For instance, Sunúz contains terms to describe movement in a six dimensional multi-planar space, something of use to beings who visit the other planar realms where " demons " live.

For example, a common method of describing a knot is a planar diagram called a knot diagram.

For spatial or planar topologic or topographic sets see map.

For example in 1880 he worked on the Four color theorem and proved that it was true if and only if no snarks were planar.

For example, the planar graphs are closed under taking minors: contracting an edge in a planar graph, or removing edges or vertices from the graph, cannot destroy its planarity.

For k > 1 a planar embedding is said to be k-outerplanar if removing the vertices on the outer face results in a ( k − 1 )- outerplanar embedding.

For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a planar graph is just a vertex coloring of its planar dual.

Several different approaches lead to optimal data structures, with O ( n ) storage space and O ( log n ) query time, where n is the total number of vertices in S. For simplicity, we assume that the planar subdivision is contained inside a square bounding box.

For example, an sp or sp < sup > 2 </ sup > hybridized molecule would be planar, with a mirror plane.

For simple polyhedra, the medial graph and the line graph coincide, but for non-simple graphs the medial graph remains planar.

For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always, and for multigraphs, the number of colors may be as large as.

For a planar surface Σ, a positive path element dℓ of curve ∂ Σ is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal n to the surface Σ.

For midrange beaming control, the outer mouth was expanded further with a short, flared flange in the Keele style, or with added planar sides of a greater flare angle.

( The same factor applies to other magnetic quantities such as H and M .) For example, in a planar light wave in vacuum, | E ( r, t )|=| B ( r, t )| in Gaussian units, while | E ( r, t )|= c | B ( r, t )| in SI units.

For example, BH < sub > 3 </ sub > has a trigonal planar geometry, three 120 ° bond angles, three equivalent hybrids about the boron atom, and thus 1 + < sup > 2 </ sup > cos ()

For example, Nostoceras started off with a helix like a Turrilites, but had a planar hook hanging underneath, more like an Ancyloceras.

For example, a S < sub > N </ sub > 1 substitution reaction converts a molecule that is chiral by merit of non-planarity into a planar molecule, which has no handedness.

For many purposes, such a wavefront can be considered planar.

For example, reaction of a square planar complex with methyl iodide:

For simple element geometries ( e. g. planar triangles ) analytical integration can be used.

For higher bias voltages, the predictions of simple planar tunneling models using the Wentzel-Kramers Brillouin ( WKB ) approximation are useful.

For instance, in biology, Fitzhugh and Nagumo extended the equation in a planar field as a model for action potentials of neurons.

: 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 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..