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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 middle layer graph of an odd-dimensional hypercube graph Q 2n+1 (n,n+1) is a subgraph whose vertex set consists of all binary strings of length 2n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices for which the corresponding binary strings differ in exactly one bit. Every middle layer graph is ...
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.
In graph theory, a partial cube is a graph that is an isometric subgraph of a hypercube. [1] In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube.
A 4D hypercube network can be created by duplicating two 3D networks, and adding a most significant bit. The new added bit should be ‘0’ for one 3D hypercube and ‘1’ for the other 3D hypercube. The corners of the respective one-bit changed MSBs are connected to create the higher hypercube network. This method can be used to construct ...
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]
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.