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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.
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]
The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on n {\displaystyle n} vertices can have.
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.
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]
The product C 4 C 4 is a four-dimensional hypercube graph; it has 16 vertices, and any single vertex can only dominate itself and four neighbors, so three vertices could only dominate 15 of the 16 vertices. Therefore, at least four vertices are required to dominate the entire graph, the bound given by Vizing's conjecture.