Graph ( mathematics ) | Graphs like this are among the objects studied by discrete mathematics, for their interesting graph property | mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithm s.

Graphs are the basic subject studied by graph theory.

Graphs are converted into factor graph form to perform belief propagation.

Graphs as used in graph theory.

For graphs as charts, plots and drawings, see: Category: Graphs ( images ).

( Graphs which show just the length of the rule in the two dynasties are the most widely known ; however, Fomenko's conclusions are also based on other parameters, as described above.

Graphs are switching equivalent if one can be obtained from the other by switching.

Lists, Trees and Directed Acyclic Graphs are other possible and common alternatives.

Current research activities of the Department are in the areas of Analysis, Algebra, Operator Theory, Functional Analysis, General topology, Fuzzy mathematics, Graph Theory, Combinations, Convexity Theory, Fluid Dynamics, Non-linear waves, Stability, Stochastic Processes in general and Random Graphs, Operations Research and the History of Mathematics.

Individual studies include Moretti, Maps, Graphs, Trees ( 2005 ), John Pizer, The Idea of World Literature ( 2006 ), Mads Rosendahl Thomsen, Mapping World Literature ( 2008 ), and Theo D ' haen, The Routledge Concise History of World Literature ( 2011 ).

In practice it is often difficult to decide if two drawings represent the same graph.

Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.

As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph.

If a DAG G represents a partial order ≤, then the transitive reduction of G is a subgraph of G with an edge u → v for every pair in the covering relation of ≤; transitive reductions are useful in visualizing the partial orders they represent, because they have fewer edges than other graphs representing the same orders and therefore lead to simpler graph drawings.

Alternative conventions to node-link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions ; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections ; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other ; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks ; and visualizations of the adjacency matrix of the graph.

Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.

Some layout methods automatically lead to symmetric drawings ; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.

Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using graph drawing techniques.

In mathematics, a dessin d ' enfant ( French for a " child's drawing ", plural dessins d ' enfants, " children's drawings ") is a type of graph drawing used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.

In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.

One can also consider graph drawings in the three-dimensional grid.

Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, and bioinformatics.

He also gives his name to the Pappus chain, and to the Pappus configuration and Pappus graph arising from his hexagon theorem.

* Misleading graph, discusses problems arising from graphical presentations.

The irreducible ( M, M ) and ( M, N ) bimodules arising in this way form the vertices of the principal graph, a bipartite graph.

Many other decision problems, such as graph coloring problems, planning problems, and scheduling problems, can be easily encoded into SAT.

One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

** In power engineering, a " bus " is any graph node of the single-line diagram at which voltage, current, power flow, or other quantities are to be evaluated.

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 ).

A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another ; see graph ( mathematics ) for more detailed definitions and for other variations in the types of graph that are commonly considered.

The graphs studied in graph theory should not be confused with the graphs of functions or other kinds of graphs.

Still other methods in phonology ( e. g. Optimality Theory, which uses lattice graphs ) and morphology ( e. g. finite-state morphology, using finite-state transducers ) are common in the analysis of language as a graph.

( However, there are other, similar matrices that are also called " Laplacian matrices " of a graph.

This happens whenever the molecular graph is symmetrical, as in the 3-ketopentoses H ( CHOH )< sub > 2 </ sub >( CO )( CHOH )< sub > 2 </ sub > H, and the two halves are mirror images of each other.

In addition to the aforementioned simple libraries, other higher-level object-oriented scene graph retained mode libraries exist such as PLIB, OpenSG, OpenSceneGraph, and OpenGL Performer.

In the theory of quadratic forms, the parabola is the graph of the quadratic form ( or other scalings ), while the elliptic paraboloid is the graph of the positive-definite quadratic form ( or scalings ) and the hyperbolic paraboloid is the graph of the indefinite quadratic form Generalizations to more variables yield further such objects.

The connected component containing the special vertex contains the objects that can't be collected, while other connected components of the graph only contain garbage.

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.

* Graph center, the vertices in a graph that minimize the maximal distance from all other vertices

* Cycle ( graph theory ), a closed path, with no other repeated vertices than the starting and ending vertices

Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph.

This algorithm is often used in routing and as a subroutine in other graph algorithms.

For a given source vertex ( node ) in the graph, the algorithm finds the path with lowest cost ( i. e. the shortest path ) between that vertex and every other vertex.

For example, if the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities.

In other words, any connected graph without cycles is a tree.