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The generating function of the Bernoulli polynomials is given by: = = ()! These polynomials are given in terms of the Hurwitz zeta function: (,) = = (+)by (,) = for .Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation: [6]
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.
Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
Floor function: Largest integer less than or equal to a given number. Ceiling function: Smallest integer larger than or equal to a given number. Sign function: Returns only the sign of a number, as +1, −1 or 0. Absolute value: distance to the origin (zero point)
Given this information, the degree-sum formula can be used to recover the number of edges in the given graph and the degrees of each vertex. From this, it is possible to determine whether the given graph is a regular graph , and if so to determine it uniquely from any vertex-deleted subgraph by adding a new neighbor for all the subgraph ...
In a graph of order n, the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops are allowed). The edges of a graph define a symmetric relation on the vertices, called the adjacency relation.
Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics , a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values is bounded .
To decide if a graph has a Hamiltonian path, one would have to check each possible path in the input graph G. There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow.