Graph theory - it is one of the subsections of mathematics, the main feature of which is the geometrical method in the study of objects.It is considered to be the founder of the famous mathematician Euler.
Application of graph theory to the late 19th century, was reduced to the solution of entertaining problems and did not attract significant attention.Since the 20th century, when the graph theory was formed as an independent mathematical discipline, it has been widely used in the fields of science, cybernetics, physics, logistics, programming, biology, electronics, transportation and communication systems.
Basic concepts of graph theory
Base is Earl.The terminology can be found such a thing as a network of identical graph.Last - is a non-empty number of points, that is, vertices and segments, ie edges, both ends of which correspond to a given number of points.Graph theory does not put a definite meaning to the values of edges and vertices.For example, the city and the roads connecting them, where the first - it is the top of the graph, and the second - the ribs.Greater importance is given to the theory of the arcs.If the edges have a direction, it is called the arc, if the graph with oriented edges, it is called a digraph.
In the terminology of the theory of the same concepts are the following:
subgraph is a graph, all edges and vertices are among the vertices and edges.
connected graph - one that has two different peaks exist chain connecting them.
weighted connected graph - one that set the weighting function.
tree - a connected graph without cycles.
skeleton - subgraph which is a tree.
When the image of the graph on the plane using a specific notation: top corresponds to the selected point on the surface of the simplest, and if there is an edge between vertices, the corresponding points are combined segment.If the graph-oriented, these segments are replaced by the arrows.
But it is not necessary to compare the image of the graph with him, ie with an abstract structure, because one count can be given more than one graphical representation.Drawing on the plane is given in order to see which pair of vertices edges together and which are not.
Among some problems in the theory of graphs release:
- problem of the shortest circuit (replacement of equipment, accommodation places ambulances and telephone exchanges).
- maximum flow problem (ordered motion in a dynamic network, distribution of work, the organization of capacity).
- covering problem and packages (accommodation dispatch centers).
- coloring in columns (memory allocation on electronic computers).
- Communication networks and graphs (a communication network, the analysis of communication networks).
is currently not possible to program the majority of tasks without the knowledge of graph theory.This facilitates and simplifies the work with a computer.
Program uses a variety of structures and universal methods for solving problems and one of them is the theory of graphs.Its importance is hard to overestimate.Graph theory in programming simplifies the search for information, to optimize the program, convert and distribute the data.Through the theory of algorithms, there is a possibility of application and assessment to use for specific tasks, to carry out a modification of the algorithm, without decreasing the degree of mathematical certainty of the final version of the program.
important feature of the control system or model is a set of binary relations with the set of actions and data units.These structures are the only part of the program and converts them information.Therefore, the graphs are the basis of design for the programmer.
