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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.
If a graph is 1-factorable then it has to be a regular graph. However, not all regular graphs are 1-factorable. A k-regular graph is 1-factorable if it has chromatic index k; examples of such graphs include: Any regular bipartite graph. [1] Hall's marriage theorem can be used to show that a k-regular bipartite graph contains a perfect matching.
Let f : X → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y. [4] If X is any space then the identity map Id : X → X is continuous but its graph, which is the diagonal Gr Id := { (x, x) : x ∈ X }, is closed in X × X if and only if X is Hausdorff. [7]
In mathematics, a functional equation [1] [2] [irrelevant citation] is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations.
This formula can fail when one of these conditions is not true. For example, consider g(x) = x 3. Its inverse is f(y) = y 1/3, which is not differentiable at zero. If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1/g′(f(0)). Since f(0) = 0 and g′(0) = 0, we must evaluate 1/0, which is ...
The points P 1, P 2, and P 3 (in blue) are collinear and belong to the graph of x 3 + 3 / 2 x 2 − 5 / 2 x + 5 / 4 . The points T 1, T 2, and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.
Let and be the eigenvalue and eigenvector of the Laplacian matrix (the eigenvalues are sorted in an increasing order, i.e., = [2]), the graph Fourier transform (GFT) ^ of a graph signal on the vertices of is the expansion of in terms of the eigenfunctions of . [3] It is defined as: [1] [4]
An important special case of the maximum cardinality matching problem is when G is a bipartite graph, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general ...