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An abstract graph is said to be a unit distance graph if it is possible to find distinct locations in the plane for its vertices, so that its edges have unit length and so that all non-adjacent pairs of vertices have non-unit distances. When this is possible, the abstract graph is isomorphic to the unit distance graph of the chosen locations ...
If k is sufficiently large, it is known that G has to be 1-factorable: If k = 2n − 1, then G is the complete graph K 2n, and hence 1-factorable (see above). If k = 2n − 2, then G can be constructed by removing a perfect matching from K 2n. Again, G is 1-factorable. Chetwynd & Hilton (1985) show that if k ≥ 12n/7, then G is 1-factorable.
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation y = f ( x ) {\displaystyle y=f(x)} in which the functional form is explicitly stated; this is called an explicit representation.
The graph of a function on its own does not determine the codomain. It is common [3] to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective. Graph of the function () = over the interval [−2,+3]. Also shown are the two real roots and the local minimum ...
By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [−2, 2] and the image is [−1, 1]; for the second one, the domain is [−2, ∞) and the image is [1, ∞); for the last one, the domain is (−∞, 2] and the image is (−∞, −1]. As the three graphs together form a ...
At =, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function. y = e x {\displaystyle y=e^{x}} (for real x ) has inverse x = ln y {\displaystyle x=\ln {y}} (for positive y {\displaystyle y} )
A graph is said to be k-factor-critical if every subset of n − k vertices has a perfect matching. Under this definition, a hypomatchable graph is 1-factor-critical. [13] Even more generally, a graph is (a,b)-factor-critical if every subset of n − k vertices has an r-factor, that is, it is the vertex set of an r-regular subgraph of the given ...