In mathematics, graph theory is the study of graphs , which is the mathematical structure used to model the paired relationships between objects. The graphs in this context consist of vertices , nodes , or points linked by edge , arcs , or the line . The graph may be not directed , which means that there is no difference between two vertices associated with each edge, or the end may be redirected from one point to another; see Graphs (discrete math) for more detailed definitions and for other variations in commonly considered graph types. Graphics is one of the main objects of study in discrete mathematics.
See the graph theory glossary for basic definitions in graph theory.
Video Graph theory
Definition
Definitions in graph theory vary. The following are some basic ways to define graphs and related mathematical structures.
Graph
In the most general sense of the term, the graph is the ordered pair G = ( V , E ) which consists of sets V of the vertices or node or points together with a set of E from edge or arcs or lines , which is a 2-element subset V (ie an edge is associated with two vertices, and the association takes the form of an unordered pair consisting of two such vertices). To avoid ambiguity, this type of graph can be properly described as not directed and simple.
Another feeling of graphics comes from a different concept from the edge set. In a general sense, V is a set along with the event relation associated with each end of two vertices. In another general sense, E is a multiset of irregular pairs (not necessarily different). Many authors call this type of object as a multigraf or pseudograph.
All these and other variants are described more fully below.
The node belonging to an edge is called the edge or end of the node from the edge. The point may be in the graph instead of the edge.
V and E are usually considered limited, and many well-known results are incorrect (or somewhat different) for unlimited graphics because many arguments fail in unlimited cases. The Sequence of a graph is | V |, the number of vertices. The size of of the graph is | E |, the number of sides. The degrees or valence of the point is the number of sides connected to it, where the edge connecting the dot for itself (one circle) is counted twice.
For the edge of { x , y }, graphical theorists usually use a somewhat shorter notation xy .
Maps Graph theory
Applications
Graphics can be used to model many kinds of relationships and processes in physical, biological, social and information systems. Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term network is sometimes defined as a graph in which attributes (eg names) are associated with nodes and/or ledges.
In computer science, graphics are used to represent communications networks, data organizations, computing devices, computing streams, etc. For example, the structure of a website link can be represented by a directed graph, where the node represents the web page and the directional end shows links from one page to another. A similar approach can be taken to problems in social media, travel, biology, computer chip design, mapping of neuro-degenerative disease progression, and many other fields. The development of algorithms to handle charts is therefore a major interest in computer science. Graphical transformation is often formalized and represented by a rewriting chart system. Complementing a graphical transformation system that focuses on manipulation of manipulation in rule-based memory is a graphical database that is geared toward storing, storing and querying secure graph-structured data from transactions.
Graphic-theoretical methods, in various forms, have proved very useful in linguistics, since natural languages ​​often match discrete structures. Traditionally, the syntax and semantics of the composition follow a tree-based structure, whose expressive power lies in the principle of compositionality, modeled in a hierarchical graph. More contemporary approaches such as the phrase-head-driven syntactic language grammar model of natural language using a typed feature structure, which is directed acyclic graphics. In lexical semantics, especially those applied to computers, modeling the meaning of a word is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. However, other methods in phonology (eg optimality theory, which uses grid graphs) and morphology (eg to-state morphology, using finite transducers state) are common in language analysis as charts. Indeed, the utility of this field of mathematics for linguistics has spawned organizations such as TextGraphs, as well as various 'Net' projects, such as WordNet, VerbNet, and others.
Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the complex three-dimensional structure of complex simulated atomic structures can be studied quantitatively by collecting statistics on the graph-theoretical properties associated with the atomic topology. In graphical chemistry create a natural model for a molecule, in which vertices represent atoms and edge bonds. This approach is mainly used in computer processing of molecular structures, from chemical editors to database searches. In statistical physics, graphs can represent local connections between the interacting parts of a system, as well as the physical process dynamics of the system. Similarly, in the graph of computational neuroscience can be used to represent the functional relationship between the areas of the brain that interact to induce various cognitive processes, where the vertices represent different areas of the brain and the ends represent the relationship between the areas. Graph theory plays an important role in electric network power modeling, here, the weights are associated with the resistance of the wire segment to obtain the electrical properties of the network structure. Graphics are also used to represent micro-scale channels from porous media, where vertices represent pores and the edges represent smaller channels connecting pores.
Graph theory is also widely used in sociology as a way, for example, to measure the prestige of actors or to explore the spread of rumors, especially through the use of social network analysis software. Under the umbrella of social networking there are many different types of charts. Graphs of acquaintances and companions describe whether people know each other. The graph model of the influence of whether a particular person can influence the behavior of others. Finally, a collaboration graph model does two people work together in a certain way, such as acting in a shared movie.
Likewise, graph theory is useful in biological and conservation efforts where vertices can represent areas where certain species exist (or inhabit) and ultimately represent migration paths or inter-regional movements. This information is important when looking at patterns of breeding or tracking the spread of disease, parasites or how changes in movement can affect other species.
In mathematics, graphs are useful in geometry and certain parts of topologies such as the knot theory. Algebra graph theory has a close association with group theory.
The graph structure can be extended by assigning weights to each side of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which paired connections have multiple numeric values. For example, if the graph shows a road network, weights can represent the length of each path. There may be some weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. These weighted graphs are commonly used to program GPS, and search engine travel planning comparing flight time and cost.
History
The paper written by Leonhard Euler on the Seven Bridges KÃÆ'¶nigsberg and published in 1736 is considered to be the first paper in the history of graph theory. This paper, as well as the one written by Vandermonde on the subject of the knight, was carried out with Leibniz's analytics site . The Euler formula associated with the number of edges, vertices, and faces of the convex polyhedron is studied and generalized by Cauchy and L'Huilier, and represents the beginning of a branch of mathematics known as topology.
More than a century after Euler's paper on the KÃÆ'¶nigsberg bridge and when the List introduced the concept of topology, Cayley was led by an interest in a particular analytical form arising from differential calculus to study a special class of graphs, trees . This research has many implications for the chemical theory. The techniques he uses mainly relate to calculating graphs with certain properties. The enumerative graphing theory then emerged from Cayley's results and the fundamental results published by PÃÆ'³lya between 1935 and 1937. It was generalized by De Bruijn in 1959. Cayley attributes the results to a tree with a contemporary study of chemical composition. The incorporation of ideas from mathematics with those derived from chemistry begins what has become part of the standard terminology of graph theory.
In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, in which he drew the analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:
- "[...] Every invariant and co-variant thus becomes expressed by a graph exactly identical to the KekulÃÆ'Â © an diagram or chemicograph. [...] I provides rules for the geometric multiplication of the graph, ie to build a graphic to the product from within or a separate graph co-variant given [...] "(italics like original).
The first textbook on graph theory was written by DÃÆ' Â © nes K n nig, and published in 1936. Another book by Frank Harary, published in 1969, "considered the world as the definitive textbook on this subject", and allows mathematicians, experts chemists, electrical engineers and social scientists to talk to each other. Harary donated all the royalties to fund the PÃÆ'³lya Prize.
One of the most famous and stimulating issues in graph theory is the four-color problem: "Is it true that every map drawn on a plane may have its territory colored with four colors, such that every two areas have shared borders of different colors?" This problem was first proposed by Francis Guthrie in 1852 and his first written record was in De Morgan's letter addressed to Hamilton in the same year. Much of the wrong evidence has been put forward, including those by Cayley, Kempe, and others. The study and generalization of this issue by Tait, Heawood, Ramsey and Hadwiger led to the study of graphic staining embedded on the surface with a mutable genus. The Tait reforms resulted in a new class of problems, the factorization problem , mainly studied by Petersen and K? Nig. Ramsey's works on dyeing and more specifically the results obtained by TurÃÆ'¡n in 1941 are the origin of other branches of graph theory, extreme graph theory.
The problem of four colors remains unsolved for more than a century. In 1969 Heinrich Heesch published a method for solving computer problems. Computer-assisted evidence produced in 1976 by Kenneth Appel and Wolfgang Haken makes a fundamental use of the "usage" idea developed by Heesch. The evidence involved examined the properties of the 1936 configuration by the computer, and was not fully accepted at that time due to its complexity. The evidence is simpler considering only 633 configurations were given twenty years later by Robertson, Seymour, Sanders and Thomas.
The development of autonomous topologies from 1860 and 1930 graph theory was fertilized through the works of Jordan, Kuratowski and Whitney. Another important factor of the general development of graphic theory and topography comes from the use of modern algebraic techniques. The first example of such use comes from the work of physicist Gustav Kirchhoff, published in 1845 Kirchhoff's circuit law to calculate the voltage and current in the electrical circuit.
The introduction of probabilistic methods in graph theory, especially in the study of Erd? And RÃÆ' Â © nyi from the asymptotic probability of graph connectivity, elicits another branch, known as random graph theory, which has been the source of useful graphic-theoretical results.
Graphics
Graphs are represented visually by drawing a point or circle for each dot, and drawing the bow between two nodes if they are connected by an edge. If the graph is directed, directions are indicated by drawing arrows.
Graphic images should not be equated with the graph itself (abstract, non-visual structure) because there are several ways to arrange graphic images. The important thing is which nodes are connected to the other by how many sides and not the proper layout. In practice, it is often difficult to decide whether two images represent the same graph. Depending on the problem domain, some layouts may be more appropriate and easier to understand than others.
The pioneering work of W. T. Tutte is very influential on the subject of graphic images. Among other achievements, he introduced the use of linear algebraic methods to obtain graphic images.
The graphic image can also be said to include problems related to junction numbers and various generalizations. The intersection number of a graph is the minimum number of intersections between the sides to which the graphic image on the plane should contain. For a planar graph, the junction number is zero by definition.
Images on surfaces other than planes are also studied.
The graph-theoretical data structure
There are various ways to store graphics in a computer system. The data structure used depends on the graph structure and the algorithm used to manipulate the graph. Theoretically one can distinguish between lists and matrix structures but in concrete applications the best structure is often a combination of both. List structure is often preferred for rare graphs because they have smaller memory requirements. The matrix structure on the other hand provides faster access for some applications but can consume large amounts of memory.
The list structure includes a list of events, an array of vertex pairs, and an adjacency list, which separately lists the neighbors of each vertex: Just like the incident list, each vertex has a list of which nodes are adjacent.
The matrix structure includes incidence matrix, a matrix of 0 and 1 whose rows represent vertices and whose columns represent edges, and neighboring matrices, in which both rows and columns are indexed by nodes. In both cases 1 shows two adjacent objects and 0 shows two non-adjacent objects. Laplace matrix is ​​a modified form of an adjacency matrix that combines information about the degree of a node, and is useful in some calculations such as Kirchhoff's theorem on the number of tree ranges of graphs. The distance matrix, like the neighboring matrix, has both rows and columns indexed by nodes, but rather than containing 0 or 1 in each cell, it contains the shortest path length between two nodes.
Problem
Enumeration
There is a lot of literature on graphic enumeration: the problem of calculating the graph meets the specified requirements. Some of these works were found in Harary and Palmer (1973).
Subgraph, induce subgraph, and minors
A common problem, called the subgraph isomorphism problem, is finding a fixed graph as a subgraph in the given graph. One reason to be interested in such a question is that many graphic properties are offspring for subgraph, which means that the graph has properties if and only if all subgraphs have them as well. Unfortunately, finding the maximum subgraph of a particular type is often a NP-complete problem. As an example:
- Finding the largest complete subgraph is called a click problem (NP-complete).
A similar problem is finding the subgraph induced in the given graph. Again, some important graphic properties are hereditary in relation to the induced subgraph, which means that the graph has properties if and only if all the induced subgraphs also have them. Finding the maximum induced subgraph of a particular type is also often NP-complete. As an example:
- Finding the largest unknown undescribed subgraph or independent circuit is called an independent setting problem (NP-complete).
Still another such problem, the small hold problem, is to find the fixed graph as the minor of the given graph. Minor or subcontracted graphs are graphs obtained by taking subgraphs and contracting some (or not) edges. Many graph properties are hereditary for minors, which means that the graph has properties if and only if all minors have them as well. For example, Wagner's Theorem states:
- The graph is planar if it contains as a minor both full bipartite graphics K 3.3 (see Three-cottage problem) or complete chart K 5 .
A similar problem, the problem of containment sharing, is to find a fixed graph as part of a given graph. Subdivisions or homeomorphism charts are graphs obtained by grouping some (or not) edges. The sharing restrictions are related to graph properties like planarity. For example, Kuratowski's Theorem states:
- The graph is planar if it contains as a subdivision, both complete bipartite graphics K 3.3 and the full graph of K 5 .
Another problem in the detention of the subdivision is the alleged Kelmans-Seymour:
- Each non-planar 5-dots graph contains a 5-dots complete graph of K 5 .
Another class of problem should be done by the extent to which the various species and graphic generalizations are determined by their point-deleted subgraph . As an example:
- Alleged reconstruction
Coloring chart
Many problems and theorems in graph theory are related to various ways of graphic coloring. Usually, a person is interested in coloring the graph so that no two adjacent vertices have the same color, or with other similar restrictions. One can also consider the edge coloring (perhaps so that no two coincide edges are of the same color), or other variations. Among the results and notable guesses about graph coloring are as follows:
- Four color theorist
- Strong perfect graphical theorem
- Erd? s-Faber-LovÃÆ'¡sz is speculating (unsolved)
- Total coloring cheat, also called Behzad's (unsolved) guess.
- Coloring list (unsolved)
- Hadwiger conjecture (graph theory) (unsolved)
Subsumption and unification
The theory of constraint modeling concerns the family of directed graphs associated with a partial sequence. In this application, graphs are sorted by specificity, which means that the more limited graphs - which are more specific and thus contain more information - are absorbed by their more general ones. Operations between graphs include evaluating the direction of the subsumption relationship between two graphs, if any, and the computation of the computational graph. The union of two argument graphs is defined as the most common graph (or calculation) consistent with (ie containing all information in) the input, if such a graph exists; an efficient unification algorithm is known.
For a strictly constrained framework of composition, the unification of the graph is quite satisfactory and a combination function. Famous applications include automated theorems that prove and model the elaboration of linguistic structures.
- Hamiltonian path problem
- The minimum range tree
- Route check issue (also called "Chinese postman problem")
- Seven KÃÆ'¶nigsberg bridges
- The shortest path problem
- Steiner Tree
- The three cottage issues
- Travel salesman problem (NP-hard)
Network stream
There are many problems that arise primarily from applications that have to do with various current ideas in the network, for example:
- Max stream cuts off the theorem
Visibility issues
- The museum keeper issue
Includes problems
Including problems in graphs are specific examples of subgraph-finding problems, and they tend to be closely related to click problems or independent regulatory issues.
- Set closing problem
- Vertex closing problem
Decomposition issue
Decomposition, defined as a partition set of graph edges (with many nodes required along the edges of each partition part), has a wide range of questions. Often, it is necessary to decipher the graph into an isomorphic subgraph to a fixed graph; for example, outlining the complete graph into the Hamilton cycle. Another problem determines the graph family in which a particular graph should be parsed, for example, family cycles, or outlines the full graph of K n to n - 1 specified tree has, respectively, 1, 2, 3,..., n - 1 the edges.
Some of the specific decomposition issues that have been studied include:
- Arboricity, decomposition to be as little as possible forest
- Double cover cycle, decomposition into a set of cycles covering each side exactly twice
- Dye edges, decomposition to as little as possible as possible
- Factor graphs, ordinary graph decomposition into regular subgraphs of a certain degree
Graph class
Source of the article : Wikipedia
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