These are graphs of ψ ( x, y, z ) functions which depend on the coordinates of one electron.

Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

* Cartesian product of graphs, a binary operation on graphs

The graphs on the right side depict the ( finite ) coefficients that modulate the infinite amplitudes of a comb function whose teeth are spaced at the reciprocal of the time-domain periodicity.

The graphs on the right side depict the ( finite ) coefficients that modulate the infinite amplitudes of a comb function whose teeth are spaced at intervals of 1 / P.

Complementary to graph transformation systems focussing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.

In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.

The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.

There is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions.

* Goshen on citydata. com – collection of statistics and graphs of Goshen demographics.

* Study of the eigenvalues and eigenvectors of the Laplacian on domains, manifolds, and ( to a lesser extent ) graphs is also considered a branch of harmonic analysis.

** for his fundamental and broad contributions to discrete optimization including his deep research on balanced and ideal matrices, perfect graphs and cutting planes for mixed-integer optimization.

Pascal has its roots in the ALGOL 60 language, but also introduced concepts and mechanisms which ( on top of ALGOL's scalars and arrays ) enabled programmers to define their own complex ( structured ) datatypes, and also made it easier to build dynamic and recursive data structures such as lists, trees and graphs.

* A tutorial-style dissertation by Volker Koch that introduces message-passing on factor graphs to decompose EMG signals

In 2007, work was completed on renovations of the press briefing room, adding fiber optic cables and LCD screens for the display of charts and graphs.

The points where this is not true are determined by a condition on the derivative of f. The circle, for instance, can be pasted together from the graphs of the two functions.

He is a leading expert in splittings and accessibility of discrete groups, Groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups.

This work has been developed to an important theory in the book " Groups acting on graphs ", Cambridge University Press, 1989, with Warren Dicks.

For many natural P-complete graph problems, where the graph is expressed in a natural representation such as an adjacency matrix, solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller ; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.

Three of these graphs are shown below, and they clearly show why Medicare ( as currently formulated ) is on the path to fiscal insolvency: No matter what the wage level, marital status, or retirement date, a man or woman can expect to receive benefits that will cost the system far more than the taxes he or she paid into the system.

The problem is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces.

This is illustrated in the vapor pressure chart ( see right ) that shows graphs of the vapor pressures versus temperatures for a variety of liquids.

The plot is typically represented as a three dimensional graph ( as at right ), or as separate graphs in the vertical plane and horizontal plane.

So from the above examples, we can conclude right away that the categories of directed graphs, G-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of G, modules over the ring R, and presheaves of abelian groups on a topological space X are all abelian, complete and cocomplete.

This was true of normal as well as extra complex identificational graphs, such as the hǔ 虎 " tiger " clan emblem at right, which was turned 90 degrees clockwise on its bronze.

That perspective might help to interpret the different noise-floor levels between the two graphs in the figure on the right.

This mix of state diagrams and flow graphs is illustrated in the figure on the right, which shows the control logic for a simple stopwatch.

For graphs with minimum degree at least 5, only the six subgraphs in the left and right columns of the figure are needed in the characterization.

The graphs to the right show the magnitude response for various values of, demonstrating this periodicity.

Lott and Whitley argue that if Donohue and Levitt are right that 80 percent of the drop in murder rates during the 1990s is due solely to the legalization of abortion, their results should be seen in these graphs without anything being controlled for, and that in fact the opposite is true.

( See the graphs to the right.

Lott and Whitley argue that if Donohue and Levitt are right that 80 percent of the drop in murder rates during the 1990s is due solely to the legalization of abortion, their results should be seen in these graphs without anything being controlled for, and that in fact the opposite is true.

Binary De Bruijn graphs can be drawn ( below, left ) in such a way that they resemble objects from the theory of dynamical systems, such as the Lorenz attractor ( below, right ):

The directed edges of these graphs describe the way an irreducible bimodule decomposes when tensored with X and X * on the right.

The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1 /√ x ( see also their asymptotic forms below ), although their roots are not generally periodic, except asymptotically for large x.

Some graphs are given for: 1N400x series, and CY7 cryogenic temperature sensor.

* USING THE ANALOG DEVICES ACTIVE FILTER DESIGN TOOL: a similar application note from Analog Devices with extensive graphs, active RC filter topologies, and tables for practical design.

Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

Historically they were sometimes called Feynman-Dyson diagrams or Dyson graphs, because when they were introduced the path integral was unfamiliar, and Freeman Dyson's derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods.

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.

[...] I give a rule for the geometrical multiplication of graphs, i. e. for constructing a graph to the product of in-or co-variants whose separate graphs are given.

List structures are often preferred for sparse graphs as they have smaller memory requirements.

Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their point-deleted subgraphs, for example:

Many problems have to do with various ways of coloring graphs, for example:

The European Commission had some difficulty funding the project's next stage, after several allegedly " per annum " sales projection graphs for the project were exposed in November 2001 as " cumulative " projections ( which for each year projected, necessarily included all previous years of sales ).