: For example, graph 1 above expresses the abstraction " agent sits on location ".

For curves given by the graph of a function, horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound.

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For example, if a graph has 17 valid 3-colorings, the SAT formula produced by the reduction will have 17 satisfying assignments.

For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

For example, they allow representation of a directed graph with trees on the nodes.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

For the case of power-of-two, Papadimitriou ( 1979 ) argued that the number of complex-number additions achieved by Cooley – Tukey algorithms is optimal under certain assumptions on the graph of the algorithm ( his assumptions imply, among other things, that no additive identities in the roots of unity are exploited ).

For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries-Wong graph.

For any positive integers and, there exists a graph with girth at least and chromatic number at least ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number.

For example if a graph represents a road network, the weights could represent the length of each road.

For a planar graph, the crossing number is zero by definition.

For contraint frameworks which are strictly compositional, graph unification is the sufficient satisfiability and combination function.

For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph.

For example, a similar geometric system was published in 1987 by Kjell Gustafson, whose method represents a rhythm as a two-dimensional graph.

For example, the figure shows sin θ for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle ( top ) and as a graph ( bottom ).

For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O ( n log n ) time for n points ( considerably less than the number of edges ).

For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function ; in other words, the function's graph.

For a discussion of tree structures in specific fields, see Tree ( data structure ) for computer science: insofar as it relates to graph theory, see tree ( graph theory ), or also tree ( set theory ).

For instance, if I is the category of the directed graph, then C < sup > I </ sup > has as objects the morphisms of C, and a morphism between and in C < sup > I </ sup > is a pair of morphisms and in C such that the " square commutes ", i. e..

For a two-dimensional array, the element with indices i, j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

For convenience, pixels are normally arranged in a regular two-dimensional grid.

For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.

For example, the surface of the Earth is ( ideally ) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it ( except at the poles and along the 180th meridian ).

For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: ΣH

For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional ( 2D ) surface.

For two-dimensional signals, e. g., images, the spectrum is reciprocal to f < sup > 2 </ sup >.

For two-dimensional rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:

For example, is a meromorphic function on the two-dimensional complex affine space.

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent.

For example, the ordinary generating function of a two-dimensional array a < sub > m, n </ sub > ( where n and m are natural numbers ) is

For instance, a "< big > b </ big >" and a "< big > d </ big >" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written.

For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder.

For example, a monatomic gas with a fixed number of particles is a simple case of a two-dimensional system ().

For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives.

For a two-dimensional flow the vorticity acts as a measure of the local rotation of fluid elements.

For a two-dimensional analysis a plane stress or a plane strain condition can be assumed.

For example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the two-dimensional plane.

For such models, economists often use two-dimensional graphs instead of functions.

For example, a two-dimensional real torus has a SL ( 2, Z ) group of large diffeomorphisms by which the one-cycles of the torus are transformed into their integer linear combinations.