Graph concept in history. Graph theory

Leonard Euler is considered the founder of graph theory. In 1736, in one of his letters, he formulates and proposes a solution to the problem of the seven Königsberg bridges, which later became one of the classical problems graph theory.

The first problems in graph theory were related to solving mathematical recreational problems and puzzles. Here is a retelling of an excerpt from Euler’s letter dated March 13, 1736: “I was given a problem about an island located in the city of Königsberg and surrounded by a river with 7 bridges across it. The question is whether someone can go around them continuously, passing only once over each bridge. And then I was informed that no one had yet been able to do this, but no one had proven that it was impossible. This question, although trivial, seemed to me, however, worthy of attention in that neither geometry, nor algebra, nor combinatorial art are sufficient to solve it. After much thought, I found an easy rule, based on a completely convincing proof, with the help of which it is possible, in all problems of this kind, to immediately determine whether such a detour can be made through any number and any number of bridges located in any way or not.” The Königsberg bridges can be schematically depicted as follows:

Euler's rule:

1. In a graph that does not have vertices of odd degrees, there is a traversal of all edges (and each edge is traversed exactly once) starting at any vertex of the graph.

2. In a graph that has two and only two vertices with odd degrees, there is a traversal starting at one vertex with odd degree and end to the other.

3. In a graph that has more than two vertices with odd degrees, such a traversal does not exist.

There is another type of problem related to traveling along graphs. We are talking about problems in which it is necessary to find a path passing through all vertices, and no more than once through each. A cycle that passes through each vertex once and only once is called a Hamiltonian line (after William Rowan Hamilton, the famous Irish mathematician of the last century who was the first to study such lines). Unfortunately, a general criterion has not yet been found with the help of which one could decide whether a given graph is Hamiltonian, and if so, then find all Hamiltonian lines on it.

Formulated in the mid-19th century. The four color problem also looks like an entertaining problem, but attempts to solve it have led to some research on graphs that have theoretical and applied value. The four-color problem is formulated as follows: “Can an area of ​​any flat map be colored with four colors so that any two adjacent areas are colored with different colors?” The hypothesis that the answer is affirmative was formulated in the mid-19th century. In 1890, a weaker statement was proven, namely that any flat map can be colored in five colors. By associating any planar map with its dual planar graph, we obtain an equivalent formulation of the problem in terms of graphs: Is it true that the chromatic number of any planar graph is less than or equal to four? Numerous attempts to solve the problem influenced the development of a number of areas of graph theory. In 1976, a positive solution to the problem using a computer was announced.

Another old topological problem that has been particularly resistant to solution for a long time and has haunted the minds of puzzle lovers is known as the “electricity, gas and water supply problem.” In 1917, Henry E. Dudeney gave it this formulation. Gas, electricity and water must be installed in each of the three houses shown in the figure.

Graph theory. 1

The history of the emergence of graph theory. 1

Euler's rule. 1

Literature

1. Belov Graph Theory, Moscow, "Science", 1968.

2. New pedagogical and information Technology E.S. Polat , Moscow, "Akademia" 1999

3. Kuznetsov O.P., Adelson-Velsky G.M. Discrete mathematics for the engineer. – M.: Energoatomizdat, 1988.

4. Cook D., Baze G. Computer mathematics. – M.: Science, 1990.

5. Nefedov V.N., Osipova V.A. Discrete mathematics course. – M.: MAI Publishing House, 1992.

6. Ore O. Graph theory. – M.: Science, 1980.

7. Ismagilov R.S., Kalinkin A.V. Materials for practical classes in the course: Discrete Mathematics

German Graf), title of nobility. Introduced in Russia by Peter I (B.P. Sheremetev was the first to receive it in 1706). At the end of the 19th century. over 300 count families were taken into account. Liquidated by Decree of the All-Russian Central Executive Committee and the Council of People's Commissars of November 11, 1917.

Excellent definition

Incomplete definition ↓

Graph

Anton (Graf, Anton) 1736, Winterthur - 1813, Dresden. German painter. He studied in 1753-1756 with I. W. Schellenberg in Winterthur, then with I. Ya. Hyde in Augsburg. He worked as a portrait painter in Regensburg, Winterthur, Augsburg, Munich, Zurich. From 1766 - court artist in Dresden. Since 1789 - professor at the Dresden Academy of Arts. Member of the Berlin, Vienna, Munich academies of arts. Traveled extensively in Germany and Switzerland. He was a portrait painter, also painted landscapes, and was involved in miniatures. The artist's early works were executed in the tradition of ceremonial Baroque portraiture. The images of noble persons of the royal palace of Prussia are full of solemnity and representativeness in the portraits of Frederick, Prince of Prussia (1777-1778), Frederica, Princess of Prussia (1787), Frederick William II, King of Prussia (1788, all - Berlin, Charlottenburg). Strong chiaroscuro and warm color scheme indicate the young artist’s passion for Rembrandt’s style. In the 1780-1790s, the Count often painted models against the backdrop of a landscape, which somewhat softened the tension and static figures in his portraits ( Henry VIII, 1804, Germany, private collection; I. F. von Tilman, Nuremberg, German National. museum). In the spirit of neoclassical tastes of the era, he depicts those portrayed as ancient graces in a landscape (Frederica Hillendorff, 1803, Germany, private collection). The portraits of people close to the artist are more profound in conveying their inner state: the Artist K. K. Lunwig (1808, Hamburg, Kunsthalle), lyrical female images - Louise Elisabeth Funk (1790, Leipzig, Museum fine arts), Caroline Susanna Graf (1805, Hamburg, Kunsthalle). Subtle light-and-shade modeling emphasizes the clear plasticity of the figures inherent in the Count’s images. The airy sfumato enveloping the figures testifies to the study of the techniques of English portraiture of the 18th century. Portraits of outstanding figures of the Enlightenment - S. Gessner (1765-1766, Zurich, Kunsthalle), G. E. Lessing (1771, Leipzig, University Library), K. M. Wieland (1794, Weimar, Goethe Museum), I. G. Sulzer (1771, Winterthur, Kunsthalle) - perhaps the most significant thing that was created by the artist. In the portraits of the artist’s father-in-law I. G. Sulzer, a famous German philosopher, aesthetics and mathematician, and S. Gessner, a Swiss poet, author of the poetry collection Idylls (1756), the Count uses the scheme of a baroque portrait, depicting models at a moment of seemingly interrupted movement. A true artist of the Age of Enlightenment, the Count strives to reveal the spirituality and bright mind of people who have become the cultural heritage of the nation. The portraits are painted on a dark background, like a number of other later works (H. I. Medem, 1796; G. L. Gogel, 1796, both St. Petersburg, State Hermitage Museum). Interest in the psychologically in-depth development of an image is also inherent in the artist’s self-portraits. In early self-portraits of 1765 (New York, Historical Society) and 1766 (Dresden, Art Gallery) the motif of interrupted movement brings some traditionality to the compositional solution. Late works (1794-1795, Dresden, Art Gallery; 1808, Winterthur, Kunsthalle) create the image of an artist whose work marked many important phenomena of German culture of the 18th century, laying the traditions of realistic imagery of the subsequent century. IN late period the artist painted a number of landscapes that characterize his excellent command of drawing from life, his interest in plein air, and the development of the problem of “mood landscape” (View of the surroundings of Dresden, 1800; Morning, ca. 1800; Noon, ca. 1800; Evening, ca. 1800, all - Dresden, Art Gallery).

VLADIMIR STATE PEDAGOGICAL UNIVERSITY

ABSTRACT

"GRAPH THEORY"

Performed:

Zudina T.V.

Vladimir 2001

1. Introduction

2. History of the emergence of graph theory

3. Basic definitions of graph theory

4. Basic theorems of graph theory

5. Problems on the application of graph theory

6. Application of graph theory in a school mathematics course

7. Application of graph theory in various fields of science and technology

8. Recent advances in graph theory

§1. HISTORY OF THE APPEARANCE OF GRAPH THEORY.

The founder of graph theory is considered to be the mathematician Leonhard Euler (1707-1783). The history of this theory can be traced through the correspondence of the great scientist. Here is a translation of the Latin text, which is taken from Euler’s letter to the Italian mathematician and engineer Marinoni, sent from St. Petersburg on March 13, 1736 [see. pp. 41-42]:

“I was once asked a problem about an island located in the city of Königsberg and surrounded by a river across which seven bridges are thrown. The question is whether anyone can go around them continuously, passing only once through each bridge. And then I was informed that no one still have not been able to do this, but no one has proven that it is impossible. This question, although trivial, seemed to me, however, worthy of attention in that neither geometry, nor algebra, nor combinatorial art are sufficient to solve it... After much thought, I found an easy rule, based on a completely convincing proof, with the help of which it is possible to immediately determine in all problems of this kind whether such a detour can be made through any number of bridges located in any way or not. so that they can be represented in the following figure[fig.1] , on which A denotes an island, and B , C and D - parts of the continent separated from each other by river branches. The seven bridges are indicated by letters a , b , c , d , e , f , g ".

(FIGURE 1.1)

Regarding the method he discovered to solve problems of this kind, Euler wrote [see. pp. 102-104]:

“This solution, by its nature, apparently has little to do with mathematics, and I do not understand why one should expect this solution from a mathematician rather than from any other person, for this decision is supported by reasoning alone, and there is no need to involve to find this solution, any laws inherent in mathematics. So, I do not know how it turns out that questions that have very little to do with mathematics are more likely to be solved by mathematicians than by others."

So is it possible to get around the Königsberg bridges by passing only once over each of these bridges? To find the answer, let's continue Euler's letter to Marinoni:

"The question is to determine whether it is possible to bypass all these seven bridges, passing through each only once, or not. My rule leads to the following solution to this question. First of all, you need to look at how many areas there are, separated by water - such , which have no other transition from one to another except through a bridge. In this example, there are four such sections - A , B , C , D . The next thing to distinguish is whether the number of bridges leading to these individual sections is even or odd. So, in our case, five bridges lead to section A, and three bridges each lead to the rest, i.e. The number of bridges leading to individual sections is odd, and this alone is enough to solve the problem. Once this has been determined, we apply the following rule: if the number of bridges leading to each individual section were even, then the detour in question would be possible, and at the same time it would be possible to start this detour from any section. If two of these numbers were odd, for only one cannot be odd, then even then the transition could take place, as prescribed, but only the beginning of the circuit must certainly be taken from one of those two sections to which the odd number leads bridges. If, finally, there were more than two sections to which an odd number of bridges lead, then such a movement is generally impossible ... if other, more serious problems could be brought here, this method could be of even greater benefit and should not be neglected." .

The rationale for the above rule can be found in a letter from L. Euler to his friend Ehler dated April 3 of the same year. We will retell below an excerpt from this letter.

The mathematician wrote that the transition is possible if there are no more than two areas in the fork of the river, to which an odd number of bridges lead. To make it easier to imagine this, we will erase the already traversed bridges in the figure. It’s easy to check that if we start moving in accordance with Euler’s rules, cross one bridge and erase it, then the figure will show a section where again there are no more than two areas to which an odd number of bridges leads, and if there are areas with an odd number bridges we will be located in one of them. Continuing to move on like this, we will cross all the bridges once.

The story of the bridges of the city of Königsberg has a modern continuation. Let's open, for example, a school textbook on mathematics edited by N.Ya. Vilenkina for the sixth grade. In it, on page 98, under the heading of developing attentiveness and intelligence, we will find a problem that is directly related to the one that Euler once solved.

Problem No. 569. There are seven islands on the lake, which are connected to each other as shown in Figure 1.2. Which island should a boat take travelers to so that they can cross each bridge and only once? Why can't travelers be transported to the island? A ?

(FIGURE 1.2)

Solution. Since this problem is similar to the problem of the Königsberg bridges, when solving it we will also use Euler’s rule. As a result, we get the following answer: the boat must deliver travelers to the island E or F so that they can cross each bridge once. From the same Euler rule it follows that the required detour is impossible if it starts from the island A .

In conclusion, we note that the problem of the Königsberg bridges and similar problems, together with a set of methods for their study, constitute a very important branch of mathematics in practical terms, called graph theory. The first work on graphs belonged to L. Euler and appeared in 1736. Subsequently, Koenig (1774-1833), Hamilton (1805-1865), and modern mathematicians C. Berge, O. Ore, A. Zykov worked on graphs.

§2. BASIC THEOREMS OF GRAPH THEORY

Graph theory, as mentioned above, is a mathematical discipline created by the efforts of mathematicians, therefore its presentation includes the necessary strict definitions. So, let's proceed to an organized introduction of the basic concepts of this theory.

Definition 2.01. Count is a collection of a finite number of points called peaks graph, and pairwise lines connecting some of these vertices, called ribs or arcs graph.

This definition can be formulated differently: count is called a non-empty set of points ( peaks) and segments ( ribs), both ends of which belong to a given set of points (see Fig. 2.1).

(FIGURE 2.1)

In what follows, we will denote the vertices of the graph by Latin letters A , B ,C ,D. Sometimes the graph as a whole will be denoted by one capital letter.

Definition 2.02. The vertices of a graph that do not belong to any edge are called isolated .

Definition 2.03. A graph consisting only of isolated vertices is called zero - count .

Designation: O " – a graph with vertices that has no edges (Fig. 2.2).

(FIGURE 2.2)

Definition 2.04. A graph in which every pair of vertices is connected by an edge is called complete .

Designation: U " – graph consisting of n vertices and edges connecting all possible pairs of these vertices. Such a graph can be represented as n– a triangle in which all diagonals are drawn (Fig. 2.3).

(FIGURE 2.3)

Definition 2.05. Degree peaks is the number of edges to which a vertex belongs.

Designation: p (A) – vertex degree A . For example, in Figure 2.1: p (A)=2, p (B)=2, p (C)=2, p (D)=1, p (E)=1.

Definition 2.06. Count, degrees of all k whose vertices are identical is called homogeneous count degrees k .

Figures 2.4 and 2.5 show homogeneous graphs of the second and third degree.

(FIGURE 2.4 and 2.5)

Definition 2.07. Supplement given graph is a graph consisting of all the edges and their ends that must be added to the original graph to obtain a complete graph.

Figure 2.6 shows the original graph G , consisting of four vertices and three segments, and in Figure 2.7 - the complement of this graph - the graph G " .

(FIGURE 2.6 and 2.7)

We see that in Figure 2.5 there are ribs A.C. And BD intersect at a point that is not a vertex of the graph. But there are cases when a given graph needs to be represented on a plane in such a way that its edges intersect only at the vertices (this issue will be discussed in detail further, in paragraph 5).

Definition 2.08. A graph that can be represented on a plane in such a way that its edges intersect only at the vertices is called flat .

For example, Figure 2.8 shows a planar graph that is isomorphic (equal) to the graph in Figure 2.5. However, note that not every graph is planar, although the converse is true, that is, any planar graph can be represented in the usual form.

(FIGURE 2.8)

Definition 2.09. A polygon of a planar graph that does not contain any vertices or edges of the graph is called edge .

2016 school year Year

1. Introduction

2. History of the emergence of graph theory

3. Basic concepts of graph theory

4. Basic theorems of graph theory

5. Methods of representing graphs in a computer

6. Review of graph theory problems

7. Conclusion

8. References

Introduction

Recently, research in areas traditionally related to discrete mathematics has become increasingly prominent. Along with such classical branches of mathematics as mathematical analysis, differential equations, V curriculum In the specialty "Applied Mathematics" and many other specialties, sections on mathematical logic, algebra, combinatorics and graph theory appeared. The reasons for this are not difficult to understand by simply identifying the range of problems solved on the basis of this mathematical apparatus.

The history of the emergence of graph theory.

1. Problem about the Königsberg bridges. In Fig. 1 shows a schematic plan of the central part of the city of Koenigsberg (now Kaliningrad), including two banks of the Pergola River, two islands in it and seven connecting bridges. The task is to go around all four parts of the land, crossing each bridge once, and return to the starting point. This problem was solved (it was shown that there was no solution) by Euler in 1736.

rice. 1

2. The problem of three houses and three wells. There are three houses and three wells, somehow located on a plane. Draw a path from each house to each well so that the paths do not intersect (Fig. 2). This problem was solved (it was shown that there is no solution) by Kuratovsky in 1930.

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3. The four color problem. A division of a plane into non-overlapping areas is called a map. Areas on a map are called adjacent if they have a common border. The task is to color the map in such a way that no two adjacent areas are painted with the same color (Fig. 3). Since the end of the century before last, the hypothesis has been known that four colors are enough for this. In 1976, Appel and Heiken published a solution to the four-color problem, which was based on a computer search. The solution to this problem “programmatically” was a precedent that gave rise to a heated debate, which is by no means over. The essence of the published solution is to try a large but finite number (about 2000) types of potential counterexamples to the four-color theorem and show that not a single case is a counterexample. This search was completed by the program in about a thousand hours of supercomputer operation. It is impossible to check the resulting solution “manually” - the scope of enumeration goes far beyond human capabilities. Many mathematicians raise the question: can such a “program proof” be considered a valid proof? After all, there may be errors in the program... Methods for formally proving the correctness of programs are not applicable to programs of such complexity as the one being discussed. Testing cannot guarantee the absence of errors and in this case is generally impossible. Thus, we can only rely on the programming skills of the authors and believe that they did everything right.

Rice. 3

Basic concepts of graph theory

1) Graph G(V,E) is a collection of two sets – a non-empty set V (set of vertices) and a set E of two-element subsets of the set V (E – set of edges).

2) Oriented is a graph in which is a set of ordered pairs of vertices of the form (x,y), where x is called the beginning, and y is the end of the arc. The arc (x, y) is often written as . They also say that the arc leads from vertex x to vertex y, and vertex y adjacent with vertex x.

3) If an element of the set E can be a pair identical(not distinct) elements of V, then such an element of the set E is called loop, and the graph is called graph with loops(or pseudograph).

4) If E is not a set, but set containing several identical elements, then these elements are called multiple edges, and the graph is called multigraph.

5) If the elements of the set E are not necessarily two-element, but any subsets of the set V, then such elements of the set E are called hyperarcs, and the graph is called hypergraph.

6) If the function is specified F: V → M and/or F: E → M, then the set M called a set notes, and the graph is called marked(or loaded). The set of marks is usually letters or integers. If the function F is injective, that is, different vertices (edges) have different labels, then the graph is called numbered.

7) Subgraph is called the graph G′(V′,E′), where and/or .

a) If V′ = V, then G′ is called core subgraph G.

b) If , then the graph G′ is called own subgraph of graph G.

c) A subgraph G′(V′,E′) is called a regular subgraph of the graph G(V,E) if G′ contains all possible edges of G.

8) Degree (valence) vertices is the number of edges incident to this vertex (the number of vertices adjacent to it).

9) Route in a graph is an alternating sequence of vertices and edges in which any two adjacent elements are incident.

a) If , then the route closed, otherwise open.

b) If all edges are different, then the route is called chain.

c) If all vertices (and therefore edges) are different, then the route is called simple chain.

d) A closed circuit is called cycle.

e) A closed simple circuit is called simple loop.

f) A graph without cycles is called acyclic.

g) For digraphs, the chain is called by, and the cycle is contour.

rice. 4. Routes, chains, cycles

Example

In the graph, the diagram of which is shown in Fig. 4:

1. v 1, v 3, v 1, v 4 – a route, but not a chain;

2. v 1, v 3, v 5, v 2, v 3, v 4 – a chain, but not a simple chain;

3. v 1, v 4, v 3, v 2, v 5 – simple chain;

4. v 1, v 3, v 5, v 2, v 3, v 4, v 1 – a cycle, but not a simple cycle;

5. v 1 , v 3 , v 4 , v 1 – a simple cycle.

10) If a graph has a cycle (not necessarily simple) containing all the edges of the graph once, then such a cycle is called Eulerian cycle.

11) If a graph has a simple cycle containing all the vertices of the graph (one at a time), then such a cycle is called Hamiltonian cycle.

12) tree called a connected graph without cycles.

13) skeleton A tree containing all the vertices of a graph is called.

14) Matching is a set of edges in which no two are adjacent.

15) The matching is called maximum, if no superset of it is independent.

16) Two vertices in a graph connected, if there is a simple chain connecting them.

17) A graph in which all vertices are connected is called coherent.

18) A graph consisting only of isolated vertices is called quite incoherent.

19) Route length the number of edges in it is called (with repetitions).

20) Distance between vertices u and v is called the length of the shortest chain, and the shortest chain itself is called geodetic.

21) Diameter of a graph G is called the length of the longest geodesic.

22) Eccentricity vertex v in a connected graph G(V,E) is the maximum distance from vertex v to other vertices of the graph G.

23) Radius of a graph G is called the smallest of the eccentricities of the vertices.

24) Vertex v is called central, if its eccentricity coincides with the radius of the graph.

25) The set of central vertices is called center graph.

rice. 5 Eccentricities of vertices and centers of graphs (highlighted).

Related information.

Material from Wikipedia - the free encyclopedia

Graph theory- a branch of discrete mathematics that studies the properties of graphs. In a general sense, a graph is represented as a set peaks(nodes) connected ribs. In a strict definition, such a pair of sets is called a graph. $G\; =\; (V,\; E)$, Where $V$ is a subset of any countable set, and $E$- subset $V\backslash times\; V$.

Graph theory finds application, for example, in geographic information systems (GIS). Existing or newly designed houses, structures, blocks, etc. are considered as vertices, and the roads, utility networks, power lines, etc. connecting them are considered as edges. The use of various calculations performed on such a graph allows, for example, to find the shortest detour route or the nearest grocery store, or to plan the optimal route.

Graph theory contains a large number of unsolved problems and as yet unproven hypotheses.

History of graph theory

Leonard Euler is considered the founder of graph theory. In 1736, in one of his letters, he formulated and proposed a solution to the problem of the seven bridges of Königsberg, which later became one of the classical problems of graph theory.

Graph theory terminology

Representation of graphs on a plane

When depicting graphs in pictures, it is most often used next system notations: the vertices of the graph are depicted by dots or, when specifying the meaning of the vertex, by rectangles, ovals, etc., where the meaning of the vertex is revealed inside the figure (graphs of algorithm flowcharts). If there is an edge between the vertices, then the corresponding points (shapes) are connected by a line or arc. In the case of a directed graph, arcs are replaced by arrows, or the direction of an edge is explicitly indicated. Sometimes explanatory inscriptions are placed next to the edge, revealing the meaning of the edge, for example, in transition graphs of finite state machines. There are planar and non-planar graphs. A planar graph is a graph that can be depicted in a picture (plane) without intersecting edges (the simplest are a triangle or a pair of connected vertices), otherwise the graph is non-planar. In the event that the graph does not contain cycles (containing at least one path one-time traversal of edges and vertices with a return to the original vertex), it is usually called a “tree”. Important types of trees in graph theory are binary trees, where each vertex has one incoming edge and exactly two outgoing ones, or is finite - has no outgoing edges and contains one root vertex with no incoming edge.

The image of a graph should not be confused with the graph itself (abstract structure), since more than one graphical representation can be associated with one graph. The image is intended only to show which pairs of vertices are connected by edges and which are not. It is often difficult in practice to answer the question whether two images are models of the same graph or not (in other words, whether the graphs corresponding to the images are isomorphic). Depending on the task, some images may provide more clarity than others.

Some problems of graph theory

• The Seven Bridges of Königsberg Problem is one of the first results in graph theory, published by Euler in .
• The four-color problem was formulated in 1852, but a non-classical proof was obtained only in 1976 (4 colors are enough for a map on a sphere (plane)).
• The traveling salesman problem is one of the most famous NP-complete problems.
• The clique problem is another NP-complete problem.
• Finding the minimum spanning tree.
• Graph isomorphism - is it possible to obtain another by renumbering the vertices of one graph?
• Planarity of the graph - is it possible to depict the graph on a plane without intersections of edges (or with a minimum number of layers, which is used when tracing interconnections of elements of printed circuit boards or microcircuits).

Application of graph theory

see also

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Literature

• Distel R. Graph theory Trans. from English - Novosibirsk: Publishing House of the Institute of Mathematics, 2002. - 336 p. ISBN 5-86134-101-X.
• Diestel R.. - NY: Springer-Verlag, 2005. - P. 422.
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• Berge K.
• Emelichev V. A., Melnikov O. I., Sarvanov V. I., Tyshkevich R. I. Lectures on graph theory. M.: Nauka, 1990. 384 p. (Ed. 2, revised M.: URSS, 2009. 392 p.)
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• Chemical applications of topology and graph theory. Ed. R. King. Per. from English M.: Mir, 1987.
• Kirsanov M. N. Graphs in Maple. M.: Fizmatlit, 2007. 168 p. vuz.exponenta.ru/PDF/book/GrMaple.pdf eqworld.ipmnet.ru/ru/library/books/Kirsanov2007ru.pdf
• Christofides N.
• Cormen T.H. et al. Part VI. Algorithms for working with graphs // Algorithms: construction and analysis = Introduction to Algorithms. - 2nd ed. - M.: Williams, 2006. - P. 1296. - ISBN 0-07-013151-1.
• Ore O.. - 2nd ed. - M.: Science, 1980. - P. 336.
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• Sergey Melnikov// Science and life . - 1996. - Issue. 3. - pp. 144-145. The article is about the graph game Sim, invented by Gustav Simmons.

Links

• : a program that provides the user with a wide range of tools and methods for visualizing and searching for information in graphs

Classical analysis Function theory Differential and integral equations

An excerpt characterizing Graph Theory

But before he finished these words, Prince Andrei, feeling tears of shame and anger rising in his throat, was already jumping off his horse and running to the banner.
- Guys, go ahead! – he shouted childishly.
"Here it is!" thought Prince Andrei, grabbing the flagpole and hearing with pleasure the whistle of bullets, obviously aimed specifically at him. Several soldiers fell.
- Hooray! - Prince Andrei shouted, barely holding the heavy banner in his hands, and ran forward with undoubted confidence that the entire battalion would run after him.
Indeed, he only ran a few steps alone. One soldier set off, then another, and the whole battalion shouted “Hurray!” ran forward and overtook him. The non-commissioned officer of the battalion ran up and took the banner, which was shaking from the weight in the hands of Prince Andrei, but was immediately killed. Prince Andrei again grabbed the banner and, dragging it by the pole, fled with the battalion. Ahead of him, he saw our artillerymen, some of whom fought, others abandoned their cannons and ran towards him; he also saw French infantry soldiers who grabbed artillery horses and turned the guns. Prince Andrei and his battalion were already 20 steps from the guns. He heard the incessant whistling of bullets above him, and soldiers constantly groaned and fell to the right and left of him. But he didn't look at them; he peered only at what was happening in front of him - on the battery. He clearly saw one figure of a red-haired artilleryman with a shako knocked on one side, pulling a banner on one side, while a French soldier was pulling the banner towards himself on the other side. Prince Andrey already clearly saw the confused and at the same time embittered expression on the faces of these two people, who apparently did not understand what they were doing.
"What are they doing? - thought Prince Andrei, looking at them: - why doesn’t the red-haired artilleryman run when he has no weapons? Why doesn't the Frenchman stab him? Before he can reach him, the Frenchman will remember the gun and stab him to death.”
Indeed, another Frenchman, with a gun to his advantage, ran up to the fighters, and the fate of the red-haired artilleryman, who still did not understand what awaited him and triumphantly pulled out the banner, was to be decided. But Prince Andrei did not see how it ended. It seemed to him that one of the nearby soldiers, as if swinging a strong stick, hit him in the head. It hurt a little, and most importantly, it was unpleasant, because this pain entertained him and prevented him from seeing what he was looking at.
"What is this? I'm falling? My legs are giving way,” he thought and fell on his back. He opened his eyes, hoping to see how the fight between the French and the artillerymen ended, and wanting to know whether the red-haired artilleryman was killed or not, whether the guns were taken or saved. But he didn't see anything. There was nothing above him anymore except the sky - high sky, not clear, but still immeasurably high, with gray clouds quietly creeping along it. “How quiet, calm and solemn, not at all like how I ran,” thought Prince Andrei, “not like how we ran, shouted and fought; It’s not at all like how the Frenchman and the artilleryman pulled each other’s banners with embittered and frightened faces - not at all like how the clouds crawl across this high endless sky. How come I haven’t seen this high sky before? And how happy I am that I finally recognized him. Yes! everything is empty, everything is deception, except this endless sky. There is nothing, nothing, except him. But even that is not there, there is nothing but silence, calm. And thank God!…"

On Bagration’s right flank at 9 o’clock the business had not yet begun. Not wanting to agree to Dolgorukov’s demand to start the business and wanting to deflect responsibility from himself, Prince Bagration suggested that Dolgorukov be sent to ask the commander-in-chief about this. Bagration knew that, due to the distance of almost 10 versts separating one flank from the other, if the one sent was not killed (which was very likely), and even if he found the commander-in-chief, which was very difficult, the sent one would not have time to return earlier evenings.
Bagration looked around at his retinue with his large, expressionless, sleep-deprived eyes, and Rostov’s childish face, involuntarily frozen with excitement and hope, was the first to catch his eye. He sent it.
- What if I meet His Majesty before the Commander-in-Chief, Your Excellency? - said Rostov, holding his hand to the visor.
“You can hand it over to your Majesty,” Dolgorukov said, hastily interrupting Bagration.
Having been released from the chain, Rostov managed to sleep for several hours before the morning and felt cheerful, courageous, decisive, with that elasticity of movements, confidence in his happiness and in that mood in which everything seems easy, fun and possible.
All his wishes were fulfilled that morning; a general battle was fought, he took part in it; Moreover, he was an orderly under the bravest general; Moreover, he was traveling on an errand to Kutuzov, and perhaps even to the sovereign himself. The morning was clear, the horse under him was good. His soul was joyful and happy. Having received the order, he set off his horse and galloped along the line. At first he rode along the line of Bagration’s troops, which had not yet entered into action and stood motionless; then he entered the space occupied by Uvarov’s cavalry and here he already noticed movements and signs of preparations for the case; Having passed Uvarov's cavalry, he already clearly heard the sounds of cannon and gunfire ahead of him. The shooting intensified.
In the fresh morning air there were no longer, as before, at irregular intervals, two, three shots and then one or two gun shots, and along the slopes of the mountains, in front of Pratzen, the rolls of gunfire were heard, interrupted by such frequent shots from guns that sometimes several cannon shots were no longer separated from each other, but merged into one common roar.
It was visible how the smoke of the guns seemed to run along the slopes, catching up with each other, and how the smoke of the guns swirled, blurred and merged with one another. Visible, from the shine of the bayonets between the smoke, were the moving masses of infantry and narrow strips of artillery with green boxes.
Rostov stopped his horse on a hill for a minute to examine what was happening; but no matter how hard he strained his attention, he could neither understand nor make out anything of what was happening: some people were moving there in the smoke, some canvases of troops were moving both in front and behind; but why? Who? Where? it was impossible to understand. This sight and these sounds not only did not arouse in him any dull or timid feeling, but, on the contrary, gave him energy and determination.
“Well, more, give it more!” - He turned mentally to these sounds and again began to gallop along the line, penetrating further and further into the area of ​​​​the troops who had already entered into action.
“I don’t know how it will be there, but everything will be fine!” thought Rostov.
Having passed some Austrian troops, Rostov noticed that the next part of the line (it was the guard) had already entered into action.
"All the better! I’ll take a closer look,” he thought.
He drove almost along the front line. Several horsemen galloped towards him. These were our life lancers, who were returning from the attack in disordered ranks. Rostov passed them, involuntarily noticed one of them covered in blood and galloped on.
“I don’t care about this!” he thought. Before he had ridden a few hundred steps after this, to his left, across the entire length of the field, a huge mass of cavalrymen on black horses, in shiny white uniforms, appeared, trotting straight towards him. Rostov put his horse into full gallop in order to get out of the way of these cavalrymen, and he would have gotten away from them if they had kept the same gait, but they kept speeding up, so that some horses were already galloping. Rostov heard their stomping and the clanking of their weapons more and more clearly, and their horses, figures, and even faces became more visible. These were our cavalry guards, going into an attack on the French cavalry, which was moving towards them.
The cavalry guards galloped, but still holding their horses. Rostov already saw their faces and heard the command: “march, march!” uttered by an officer who unleashed his blood horse at full speed. Rostov, fearing to be crushed or lured into an attack on the French, galloped along the front as fast as his horse could, and still did not manage to get past them.
The last cavalry guard, a huge, pockmarked man, frowned angrily when he saw Rostov in front of him, with whom he would inevitably collide. This cavalry guard would certainly have knocked down Rostov and his Bedouin (Rostov himself seemed so small and weak in comparison with these huge people and horses), if he had not thought of swinging his whip into the eyes of the cavalry guard's horse. The black, heavy, five-inch horse shied away, laying down its ears; but the pockmarked cavalry guard thrust huge spurs into her sides, and the horse, waving its tail and stretching its neck, rushed even faster. As soon as the cavalry guards passed Rostov, he heard them shout: “Hurray!” and looking back he saw that their front ranks were mingling with strangers, probably French, cavalrymen in red epaulets. It was impossible to see anything further, because immediately after that, cannons began firing from somewhere, and everything was covered in smoke.
At that moment, as the cavalry guards, having passed him, disappeared into the smoke, Rostov hesitated whether to gallop after them or go where he needed to go. This was that brilliant attack of the cavalry guards, which surprised the French themselves. Rostov was scared to hear later that out of all this mass of huge handsome people, out of all these brilliant, rich young men on thousands of horses, officers and cadets who galloped past him, after the attack only eighteen people remained.