The term of graph has two meanings in mathematics:

• the Graph of a function, and

• the combinative object of mathematics that we describe here, generalizing the binary and that of Polyèdre, and useful concept of Relation for the modeling of various problems of the " world réel" (i.e. one meets apart from mathematics, as for example those related to the concept of Réseau that it is data-processing, social, road or different…).

Formal definition

One generally makes go up the graph theory at the Sixties, since the bases posed by Claude Berge in Graphes and hypergraphes , even if the concept is definitely older. Bank primarily joined together within this theory of the results of Combinatoire, already known, and justified the study of the graphs by applications, bonds with the Game theory and the existence of conjectures (in For the honor of the human spirit , Jean Dieudonné says that a branch of mathematics is long-lived if it poses many questions…).

Graphs

Nowadays one gives an intuitive definition of the graphs (given low) which differs seemingly from the definition given by Berge, but in fact of course they are equivalent: Bank defines a graph as a triplet $(V,\; E,\; \backslash \; phi)$ where $\backslash \; phi$ is a function $\backslash \; phi:\; E\; \backslash \; rightarrow\; V\; \backslash \; times\; V$.

The vocabulary of the graph theory is then borrowed from the geometry of the Polyèdre S. a particular case of graph is a triplet $(V,\; E,\; \backslash \; phi)$ where $V$ is the whole of the tops of a polyhedron, $E$ that of the edges of the polyhedron and where $\backslash \; phi\; (E)\; =\; (U,\; v)$ if the edge $e$ connects the two tops $u$ and $v$. This vocabulary is preserved and generally one says for any graph that $V$ is its whole of tops , $E$ that of its edges (or of the arcs if the graph is directed) and $\backslash \; phi$ is its function of incidence (which with each edge (arc) associates a couple of tops). Also it will be said, when $\backslash \; phi\; (E)\; =\; (U,\; v)$, that $e$ and $u$ are incidental (idem for $e$ and $v$), that $u$ and $v$ are adjacent and that these two tops the ends of $e$. Of two edges one says that they are adjacent if they are incidental at the same top. Nowadays, the practice is taken to explicitly write that an edge is incidental at a top, therefore not to use any notation for the function of incidence and in fact of supposing its existence. One writes as follows: that is to say $G=\; (V,\; E)$ a graph… This notation is in fact strictly rigorous only for the simple graphs.

If the graph is directed, $u$ is the initial end of $e$ and $v$ is its final end (in the case not directed these tops are ends quite simply, without another precision). In the directed graphs, two edges are consecutive if they are adjacent and if their common end is initial for an edge and final for the other.

An edge (arc) is called a buckles if its two ends are identical. Two edges (arcs) are parallel if they have the same ends (even initial end and even final end for the arcs). The multiplicity of an edge (arc) is 1 if it is not parallel with any other and if not, one says that it is multiple and its multiplicity is the full number (including it even) of its parallel edges. The concept of graph directed without multiple arc corresponds precisely to that of binary Relation while the concept of graph not-directed without multiple edge corresponds to that of symmetrical binary relation.

A graph is known as simple if it has neither buckles nor multiple edge. The maximum number of edges of a simple graph is thus $n\; (n-1)\; /2$ if it not-is directed and $n\; (n-1)$ if it is directed, where N is the number of tops (see complete Graphe). These graphs correspond to the not-reflexive binary relations. The simple graphs directed without pair of arcs of the form $(U,\; v)$ and $(v,\; U)$ correspond to the not-reflexive and antisymmetric binary relations.

Hypergraphes

The concept of not-directed graph spreads by that of Hypergraphe but the concept of connexity (existence of way) badly exports graphs with the hypergraphes and even less with the directed hypergraphes of which none the various possible definitions still were essential.

A little history

The known result, undoubtedly oldest, that one can include in the graph theory is the characterization of the graphs Eulériens, moved by the Problème of the seven bridges of Königsberg, by Euler in 1736.

In 1835, Gustav Kirchhoff published its laws of the circuits to calculate the tension and the running in a Electrical circuit. This was going to be in the Fifties at the origin of the concepts of flood and cut in the graphs. These two joined together concepts made it possible to pose the bases of the Dualité in linear Programming and of the existence of the theorems min-max in combinative Optimization, giving a new lighting to old results (like the theorems of Menger and König).

It is in 1852 that Francis Guthrie posed the famous Problème of the four colors, problem solved more than one century after its stating. This major result with given its " letters of noblesse" with the graph theory. The vocabulary even of the graph theory comes from the context of the resolution of this problem which is interested only in the graphs resulting from the polyhedrons.

In 1914, " any regular graph biparti has a coupling parfait" announced by König (published in 1916).

In 1927, Theorem of Menger first result of connectivity in graph and, a posteriori, first theorem min-max.

In 1931, Theorem of König.

In 1935, Theorem of Hall (generalized by Tutte, then Tutte-Bank) on the perfect couplings in the graphs bipartis. This result was going to be at the origin of the class Co-NP, then associated with the algorithm of the perfect coupling of Edmonds, at the origin of the conjecture $\backslash \; textrm\; \{P\}\; =\; \backslash \; textrm\; \{NP\}\; \backslash \; course\; \backslash \; textrm\; \{Co-NP\}$ in Théorie of complexity.

The year 1956 is a " double-date" , it is that of the Théorème flood-max/cut-min generalizing the theorems of Menger, König and Hall, and at the origin of the linear Programming. Moreover, it is the year of the Algorithme of Kruskal, first Algorithme glouton in graph. This result is at the origin of a true rebirth for the theory of the Matroïde S which will be narrowly connected to the graph theory by Tutte.

In 1960, conjecture of Bank.

In 1976, Theorem of the four colors (resolution of the problem arising from Guthrie).

In 2002, Theorem of the perfect graphs (resolution of the conjecture of Bank).

Second half of the 20th century, will have seen the graph theory interacting with many other fields. With leaving the war, the problems of floods in the graphs were at the origin of the linear Programming and the Algorithme of the simplex. The problems of covering trees were going to be at the origin of the generalization of the concept of graph by that of Matroïde and of the parallels between the two theories, in particular at the algorithmic level (what will influence the vocabulary of the two theories). The problems of coupling were going to be at the origin at the same time of the Théorie of complexity (including the algorithms of approximation) and of the Approche polyèdrale. To analyze the effectiveness of its algorithm of coupling, Jack Edmonds defined the concept of polynomial algorithm, thus pushing Cook has to show the existence of the first Np-complete problem . The facility of the problem of coupling contrasted with the difficulty of transverse but the optimum difficult to obtain is always limited by the easy optimum multiplied by 2, thus the algorithm of coupling max is the first approximate algorithm. Edmonds then showed that the convex envelope of the couplings of any graph can be described by a certain type of linear inequalities, it is the first polyèdral result. This approach was going to be a great success when its effectiveness in practice was going to be revealed by work on the sales representative (which is by-elsewhere the first problem non-approximable), and its theoretical effectiveness by the Théorème Optimization/Séparation and the characterizations polyèdrales of the graphs bipartis and the perfect graphs.

Abstract presentation

Any directed graph can be represented by a drawing, like illustrates it figure 2. It is besides of this representation that the " term; arc" is resulting (even if " flèche" would seem more suitable). On a drawing, one can represent the tops by points (or circles) and the arcs by arrows: an arc connects a top towards another. For the not-directed graphs, one replaces the arrows by features as in figure 1.

Intuitive definition

The intuitive way (with a formalism less heavy than that of the Sixties) to present the directed simple graphs is the following one:

a simple graph directed G is a couple (S, A), where:

* S is a unit whose elements are the tops ;

* has is a whole of couples (ordered) of tops, called arcs .

For the graph of figure 2 represented higher, one would have $S\; =\; \backslash \; left\; \backslash \; \{1,2,3,4,5\; \backslash \; right\; \backslash \}$ and $A\; =\; \backslash \; left\; \backslash \; \{(1,2),\; (2,3),\; (3,1),\; (2,5),\; (4,5),\; (5,5)\; \backslash \; right\; \backslash \}$.

Same manner one can define the not-directed simple graphs:

a simple graph not-directed G is a couple (S, A), where:

* S is a unit whose elements are the tops ;

* has is a whole of even (not ordered) of tops, called edges .

For the more general graphs (not inevitably simple) one can leave oneself there while speaking about collection (or family) about pairs (or couples for the directed case) in the overall place.

Other types of graphs exist:

• One speaks about multigraphe or p-graph when with the more $p$ arcs (resp. edges) can connect two tops. One can define it by an application of $S\; \backslash \; times\; S$ in $\backslash \; mathbb\; \{NR\}$.
• a graph is valué so with any arc (resp. edge) is associated a value (for example: a weight, a cost, a distance,…). One speaks about function of definite valuation of $S\; \backslash \; times\; S$ in $\backslash \; mathbb\; \{R\}$.

Data-processing representations of the graphs

One can store a graph using various structures. One can number the tops, then to give the arcs in the shape of a list of couples. One can also use a Matrice of adjacency, more rapid but requiring more memory capacity. Lastly, one can associate with each top a list of adjacency , i.e. a list containing all the tops towards which point the edges on the basis of this top.

See also:

• static Representation of a graph.
• dynamic Representation of a graph.

Interest in the " world réel"

The graphs are used to model, inter alia:

• an highway network with large scales: each city is a top, each road between two cities is an arc (if it does not pass by another city), and even in general two arcs: one in each direction if the road is not with one way.
• an highway network with small scales: each intersection is a top, each section of street between two intersections is an arc.
• a network of bus, a rail network…
• In analyzes social networks the graphs make it possible to observe and to describe the formal aspects of the Social networks to make the thereafter of it analyzes sociological.
• the Web: each page is a top, each hypertext link is an arc (of the page which contains it towards the pointed page).
• the network Internet is a graph whose tops are the waiters and the users, and the edges the various interconnections.
• Much of discrete systems: who pass from a state to another in a discontinuous way during time, by jumps. See Finite-state machine and System of transition from states.
• In mechanics of the solid, the Graphe of the connections is a tool of assistance to the kinematic Modélisation of the mechanisms. The properties of the graph have sometimes a significance (number of cycle, classifies, etc).
• In electronics, they represent the integrated circuits.
• a network of Neuron S formal can be represented by a directed graph, valué - each arc carries a value - and whose each top is also valué of sound threshold of activation .

The graphs are used much in data processing. In addition to their effectiveness in modeling programming science of structure of complex data, one meets them for example for:

• the databases: a relational model of data is representable by a gathering directed graph of the relations (tops of the graph) and of the dependences (arcs of the graph). One speaks in particular about semantic graph standardized to indicate a diagram of data relational resulting from the process of standardization;
• the semantic Web: an ontology is described like a whole of concepts (tops of the directed graph) and relations (arcs of the graph);
• the parallelism: the techniques of optimization of algorithm or determination of order of coherent execution in this field often take in entry graphs of dependence of flood of instructions or data, where the tops are respectively instructions (of code to be carried out) and data (initial or calculated) and the arcs of the relations of temporal dependence (such instruction must be carried out after such other; such data must be calculated before such other).

Criticisms

Alain Connes considers that knowledge on the graph theory does not form a theory but " a knowledge, a series of faits". It is in fact rather largely accepted that the graph theory is a theory " horizontale" rather than " verticale" , in the direction where it disperses around topics apparently without bond (like the connexity and coloring) instead of erecting scaffolding a pyramid of strongly dependant results.

Algorithmic and graph theory

The graph theory contains many objects (ways, trees, couplings, click, colorings…) one can give a simple representation (by the drawing). From the Algorithmic point of view, the problems (generally of Optimization) involved in these objects can be very difficult (from nature " explosive" or " exponentielle" combinative objects) or being of greatest simplicity (when the objects have very strong properties). Thus the problem of the covering tree is simpler (the simplicity of this structure is collected by the more general concept of Matroïde) while the problem of coloring belongs to hardest to solve. It is in this context, even if its bases belong without question to the Logique, that the theory of the Complexité developed, until applying nowadays to many fields.

Among the traditional problems which one knows algorithmic complexity let us quote:

• the search for Eulérien course (as in the Problem of the seven bridges of Königsberg): easy (i.e. one knows a polynomial algorithm of resolution)
• the connexity: does there exist a way connecting two tops? : easy
• the Tree covering with minimal weight: easy
• the shortest way between two tops of a valué graph: easy
• more the long way between two tops of a valué graph: difficult (i.e. Np-complete)
• the Coloring of graph with a fixed number of colors: difficult starting from three colors
• maximum dense subgraphs (sometimes called Clicks S): difficult
• problems of maximum or minimal floods: easy if the flood can be fractional, difficult if not starting from three whole floods
• the problem of assignment of tasks: easy
• the Problem of the sales representative: difficult
• the problem of the Chinese post-office employee: easy
• the Project management with the Network PERT: easy
• the Hamiltonian graphs and hypo-Hamiltonians: difficult

There of course exist problems whose difficulty (theoretical complexity) is not established yet, it is the case of the fundamental problems of the isomorphism of graph. One says of two graphs $G=\; (S,\; A)$ and $G\text{'}=\; (,\; A\text{'})$ which they are isomorphous if

there exists a Bijection $f:\; S\; \backslash \; leftrightarrow\; S\text{'}$ such as any couple of tops $(U,\; v)\; \backslash \; in\; S^2$ one has $(U,\; v)\; \backslash \; in\; has$ if and only if $(F\; (U),\; F\; (v))\backslash \; in\; A\text{'}$; i.e. if the tops of $G\text{'}$ well are re-elected, $G\text{'}=G$ then is obtained.

Ironically one cannot (one does not know a good algorithm for) recognize two identical graphs, worse it is not known if this problem is easy or difficult!

• Isomorphism of graphs: unknown difficulty (no known polynomial algorithm and no known polynomial reduction of a Np-complete problem to this problem)

Internal bonds

• Lexicon of the graph theory.
• List of the algorithms of the graph theory.
• Automat.
• Graph of Hoffman-Singleton.
• dynamic Programming.
• RDF : model of graph to describe them (méta-) given.

Software

• Graphviz : software continuation of handling and visualization of graphs.
• software Pajek allowing to visualize broad networks

External bonds

• Graph theory Course of graph theory.
• Course of graphs lodged on the site of the Analysis laboratory and architecture of the systems
• interactive Course of the ENSEM on the Graph theory
• Polytechnic Course
• Course of the ESIEE
• Diestel - Graph Theory, Electronic Edition.

Random links:

Gertrud | Mareilles | Amy Yasbeck | Mordecai Kaplan | Etienne Poncher (archbishop of Turns)