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In the special case when k is the function field of an algebraic curve over a finite field and f is any character that is trivial on k, this recovers the geometric Riemann–Roch theorem. [12] Other versions of the arithmetic Riemann–Roch theorem make use of Arakelov theory to resemble the traditional Riemann–Roch theorem more exactly.
Furthermore, this property remains true throughout the rest of the process. Now, suppose for some k ≤ n ⁄ 2 that more than εk of the inputs (1, …, k) are in B. Then by expansion properties of the graph, the registers of these inputs in Y are connected with at least 1 – ε / ε k registers in X.
k = 1 and v = 2 yields a trivial graph of two vertices joined by an edge, k = 3 and v = 10 yields the Petersen graph, k = 7 and v = 50 yields the Hoffman–Singleton graph, discovered by Hoffman and Singleton in the course of this analysis, and; k = 57 and v = 3250 predicts a famous graph that has neither been discovered since 1960, nor has its ...
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...
(For a graph with n vertices and r terminals, they use t = n − r − 1 added vertices per tree.) Then, they ask for the k -minimum spanning tree in this augmented graph with k = rt . The only way to include this many vertices in a k -spanning tree is to use at least one vertex from each added tree, for there are not enough vertices remaining ...
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective ...
The Bode phase plot is the graph of the phase, commonly expressed in degrees, of the argument function ((=)) as a function of . The phase is plotted on the same logarithmic ω {\displaystyle \omega } -axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.