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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).
Let G be a k-regular graph with 2n nodes. 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.
As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of y that has three values for −2 < y < 2, and only one value for y ≤ −2 and y ≥ −2.
A Cartesian product of two graphs. In graph theory, the Cartesian product G H of graphs G and H is a graph such that: the vertex set of G H is the Cartesian product V(G) × V(H); and; two vertices (u,v) and (u' ,v' ) are adjacent in G H if and only if either u = u' and v is adjacent to v' in H, or; v = v' and u is adjacent to u' in G.
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
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In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function, is a member of the domain of such that () vanishes at ; that is, the function attains the value of 0 at , or equivalently, is a solution to the equation () =. [1]