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Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
Hexagonal paper shows regular hexagons instead of squares. These can be used to map geometric tiled or tesselated designs among other uses. Isometric graph paper or 3D graph paper is a triangular graph paper which uses a series of three guidelines forming a 60° grid of small triangles. The triangles are arranged in groups of six to make hexagons.
Yukio Matsumoto's 11/8 conjecture predicts that every smooth oriented 4-manifold X with even intersection form has second Betti number at least 11/8 times the absolute value of the signature. This would be optimal if true, since equality holds for a complex K3 surface, which has signature 3−19 = −16.
[11] The complete bipartite graph K m,n has a vertex covering number of min{m, n} and an edge covering number of max{m, n}. The complete bipartite graph K m,n has a maximum independent set of size max{m, n}. The adjacency matrix of a complete bipartite graph K m,n has eigenvalues √ nm, − √ nm and 0; with multiplicity 1, 1 and n + m − 2 ...
Most industry standards express the direction of the grain last when giving dimensions (that is, 17 × 11 inches is short grain paper and 11 × 17 inches is long grain paper), although alternatively the grain alignment can be explicitly indicated with an underline (11 × 17 is a short grain) or the letter "M" for "machine" (11M × 17 is a short ...
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...