1 − γ)n is an even number.Then the Frankl–Rödl graph is the graph on the 2 n vertices of an n-dimensional unit hypercube [0,1] n in which two vertices are adjacent when their Hamming distance (the number of coordinates in which the two differ) is exactly (1 − γ)n. [2]
Hypercube graphs exhibit a similar phenomenon to cycle graphs. The two- and three-dimensional hypercube graphs (the 4-cycle and the graph of a cube, respectively) have distinguishing number three. However, every hypercube graph of higher dimension has distinguishing number only two. [4] The Petersen graph has distinguishing number 3.
Many important graph families can be characterized in terms of the induced paths or cycles of the graphs in the family. Trivially, the connected graphs with no induced path of length two are the complete graphs, and the connected graphs with no induced cycle are the trees. A triangle-free graph is a graph with no induced cycle of length three.
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace X n that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n.
This 9-cube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:9:36:84:126:126:84:36:9:1.