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In graph theory, the Möbius ladder M n, for even numbers n, is formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of M 6 (the utility graph K 3,3), M n has exactly n/2 four-cycles [1] which link together by their shared edges to form a topological Möbius strip.
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph.For, if G is an apex graph with apex v, then any contraction or removal that does not involve v preserves the planarity of the remaining graph, as does any edge removal of an edge incident to v.
The Pappus graph. The Levi graph of the Pappus configuration is known as the Pappus graph.It is a bipartite symmetric cubic graph with 18 vertices and 27 edges. [3]Adding three more parallel lines to the Pappus configuration, through each triple of points that are not already connected by lines of the configuration, produces the Hesse configuration.
4) Example 4 is based on a hyperbolic spiral = /. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for φ → 0 {\displaystyle \varphi \to 0} .
If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r {\displaystyle r} decreases the genus by 1 2 r ( r − 1 ) {\displaystyle {\frac {1}{2}}r(r-1)} .
In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. [2] [3] A rotation of axes is a linear map [4] [5] and a rigid transformation.
Theorem 2. If there are no cycles of length 3, then e ≤ 2v – 4. Theorem 3. f ≤ 2v – 4. In this sense, planar graphs are sparse graphs, in that they have only O(v) edges, asymptotically smaller than the maximum O(v 2). The graph K 3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be ...