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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.
The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the (n−1)-simplex's face lattice. This can be seen by orienting the n-hypercube so that two opposite vertices lie vertically, corresponding to the (n−1)-simplex itself and the null polytope, respectively.
The Dalí cross, a net of a tesseract The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.. In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. [1]
Hypercube graph, a higher-dimensional generalization of the cube graph. 3. Folded cube graph, formed from a hypercube by adding a matching connecting opposite vertices. 4. Halved cube graph, the half-square of a hypercube graph. 5. Partial cube, a distance-preserving subgraph of a hypercube. 6. The cube of a graph G is the graph power G 3.
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set K ∗ {\displaystyle K_{*}} can be described by a collection of sets K i , i ≥ 0 {\displaystyle K_{i},\ i\geq 0} , together with face and degeneracy maps between them satisfying a ...
When the number of nodes along each dimension of a toroidal network is 2, the resulting network is called a hypercube. A parallel computing cluster or multi-core processor is often connected in regular interconnection network such as a de Bruijn graph, [1] a hypercube graph, a hypertree network, a fat tree network, a torus, or cube-connected ...
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.
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]