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Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
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.
In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form: = + (/). Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such a function as a W-function. [1]
The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can be undone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g.
Theorem [7] [8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed. The theorem is a consequence of the open mapping theorem ; see § Relation to the open mapping theorem below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).
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The y-intercept is the initial value = = at =. The slope a measures the rate of change of the output y per unit change in the input x. In the graph, moving one unit to the right (increasing x by 1) moves the y-value up by a: that is, (+) = +.