Abstract on the subject higher mathematics on the topic:

Application of graph theory in chemistry

Performed by a student from group NH-202

Moscow 2011
Graphs are the field of finite mathematics that studies discrete structures; used to solve various theoretical and applied problems.
Some basic concepts. A graph is a collection of points (vertices) and a collection of pairs of these points (not necessarily all) connected by lines (Fig. 1,a). If the lines in the graph are oriented (that is, the arrows indicate the direction of connection of the vertices), they are called arcs, or branches; if unoriented, - edges. Accordingly, a graph containing only arcs is called a directed graph, or a digraph; only edge-unoriented; arcs and ribs - mixed. A graph having multiple edges is called a multigraph; a graph containing only edges belonging to two of its disjoint subsets (parts) is bipartite; arcs (edges) and (or) vertices that correspond to certain weights or numerical values ​​of any parameters are weighted. A path in a graph is an alternating sequence of vertices and arcs in which none of the vertices is repeated (for example, a, b in Fig. 1,a); contour - a closed path in which the first and last vertices coincide (for example, f, h); loop - an arc (edge) that begins and ends at the same vertex. A graph path is a sequence of edges in which none of the vertices are repeated (for example, c, d, e); cycle - a closed chain in which its initial and final vertices coincide. A graph is called connected if any pair of its vertices is connected by a chain or path; otherwise, the graph is called disconnected.
A tree is a connected undirected graph that does not contain cycles or contours (Fig. 1,b). The spanning subgraph of a graph is a subset of it that contains all the vertices and only certain edges. The spanning tree of a graph is its spanning subgraph, which is a tree. Graphs are called isomorphic if there is a one-to-one correspondence between the sets of their vertices and edges (arcs).
To solve problems of graph theory and its applications, graphs are represented using matrices (adjacency, incidence, two-row, etc.), as well as special ones. numerical characteristics. For example, in the adjacency matrix (Fig. 1c), the rows and columns correspond to the numbers of the vertices of the graph, and its elements take the values ​​0 and 1 (respectively, the absence and presence of an arc between a given pair of vertices); in the incidence matrix (Fig. 1d), the rows correspond to the numbers of the vertices, the columns correspond to the numbers of the arcs, and the elements take the values ​​0, + 1 and - 1 (respectively, the absence, presence of an arc entering and leaving the vertex). The most common numerical characteristics: the number of vertices (m), the number of arcs or edges (n), the cyclomatic number, or the rank of the graph (n - m + k, where k is the number of connected subgraphs in a disconnected graph; for example, for the graph in Fig. 1 ,b rank will be: 10-6+ 1 =5).
The application of graph theory is based on the construction and analysis of various classes of chemical and chemical-technological graphs, which are also called topological models, i.e. models that take into account only the nature of the connections between the vertices. The arcs (edges) and vertices of these graphs display chemical and chemical-technological concepts, phenomena, processes or objects and, accordingly, qualitative and quantitative relationships or certain relationships between them.

Rice. 1. Illustration of some basic concepts: a-mixed graph; b-spanning tree (solid arcs a, h, d, f, h) and a certain subgraph (dashed arcs c, e, g, k, l) of the digraph; c, r-matrices resp. adjacency and incidence of a digraph.
Theoretical problems. Chemical graphs make it possible to predict chemical transformations, explain the essence and systematize some basic concepts of chemistry: structure, configuration, conformations, quantum mechanical and statistical-mechanical interactions of molecules, isomerism, etc. Chemical graphs include molecular, bipartite and signal graphs of kinetic reaction equations.
Molecular graphs, used in stereochemistry and structural topology, chemistry of clusters, polymers, etc., are undirected graphs that display the structure of molecules (Fig. 2). The vertices and edges of these graphs correspond, respectively, to atoms and chemical bonds between them.

Rice. 2. Molecular graphs and trees: a, b - multigraphs, respectively. ethylene and formaldehyde; they say pentane isomers (trees 4, 5 are isomorphic to tree 2).
In the stereochemistry of organic substances, molecular trees are most often used - spanning trees of molecular graphs, which contain only all vertices corresponding to C atoms (Fig. 2, a and b). Compiling sets of molecular trees and establishing their isomorphism makes it possible to determine molecular structures and find the total number of isomers of alkanes, alkenes and alkynes (Fig. 2, c).
Molecular graphs make it possible to reduce problems related to coding, nomenclature and structural features (branching, cyclicity, etc.) of molecules of various compounds to the analysis and comparison of purely mathematical features and properties of molecular graphs and their trees, as well as their corresponding matrices. To identify quantitative correlations between the structure of molecules and the physicochemical (including pharmacological) properties of compounds, more than 20 thousand names of topological indices of molecules (Wiener, Balaban, Hosoya, Plat, Randich, etc.) have been developed, which are determined using matrices and numerical characteristics of molecular trees. For example, the Wiener index W = (m 3 + m)/6, where m is the number of vertices corresponding to C atoms, correlates with molecular volumes and refractions, enthalpies of formation, viscosity, surface tension, chromatographic constants of compounds, octane numbers of hydrocarbons and even physiological activity of drugs.
Important parameters of molecular graphs used to determine the tautomeric forms of a given substance and their reactivity, as well as in the classification of amino acids, nucleic acids, carbohydrates and other complex natural compounds, are the average and total (H) information capacities. The parameter is calculated using the Shannon information entropy formula: , where p t is the probability that the vertices m of the graph belong to the i-th type, or equivalence class, k; i = , Parameter. The study of molecular structures such as inorganic clusters or Möbius strips comes down to establishing the isomorphism of the corresponding molecular graphs by placing them (embedding) in complex polyhedra (for example, polyhedra in the case of clusters) or special ones. multidimensional surfaces (for example, Riemann surfaces). Analysis of molecular graphs of polymers, the vertices of which correspond to monomer units, and the edges to chemical bonds between them, makes it possible to explain, for example, the effects of excluded volume, leading to qualitative changes in the predicted properties of polymers.

Rice. 3. Reaction graphs: a-bipartite; b-signal level of kinetics; r 1, g 2 -r-tion; a 1 -a 6 -reagents; k-rate constants p-tsny; s-complex Laplace transform variable.
Using graph theory and principles of artificial intelligence, software has been developed for information retrieval systems in chemistry, as well as automated systems for identifying molecular structures and rational planning of organic synthesis. For the practical implementation on a computer of operations for selecting rational paths of chemical transformations based on the retrosynthetic and syntonic principles, multi-level branched search graphs for solution options are used, the vertices of which correspond to the molecular graphs of reagents and products, and the arcs depict the transformations of substances.

Rice. 4. Single-circuit chemical-technological system and corresponding graphs: a-structural diagram; b, c-material flow graphs, respectively. by total mass flow rates and component A flow rate; r - thermal flow graph; d-fragment of the system of equations (f 1 - f 6) of the material balance, obtained from the analysis of the graphs in Fig. 4, b and c; e-bipartite information digraph; g-information graph, I-mixer; II-reactor; III-distillation column; IV-refrigerator; I 1 -I 8 -technol. streams; q-mass flow; H is the enthalpy of the flow; i. s and i*, s* - resp. real and fictitious sources and sinks of material and heat flows; c-concentration of the reagent; V is the volume of the reactor.
Matrix representations of molecular graphs of various compounds are equivalent (after transforming the corresponding matrix elements) to matrix methods of quantum chemistry. Therefore, graph theory is used when performing complex quantum chemical calculations: to determine the number, properties and energies of molecular orbitals, predicting the reactivity of conjugated alternant and non-alternant polyenes, identifying aromatic and anti-aromatic properties of substances, etc.
To study disturbances in systems consisting of a large number of particles in chemical physics, so-called Feynman diagrams are used - graphs whose vertices correspond to the elementary interactions of physical particles, the edges to their paths after collisions. In particular, these graphs make it possible to study the mechanisms of oscillatory reactions and determine the stability of reaction systems.
To select rational paths for the transformation of reagent molecules for a given set of known interactions, bipartite reaction graphs are used (vertices correspond to molecules and these reactions, arcs correspond to the interactions of molecules in the reaction; Fig. 3,a). Such graphs make it possible to develop interactive algorithms for selecting optimal paths of chemical transformations that require the smallest number of intermediate reactions, the minimum number of reagents from the list of acceptable ones, or achieve the highest yield of products.
Signal graphs of reaction kinetics equations display systems of kinetic equations presented in algebraic-operator form (Fig. 3b). The vertices of the graphs correspond to the so-called information variables, or signals, in the form of concentrations of reagents, arcs - to the relationships of signals, and the weights of the arcs are determined by kinetic constants. Such graphs are used in studying the mechanisms and kinetics of complex catalytic reactions, complex phase equilibria in the formation of complex compounds, as well as for calculating the parameters of the additive properties of solutions.
Applied problems. To solve multidimensional problems of analysis and optimization of chemical-technological systems (CTS), the following chemical-technological graphs are used (Fig. 4): flow, information-flow, signal and reliability graphs. Flow graphs, which are weighted digraphs, include parametric, material in terms of the total mass flow rates of physical flows and the mass flow rates of some chemical components or elements, as well as thermal graphs. The listed graphs correspond to the physical and chemical transformations of substances and energy in a given chemical system.
Parametric flow graphs display the transformation of parameters (mass flow rates, etc.) of physical flows by CTS elements; the vertices of the graphs correspond to the mathematical models of the devices, as well as the sources and sinks of the specified flows, and the arcs correspond to the flows themselves, and the weights of the arcs are equal to the number of parameters of the corresponding flow. Parametric graphs are used to develop algorithms for analyzing technological modes of multi-circuit chemical systems. Such algorithms establish the sequence of calculating systems of equations of mathematical models of individual devices of any system to determine the parameters of its output flows with known values ​​of variable input flows.
Material flow graphs display changes in the consumption of substances in chemical substances. The vertices of the graphs correspond to devices in which the total mass flow rates of physical flows and the mass flow rates of some chemical components or elements are transformed, as well as sources and sinks of substances of flows or these components; Accordingly, the arcs of the graphs correspond to physical flows or physical and fictitious (chemical transformations of substances in apparatuses) sources and sinks of any components, and the weights of the arcs are equal to the mass flow rates of both types. Thermal flow graphs display heat balances in CTS; the vertices of the graphs correspond to devices in which the heat consumption of physical flows changes, and, in addition, to the sources and sinks of thermal energy of the system; arcs correspond to physical and fictitious (physical-chemical energy conversion in devices) heat flows, and the weights of the arcs are equal to the enthalpies of the flows. Material and thermal graphs are used to compile programs for the automated development of algorithms for solving systems of equations for material and heat balances of complex chemical systems.
Information-stock graphs display the logical-information structure of systems of equations of mathematical models of CTS; are used to develop optimal algorithms for calculating these systems. A bipartite information graph (Fig. 4, e) is an undirected or oriented graph, the vertices of which correspond, respectively, to the equations f l -f 6 and the variables q 1 - V, and the branches reflect their relationship. Information graph (Fig. 4, g) - a digraph depicting the order of solving equations; the vertices of the graph correspond to these equations, sources and receivers of XTS information, and the branches correspond to information variables.
Signal graphs correspond to linear systems of equations of mathematical models of chemical technological processes and systems. The vertices of the graphs correspond to signals (for example, temperature), and the branches correspond to connections between them. Such graphs are used to analyze the static and dynamic modes of multiparameter processes and chemical systems, as well as indicators of a number of their most important properties (stability, sensitivity, controllability).
Reliability graphs are used to calculate various indicators of the reliability of chemical equipment. Among the numerous groups of these graphs (for example, parametric, logical-functional), the so-called fault trees are especially important. Each such tree is a weighted digraph that displays the interrelationship of many simple failures of individual processes and CTS devices, which lead to many secondary failures and the resulting failure of the system as a whole.
To create complexes of programs for automated synthesis of optimal highly reliable production (including resource-saving), along with the principles of artificial intelligence, oriented semantic, or semantic, graphs of CTS solution options are used. These graphs, which in a particular case are trees, depict procedures for generating a set of rational alternative CTS schemes (for example, 14 possible when separating a five-component mixture of target products by rectification) and procedures for the ordered selection among them of a scheme that is optimal according to some criterion of system efficiency.
etc.............

MUNICIPAL AUTONOMOUS EDUCATIONAL INSTITUTION SECONDARY SCHOOL No. 2

Prepared

Legkokonets Vladislav, student of class 10A

Practical Application of Graph Theory

Supervisor

L.I. Noskova, mathematics teacher

Art. Bryukhovetskaya

2011

1.Introduction…………………………………………………………………………………….………….3

2. History of the emergence of graph theory………………………………………….………..4

3. Basic definitions and theorems of graph theory…………………………….………6

4. Problems solved using graphs……………………………..………………………..8

4.1 Famous problems………………………………….………………………...8

4.2 Several interesting problems………………………………….……………..9

5. Application of graphs in various areas of people’s lives……………………………...11

6. Solving problems………………………………………………………………………...12

7. Conclusion………………….…………………………………………………………….13

8. List of references………….……………………………………………………………14

9.Appendix……………………………………………………………………………….…………15

Introduction

Born from solving puzzles and entertaining games, graph theory has now become a simple, accessible and powerful tool for solving questions related to a wide range of problems. Graphs are literally omnipresent. In the form of graphs, you can, for example, interpret road maps and electrical circuits, geographical maps and molecules of chemical compounds, connections between people and groups of people. Over the past four decades, graph theory has become one of the most rapidly developing branches of mathematics. This is driven by the demands of a rapidly expanding application field. It is used in the design of integrated circuits and control circuits, in the study of automata, logical circuits, program block diagrams, in economics and statistics, chemistry and biology, in scheduling theory. That's why relevance The topic is determined, on the one hand, by the popularity of graphs and related research methods, and on the other, an undeveloped, holistic system for its implementation.

Solving many problems in life requires long calculations, and sometimes even these calculations do not bring success. This is what research problem. The question arises: is it possible to find a simple, rational, short and elegant solution to solve them. Is problem solving easier if you use graphs? This determined topic of my research: “Practical application of graph theory”

Purpose The research was to use graphs to learn how to quickly solve practical problems.

Research hypothesis. The graph method is very important and is widely used in various fields of science and human activity.

Research objectives:

1. Study literature and Internet resources on this issue.

2.Check the effectiveness of the graph method in solving practical problems.

3. Draw a conclusion.

Practical significance of the study is that the results will undoubtedly arouse the interest of many people. Haven't any of you tried to build your family tree? How to do this correctly? The head of a transport enterprise probably has to solve the problem of more profitable use of transport when transporting goods from a destination to several settlements. Every schoolchild has encountered logical transfusion problems. It turns out that they can be easily solved using graphs.

The following methods are used in the work: observation, search, selection, analysis.

History 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.

“I was once asked a problem about an island located in the city of Königsberg and surrounded by a river with seven bridges across it.

[Appendix Fig.1] 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 because 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 or not. The Koenigsberg bridges are located in such a way that they can be represented in the following figure [Appendix Fig.2], in which A denotes an island, and B, C and D - parts of the continent separated from each other by river branches

Regarding the method he discovered to solve problems of this kind, Euler wrote:

“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, there are any laws inherent in mathematics. So, I don’t 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. Next, you need to distinguish 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. When this is 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. if they were odd, because only one cannot be odd, then even then the transition could be completed, as prescribed, but only the beginning of the detour must certainly be taken from one of those two sections to which an odd number of bridges leads. 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." .

Basic definitions and theorems of graph theory

Graph theory 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 1. A graph is a collection of a finite number of points, called the vertices of the graph, and pairwise lines connecting some of these vertices, called the edges or arcs of the graph.

This definition can be formulated differently: a graph is a non-empty set of points (vertices) and segments (edges), both ends of which belong to a given set of points

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. Vertices of a graph that do not belong to any edge are called isolated.

Definition 3. A graph consisting only of isolated vertices is called null - count .

Notation: O "– a graph with vertices that has no edges

Definition 4. A graph in which each pair of vertices is connected by an edge is called complete.

Designation: U" – a graph consisting of n vertices and edges connecting all possible pairs of these vertices. Such a graph can be represented as an n-gon in which all diagonals are drawn

Definition 5. The degree of a vertex is the number of edges to which the vertex belongs.

Definition 6. A graph whose degrees of all k vertices are the same is called a homogeneous degree graph .

Definition 7. The complement of a 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.

Definition 8. A graph that can be represented on a plane in such a way that its edges intersect only at the vertices is called planar.

Definition 9. A polygon of a planar graph that does not contain any vertices or edges of the graph is called its face.

The concepts of a planar graph and a graph face are used when solving problems on the “correct” coloring of various maps.

Definition 10. A path A to X is a sequence of edges leading from A to X such that every two adjacent edges have a common vertex, and no edge occurs more than once.

Definition 11. A cycle is a path in which the starting and ending points coincide.

Definition 12. A simple cycle is a cycle that does not pass through any of the vertices of the graph more than once.

Definition 13. Length of the path , laid on a loop , the number of edges of this path is called.

Definition 14. Two vertices A and B in a graph are called connected (disconnected) if there exists (does not exist) a path leading from A to B.

Definition 15. A graph is called connected if every two of its vertices are connected; if the graph contains at least one pair of disconnected vertices, then the graph is called disconnected.

Definition 16. A tree is a connected graph that does not contain cycles.

A three-dimensional model of a tree graph is, for example, a real tree with its intricately branched crown; the river and its tributaries also form a tree, but already flat - on the surface of the earth.

Definition 17. A disconnected graph consisting entirely of trees is called a forest.

Definition 18. A tree in which all n vertices are numbered from 1 to n is called a tree with renumbered vertices.

So, we have examined the basic definitions of graph theory, without which it would be impossible to prove theorems, and, consequently, solve problems.

Problems solved using graphs

Famous problems

Traveling salesman problem

The traveling salesman problem is one of the famous problems in the theory of combinatorics. It was put forward in 1934, and the best mathematicians broke their teeth on it.

The problem statement is as follows.
A traveling salesman (wandering merchant) must leave the first city, visit cities 2,1,3..n once in an unknown order and return to the first city. The distances between cities are known. In what order should one go around the cities so that the closed path (tour) of a traveling salesman is the shortest?

Method for solving the traveling salesman problem

Greedy algorithm “go to the nearest (which you have not yet entered) city.”
This algorithm is called “greedy” because in the last steps you have to pay severely for greed.
Consider for example the network in the figure [Appendix Fig.3], representing a narrow rhombus. Let a traveling salesman start from city 1. The “go to the nearest city” algorithm will take him to city 2, then 3, then 4; at the last step you will have to pay for your greed, returning along the long diagonal of the diamond. The result will be not the shortest, but the longest tour.

Problem about the Königsberg bridges.

The problem is formulated as follows.
The city of Koenigsberg is located on the banks of the Pregel River and two islands. The different parts of the city were connected by seven bridges. On Sundays, townspeople took walks around the city. Question: is it possible to take a walk in such a way that, when you leave the house, you return back by walking exactly once over each bridge?
Bridges across the Pregel River are located as in the picture[Appendix Fig.1].

Consider the graph corresponding to the bridge diagram [Appendix Fig. 2].

To answer the question of the problem, it is enough to find out whether the graph is Eulerian. (An even number of bridges must extend from at least one vertex). You can’t walk around the city and cross all the bridges once and come back.

Several interesting tasks

1. "Routes".

Problem 1

As you remember, the hunter for dead souls Chichikov visited famous landowners once each. He visited them in the following order: Manilov, Korobochka, Nozdryov, Sobakevich, Plyushkin, Tentetnikov, General Betrishchev, Petukh, Konstanzholgo, Colonel Koshkarev. A diagram was found on which Chichikov sketched the relative positions of the estates and the country roads connecting them. Determine which estate belongs to whom, if Chichikov did not drive any of the roads more than once [Appendix Fig. 4].

Solution:

The road map shows that Chichikov began his journey from estate E, and ended with estate O. We note that only two roads lead to estates B and C, so Chichikov had to travel along these roads. Let's mark them with a bold line. Sections of the route passing through A have been identified: AC and AB. Chichikov did not travel on the roads AE, AK and AM. Let's cross them out. Let us mark with a bold line ED; Let's cross out DK. Let's cross out MO and MN; Let's mark with a bold line MF; cross out FO; Let's mark FH, NK and KO with a bold line. Let's find the only possible route under this condition. And we get: estate E - belongs to Manilov, D - Korobochka, C - Nozdryov, A - Sobakevich, B - Plyushkin, M - Tentetnikov, F - Betrishchev, N - Petukh, K - Konstanzholgo, O - Koshkarev [Appendix Fig.5].

Problem 2

The figure shows a map of the area [Appendix Fig. 6].

You can only move in the direction of the arrows. You can visit each point no more than once. In how many ways can you get from point 1 to point 9? Which route is the shortest and which is the longest.

Solution:

We sequentially “stratify” the circuit into a tree, starting from vertex 1 [Appendix Fig.7]. Let's get a tree. The number of possible ways to get from 1 to 9 is equal to the number of “hanging” vertices of the tree (there are 14 of them). Obviously the shortest path is 1-5-9; the longest is 1-2-3-6-5-7-8-9.

2 "Groups, dating"

Problem 1

The participants of the music festival, having met, exchanged envelopes with addresses. Prove that:

a) an even number of envelopes were handed over;

b) the number of participants who exchanged envelopes an odd number of times is even.

Solution: Let the festival participants be A 1, A 2, A 3. . . , And n are the vertices of the graph, and the edges connect pairs of vertices representing the guys exchanging envelopes [Appendix Fig.8]

Solution:

a) the degree of each vertex A i shows the number of envelopes that participant A i gave to his friends. The total number of transmitted envelopes N is equal to the sum of the degrees of all vertices of the graph N = degree. A 1 + step. A 2 + + . . . + step. A n -1 + degree. And n, N =2p, where p is the number of edges of the graph, i.e. N – even. Consequently, an even number of envelopes were handed over;

b) in the equality N = degree. A 1 + step. A 2 + + . . . + step. A n -1 + degree. And n the sum of odd terms must be even, and this can only be if the number of odd terms is even. This means that the number of participants who exchanged envelopes an odd number of times is even.

Problem 2

One day Andrei, Boris, Volodya, Dasha and Galya agreed to go to the cinema in the evening. They decided to coordinate the choice of cinema and show over the phone. It was also decided that if it was not possible to contact someone by phone, then the trip to the cinema would be cancelled. In the evening, not everyone gathered at the cinema, and therefore the movie visit was cancelled. The next day they began to find out who called whom. It turned out that Andrey called Boris and Volodya, Volodya called Boris and Dasha, Boris called Andrey and Dasha, Dasha called Andrey and Volodya, and Galya called Andrey, Volodya and Boris. Who couldn't get on the phone and therefore didn't come to the meeting?

Solution:

Let's draw five dots and label them with the letters A, B, C, D, D. These are the first letters of the names. Let's connect the dots that correspond to the names of the guys who called.

[Appendix Fig.9]

From the picture it is clear that each of the guys - Andrey, Boris and Volodya - telephoned everyone else. That's why these guys came to the cinema. But Galya and Dasha were unable to get on the phone with each other (points G and E are not connected by a line segment) and therefore, in accordance with the agreement, did not come to the cinema.

Application of graphs in various areas of people's lives

In addition to the examples given, graphs are widely used in construction, electrical engineering, management, logistics, geography, mechanical engineering, sociology, programming, automation of technological processes and production, psychology, and advertising. So, from all of the above, the practical value of graph theory irrefutably follows, the proof of which was the goal of this study.

In any field of science and technology you encounter graphs. Graphs are wonderful mathematical objects with which you can solve mathematical, economic and logical problems, various puzzles and simplify the conditions of problems in physics, chemistry, electronics, and automation. Many mathematical facts can be conveniently formulated in the language of graphs. Graph theory is part of many sciences. Graph theory is one of the most beautiful and visual mathematical theories. Recently, graph theory is finding more and more applications in applied issues. Even computational chemistry has emerged - a relatively young field of chemistry based on the application of graph theory.

Molecular graphs, used in stereochemistry and structural topology, chemistry of clusters, polymers, etc., are undirected graphs displaying the structure of molecules [Appendix Fig. 10]. The vertices and edges of these graphs correspond to the corresponding atoms and chemical bonds between them.

Molecular graphs and trees: [Appendix Fig. 10] a, b - multigraphs, respectively. ethylene and formaldehyde; they say pentane isomers (trees 4, 5 are isomorphic to tree 2).

In the stereochemistry of organisms the most. Molecular trees are often used - the main trees of molecular graphs, which contain only all the vertices corresponding to the C atoms. Compilation of sets of mol. trees and the establishment of their isomorphism make it possible to determine they say. structures and find the total number of isomers of alkanes, alkenes and alkynes

Protein networks

Protein networks are groups of physically interacting proteins that function in a cell together and in a coordinated manner, controlling interconnected processes occurring in the body [attachment fig. 11].

Hierarchical system graph called a tree. A distinctive feature of a tree is that there is only one path between any two of its vertices. The tree does not contain cycles or loops.

Typically, a tree representing a hierarchical system has one main vertex, which is called the root of the tree. Each vertex of the tree (except the root) has only one ancestor - the object designated by it is included in one top-level class. Any vertex of a tree can generate several descendants - vertices corresponding to classes of the lower level.

For each pair of tree vertices, there is a unique path connecting them. This property is used when finding all ancestors, for example, in the male line, of any person whose pedigree is represented in the form of a family tree, which is a “tree” in the sense of graph theory.

Example of my family tree [Appendix Fig. 12].

Another example. The picture shows the biblical family tree [Appendix Fig. 13].

Problem solving

1.Transport task. Let there be a base in the city of Krasnodar with raw materials that need to be distributed to the cities of Krymsk, Temryuk, Slavyansk-on-Kuban and Timashevsk in one trip, spending as little time and fuel as possible and returning back to Krasnodar.

Solution:

First, let's make a graph of all possible travel routes [Appendix Fig.14], taking into account the real roads between these settlements and the distance between them. To solve this problem, we need to create another graph, a tree-like [Appendix Fig.15].

For the convenience of the solution, we designate the cities with numbers: Krasnodar - 1, Krymsk - 2, Temryuk - 3, Slavyansk - 4, Timashevsk - 5.

The result is 24 solutions, but we only need the shortest paths. Of all the solutions, only two are satisfactory, this is 350 km.

Similarly, it is possible and, I think, necessary to calculate real transportation from one locality to another.

Logical problem involving transfusion. The bucket contains 8 liters of water, and there are two pans with a capacity of 5 and 3 liters. you need to pour 4 liters of water into a five-liter pan and leave 4 liters in the bucket, i.e. pour the water equally into the bucket and a large pan.

Solution:

The situation at any moment can be described by three numbers [Appendix Fig. 16].

As a result, we get two solutions: one in 7 moves, the other in 8 moves.

Conclusion

So, in order to learn how to solve problems, you need to understand what they are, how they are structured, what components they consist of, what are the tools with which problems are solved.

Solving practical problems using graph theory, it became clear that in every step, in every stage of their solution, it is necessary to apply creativity.

From the very beginning, at the first stage, it lies in the fact that you need to be able to analyze and encode the condition of the problem. The second stage is a schematic notation, which consists of a geometric representation of the graphs, and at this stage the element of creativity is very important because it is far from easy to find correspondences between the elements of the condition and the corresponding elements of the graph.

While solving a transport problem or a task of drawing up a family tree, I came to the conclusion that the graph method is certainly interesting, beautiful and visual.

I became convinced that graphs are widely used in economics, management, and technology. Graph theory is also used in programming. This was not discussed in this work, but I think it’s only a matter of time.

This scientific work examines mathematical graphs, their areas of application, and solves several problems using graphs. Knowledge of the basics of graph theory is necessary in various areas related to production and business management (for example, network construction schedule, mail delivery schedules). In addition, while working on a scientific paper, I mastered working on a computer using the WORD text editor. Thus, the objectives of the scientific work have been completed.

So, from all of the above, the practical value of graph theory irrefutably follows, the proof of which was the goal of this work.

Literature

Berge K. Graph theory and its applications. -M.: IIL, 1962.

Kemeny J., Snell J., Thompson J. Introduction to finite mathematics. -M.: IIL, 1963.

Ore O. Graphs and their application. -M.: Mir, 1965.

Harari F. Graph theory. -M.: Mir, 1973.

Zykov A.A. Finite graph theory. -Novosibirsk: Science, 1969.

Berezina L.Yu. Graphs and their application. -M.: Education, 1979. -144 p.

"Soros Educational Journal" No. 11 1996 (article "Flat graphs");

Gardner M. "Mathematical leisure", M. "World", 1972 (chapter 35);

Olehnik S. N., Nesterenko Yu. V., Potapov M. K. “Old entertaining problems”, M. “Science”, 1988 (part 2, section 8; appendix 4);

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Translation Editor's Preface
Preface to the Russian edition
Preface
TOPOLOGY OF A FINITE POINT SET AND MOLECULAR STRUCTURE. R. Merrifield, X. Simmons
1. Introduction
2. Final topology
2.1. Topology graph
2.2. Qualitative characteristics of graph topology
2.3. Quantitative characteristics of graph topology: combinatorics
3. Topology of alternant molecules
3.1. Complexity of structure
3.2. Connectivity and delocalization
4. Topology of non-alternant molecules
4.1. Duplex graph
4.2. Duplex topology
Literature
STEREOCHEMICAL TOPOLOGY. D. Volba
1. Introduction
2. An approach to the synthesis of topological stereoisomers based on Möbius strips
2.1. Complete synthesis of the first molecular Möbius strip
3. Criteria for topological stereoisomerism
3.1. Topological chirality
3.2. Topological diastereoisomerism
4. Clipping reaction and approaches to the synthesis of the molecular trefoil knot
4.1. Rupture of steps of the Mobius ladder
4.2. Molecular trefoil knot
Literature
QUALITATIVE STEREOCHEMISTRY J. Dugundji
1. Introduction
2. Permutation isomers
3. Chemical identity group
Literature
THEORY OF MOLECULAR STRUCTURE. R. Bader
1. Review of the theory
2. Some applications
Literature
ALGEBRAIC AND TOPOLOGICAL STRUCTURE OF QUANTUM CHEMISTRY, CHEMICAL KINETICS AND VISUAL RULES THAT ALLOW YOU TO MAKE QUALITATIVE PREDICTIONS FOR CHEMICAL PRACTICE. O. Sinanoglu
1. Introduction
2. Microchemistry or qualitative quantum chemical rules derived directly from structural formulas or ORTEP diagrams
2.1. Valence vector space field Vn(R) existing in Euclidean three-dimensional space (?)
2.2. The principle of linear covariance in quantum chemistry
2.3. Non-unitary classification of molecules
2.4. From structural formulas of molecules to more detailed structural-electronic formulas (and to graphs)
2.5. “Structural and deformation covariance” of molecules and graphical rules for obtaining high-quality quantum chemical results
3. Morphology of reaction mechanisms, synthesis routes and topological “stage/compound rules”
4. Features of obtaining quantum qualitative characteristics of each reaction stage of the mechanism or reaction path
Literature
REACTION TOPOLOGY: THEORY OF MANIFESTIES OF POTENTIAL SURFACES AND QUANTUM CHEMICAL DESIGN OF SYNTHESIS. P. Mezhey
1. Introduction
2. Topological manifolds, differentiable manifolds and reaction topology
3. Ratios of critical points; intersection graphs in topological space (M, Tc) and quantum chemical reaction schemes
4. Computational aspects
5. Degenerate critical points and chemical structures that do not correspond to true PES minima
6. Conclusions
Literature
TOPOLOGY OF BINDING IN POLYHEDRIC MOLECULES. R. King
1. Introduction
2. Basic concept
3. Vertex atoms
4. Polyhedral systems with localized binding
5. Systems with fully delocalized binding
6. Electron-rich polyhedral systems
7. Electron-deficient polyhedral systems
8. Anomalous peaks
9. Polyhedrans
10. Conclusions
Literature
FORMS OF CLUSTERS OF ELEMENTS OF THE MAIN SUBGROUPS: A TOPOLOGICAL APPROACH TO COUNTING SKELETAL ELECTRONS. M. McGlinchey, J. Tal
1. Introduction
2. Clusters with completely delocalized binding
3. Clusters with binding localization on edges
3.1. Hexatom clusters
3.2. Seven-atom clusters
3.3. Eight-atom clusters
4. Quantum-topological substantiation of the polyhedral model
5. Conclusions
Literature
TOPOLOGICAL PROPERTIES OF BINARY COMPOUNDS OF SULFUR WITH NITROGEN. A. Turner
1. Introduction
2. The prototype molecule is tetrasulfur tetranitride
3. Planar cyclic molecules and SnNm ions
4. Non-planar systems - equivalence of charge distribution centers
5. Application of electron density functional theory
Literature
SHOULD YOU BE DEVELOPING TOPOLOGICAL INDICES? D. Rouvray
1. Introduction
2. Wiener index
3. Index construction
4. Distance matrix indices
5. Adjacency matrix indices
6. Centric topological indices
7. Information-theoretic indices
8. Composite topological indices
9. Some mathematical relationships
10. Shape and size of molecules
11. Basic applications of indices
12. Bibliographic classification of compounds
13. Determination of physicochemical parameters
14. Development of pharmaceutical drugs
15. Conclusions
Literature
TOPOLOGICAL INDICES BASED ON THE SYMMETRY OF NEIGHBORHOODS: CHEMICAL AND BIOCHEMICAL APPLICATIONS. V. Magnuson, D. Harris, S. Beysak
1. Introduction
2. Information content of the graph
2.1. Definitions
2.2. Basic provisions
2.3. Equivalence relation
2.4. Calculation of other topological indices
3. Calculation of indices
4. Applications in quantitative structure-activity correlation (QSCA) studies
4.1. Solubility of alcohols
4.2. Inhibition of microsomal para-hydroxylation of aniline by alcohols
4.3. Toxicity (LD50) of barbiturates
Literature
GRAPH ORDERING AS AN APPROACH TO STUDYING STRUCTURE-ACTIVITY CORRELATIONS. M. Randic, J. Kraus, B. Dzonova-German-Blazic
1. Introduction
2. Basic principles of the method
3. Application to substances with antimalarial effects
3.1. Constructing a sequence of circuits
3.2. Comparison of molecules A-M
4. Discussion
Literature
MATHEMATICAL MODEL OF MOLECULAR COMPLEXITY. S. Bertz
1. Vaeding
2. Model development
2.1. Graph Theory and Molecular Topology
2.2. Information theory and molecular symmetry
3. Model verification
3.1. Model limitations
4. Model reliability
5. Conclusions
Literature
DISTANCE MATRIX FOR MOLECULES CONTAINING HETERO-ATOMS. M. Barish, J. Yashari, R. Lall, V. Srivastava, I. Trinaistich
1. Introduction
2. Relationship between adjacency matrix and distance matrix
3. Distance matrix for heterosystems
Literature
CANONICAL NUMBERING AND SYSTEM OF LINEAR NOTATION FOR CHEMICAL GRAPHS. W. Herndon
1. Introduction
2. Canonical numbering
3. Unambiguous linear notation
4. Canonical numbering of regular graphs
5. Conclusions
Literature
SYMMETRY AND SPECTRA OF GRAPHS. THEIR APPLICATIONS IN CHEMISTRY. K. Balasubramanian
1. Introduction
2. Tree pruning
3. Tree pruning and tree symmetry groups
4. Spectral polynomials of trees obtained using the pruning process
5. Applications in chemistry
Literature
GROUPS OF AUTOMORPHISMS OF SOME CHEMICAL GRAPHS. G. Jones, E. Lloyd
1. Introduction
2. Some graphs and their groups
3. Reaction graphs
3.1. Example 1: Berry mechanism
3.2. Example 2: 1,2-shifts in carbonium ions
3.3. Example 3: 1,2-shifts in homotetrahedranyl cations
3.4. Example 4: Digonal twists in octahedral complexes
3.5. Example 5: 1,3-shifts in homotetrahedranyl cations
4. Suborbital graphs
5. Conclusions
Literature
THE PROBLEM OF RECONSTRUCTION. W. Tutt
USE OF RIEMANN SURFACES IN THE GRAPHEORETICAL REPRESENTATION OF MÖBIUS SYSTEMS. A. Day, R. Mullion, M. Rigby
1. Introduction
2. Formalism of the method
3. Application
4. Conclusions
Literature
GLOBAL DYNAMICS OF SOME CLASSES OF REACTION SYSTEMS. X. Dimension
1. Introduction
2. Graph-theoretical formulation
2.1. Structure of control equations
2.2. Some concepts of graph theory
2.3. Reaction invariants
2.4. Existence of stationary states
3. Vertex-driven networks
3.1. Constant input streams
3.2. Periodic input streams
4. Conclusions
Literature
“LOGICAL DESCRIPTION” VERSUS “CONTINUOUS DESCRIPTION” OF SYSTEMS CONTAINING FEEDBACK LOOPS: RELATIONSHIP BETWEEN TIME DELAYS AND PARAMETERS. R. Thomas
1. Introduction
2. Logical description of systems containing feedback loops
2.1. “On” and “off” delays
2.2. Logic equations
2.3. State tables
2.4. Circuits (sequences of states)
2.5. Stability Analysis
3. Continuous description
3.1. Logical time delays and continuous parameters
Literature
QUALITATIVE DYNAMICS AND STABILITY OF CHEMICAL REACTION SYSTEMS. B. Clark
1. Introduction
2. Specifying a chemical system
3. Time scales - removing substances that react too quickly and too slowly
4. Chemical network theory
5. System dynamics
6. Variety of stationary states
7. Simple theorems for network analysis
8. Deeper discussion of stationary states and their existence
9. Correctness
10. Unambiguity
11. Global attraction
12. Networks in which diversity is not correct, unambiguous and globally attractive
13. Network topology and stability
14. Concluding remarks
15. Application
15.1. Versatile Features
15.2. Functions for symbolic processing and calculation of the current matrix
15.3. Theorem checking and related functions
15.4. Individual functions
Literature
HIGHER CHAOS IN SIMPLE REACTION SYSTEMS. O. Ressler, J. Hudson
1. Introduction
2. Method of generating ordinary chaos
3. Method of generating higher chaos
4. Discussion
Literature
STRANGE ATTRACTORS IN LINEAR PERIODIC TRANSFER FUNCTIONS WITH PERIODIC DISTURBANCES. X. Degn
1. Introduction
2. Results
Literature
USING SENSITIVITY ANALYSIS IN DETERMINING THE STRUCTURAL STABILITY OF MULTI-PARAMETER OSCILLATORS. R. Larter
1. Introduction
2. Method
2.1. Standard theory
2.2. Modified theory
3. Results
3.1. Initial conditions
3.2. Rate constants
3.3. More complex situations
Literature
REPRESENTATION OF n-DIMENSIONAL CHEMICAL VARIETIES USING ELECTRICAL NETWORKS. L. Puzner
1. Introduction: topological and geometric analysis of chemical processes
2. Basic geometric properties of n-dimensional metric manifolds
3. Representation as a network
4. Example for a two-dimensional system
5. Optimal paths
6. An example of using a chemical network for linear transitions between multiple states
7. Variational networks
Application: Network Analysis
Literature
LOGIC OF CHEMICAL IDEAS. P. Plyat, E. Hass
1. Introduction
2. Topology of pericyclic reactions
3. Lattices of pericyclic reactions
4. Orthomodular and Boolean reaction four-dimensional lattices
5. Conclusions
Literature
MULTIDIMENSIONAL X-MODEL. GRAPHEORETICAL AND ALGEBRAIC APPROACH TO DESCRIBING THE MECHANISMS OF COMPLEX CHEMICAL REACTIONS. E. Hass, P. Plyat
1. Introduction
2. Single-parameter X-model
3. Multidimensional X-model
3.1. Reaction routes for -cycloadditions
4. Conclusions
Literature
CLASSIFICATION OF MECHANISMS OF CHEMICAL REACTIONS FROM A GEOMETRIC POINT OF VIEW. P. Sellers
1. Introduction
2. Milner's example
3. Mechanisms without cycles
4. Other mechanisms
5. Multiple total reactions
6. Conclusions
Literature
GRAPHS, POLYMER MODELS, EXCLUDED VOLUME AND CHEMICAL REALITY. D. Klein, W. Seitz
1. Introduction
2. Isolated linear circuits
3. Counting of isomers
4. Conformations of branched polymers
5. Scaling theory
6. Transfer matrices
7. Self-similarity and renormalization
8. Discussion
Literature
VOLUME FUNCTION FOR WATER BASED ON A RANDOM LATTICE SUBGRAP MODEL. L. Quintas
1. Introduction and preliminary mathematical remarks
2. Random graph model for water
3. Volume function for water
4. Correspondence of V(p) to numerical data
5. Concluding remarks
Literature
TOPOLOGICAL ASPECTS OF ENZYME-SUBSTRATE RECOGNITION. S. Swaminathan
1. The problem of enzyme-substrate recognition
2. Edelstein-Rosen model
3. Method of phenomenological calculus
4. Hilbert description space
5. Postulates for the dynamics of complex systems
6. Model of enzyme-substrate recognition
7. Concluding remarks
Literature
DYNAMICS OF RNA SECONDARY STRUCTURE FORMATION. X. Martinets
1. Introduction
2. Energy minimization methods
3. Simulation method
4. Conclusions
Literature
PROGRAM IN LISP LANGUAGE FOR FUNCTIONAL-FRAGMENTAL REPRESENTATION OF MOLECULES AND THEIR GEOMETRY. K. Trindl, R. Givan
1. Introduction
2. Lisp - non-numerical programming language
3. Representation of molecules using Lisp language
4. Informal fragment recognition algorithm
5. Some special problems
6. Construction of a distance matrix using a fragment data bank
7. Factor analysis and Crippen’s algorithm for determining geometry through distances
8. Conclusions and prospects
Literature
Subject index

To create automated program complexes. synthesis optimal. highly reliable production (including resource-saving) along with the principles of the arts. intelligence, they use oriented semantic, or semantic, graphs of CTS solution options. These graphs, which in a particular case are trees, depict procedures for generating a set of rational alternative CTS schemes (for example, 14 possible when separating a five-component mixture of target products by rectification) and procedures for the ordered selection among them of a scheme that is optimal according to a certain criterion system efficiency (see Optimization).

Graph theory is also used to develop algorithms for optimizing time schedules for the operation of multi-product flexible equipment, optimization algorithms. placement of equipment and routing of pipeline systems, optimal algorithms. management of chemical technology processes and production, during network planning of their work, etc.

Lit.. Zykov A. A., Theory of finite graphs, [in. 1], Novosibirsk, 1969; Yatsimirsky K. B., Application of graph theory in chemistry, Kyiv, 1973; Kafarov V.V., Perov V.L., Meshalkin V.P., Principles of mathematical modeling of chemical technological systems, M., 1974; Christofides N., Graph theory. Algorithmic approach, trans. from English, M., 1978; Kafarov V.V., Perov V.L., Meshalkin V.P., Mathematical foundations of computer-aided design of chemical production, M., 1979; Chemical applications of topology and graph theory, ed. R. King, trans. from English, M., 1987; Chemical Applications of Graph Theory, Balaban A.T. (Ed.), N.Y.-L., 1976. V.V. Kafarov, V.P. Meshalkin.
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Spanish literature for the article "GRAPHS THEORY": no data

Page "GRAPHS THEORY" prepared based on materials

E. Babaev.  Candidate of Chemical Sciences.

      When talking about the mathematization of science, most often they mean only the purely pragmatic use of computational methods, forgetting the apt statement of A. A. Lyubishchev about mathematics as not so much a servant, but the queen of all sciences. It is the level of mathematization that brings this or that science into the category of exact ones, if by this we mean not the use of exact quantitative estimates, but a high level of abstraction, freedom to operate with concepts related to the categories of non-numerical mathematics.
      Among the methods of such qualitative mathematics that have found effective application in chemistry, the main role belongs to sets, groups, algebras, topological constructions and, first of all, graphs - the most general method of representing chemical structures.

Let's take, for example, four points arbitrarily located on a plane or in space, and connect them with three lines. No matter how these points (called vertices) are located and no matter how they are connected to each other by dashes (called edges), we will get only two possible graph structures, differing from each other in the mutual arrangement of connections: one graph, similar to the letters "P" " or "I", and another graph similar to the letters "T", "E" or "U". If instead of four abstract points we take four carbon atoms, and instead of dashes we take chemical bonds between them, then the two indicated graphs will correspond to two possible isomers of butane - normal and iso-structure.
      What is the reason for the growing interest of chemists in graph theory, this bizarre but very simple language of dots and lines?
      The graph has the remarkable property that it remains unchanged under any deformations of the structure that are not accompanied by a break in the connections between its elements. The structure of a graph can be distorted, completely depriving it of symmetry in the usual sense; however, the graph will still have symmetry in the topological sense, determined by the sameness and interchangeability of the end vertices. Given this hidden symmetry, one can, for example, predict the number of different isomeric amines obtained from the structures of butane and isobutane by replacing carbon atoms with nitrogen atoms; graphs make it possible to use simple physical considerations to understand patterns of the “structure property” type.
      Another, somewhat unexpected idea is to express the structural qualities of graphs (for example, the degree of their branching) using numbers. Intuitively, we feel that isobutane is more branched than normal butane; This can be expressed quantitatively, say, by the fact that in the isobutane molecule the structural fragment of propane is repeated three times, and in normal butane it is repeated only twice. This structural number (called the Wiener topological index) correlates surprisingly well with characteristics of saturated hydrocarbons such as boiling point or heat of combustion. Recently, a peculiar fashion has appeared for the invention of various topological indices; there are already more than twenty of them; Its alluring simplicity makes this Pythagorean method increasingly popular *.
      The use of graph theory in chemistry is not limited to the structure of molecules. Back in the thirties, A. A. Balandin, one of the predecessors of modern mathematical chemistry, proclaimed the principle of isomorphic substitution, according to which the same graph carries uniform information about the properties of the most diverse structured objects; it is only important to clearly define which elements are selected as vertices and what kind of relationships between them will be expressed by edges. So, in addition to atoms and bonds, you can select phases and components, isomers and reactions, macromolecules and interactions between them as vertices and edges. One can notice a deep topological relationship between the Gibbs phase rule, the stoichiometric Horiuchi rule and the rational classification of organic compounds according to the degree of their unsaturation. With the help of graphs, interactions between elementary particles, crystal fusion, cell division are successfully described... In this sense, graph theory serves as a visual, almost universal language of interdisciplinary communication.

The development of each scientific idea traditionally goes through the following stages: article review monograph textbook. The inflorescence of ideas called mathematical chemistry has already passed the stage of reviews, although it has not yet reached the status of an academic discipline. Due to the diversity of areas, the main form of publications in this area is now collections; several such collections were published in 1987-1988.
      The first collection edited by R. King “Chemical applications of topology and graph theory” (M., “Mir”, 1987) contains a translation of reports from an international symposium with the participation of chemists and mathematicians from different countries. The book gives a complete picture of the motley palette of approaches that emerged at the intersection of graph theory and chemistry. It touches on a very wide range of issues, starting from the algebraic structure of quantum chemistry and stereochemistry, the magic rules of electronic counting, and ending with the structure of polymers and the theory of solutions. Organic chemists will undoubtedly be attracted by the new strategy for the synthesis of trefoil-type molecular knots, an experimental implementation of the idea of ​​a molecular Möbius strip. Of particular interest will be review articles on the use of the topological indices already mentioned above to assess and predict a wide variety of properties, including the biological activity of molecules.
      The translation of this book is also useful because the issues raised in it may help resolve a number of debatable problems in the field of methodology of chemical science. Thus, the rejection by some chemists in the 50s of the mathematical symbolism of resonance formulas gave way in the 70s to the denial by some physicists of the very concept of chemical structure. Within the framework of mathematical chemistry, such contradictions can be eliminated, for example, using a combinatorial-topological description of both classical and quantum chemical systems.
      Although the works of Soviet scientists are not presented in this collection, it is gratifying to note the increased interest in the problems of mathematical chemistry in domestic science. An example is the first workshop “Molecular graphs in chemical research” (Odessa, 1987), which brought together about a hundred specialists from all over the country. Compared to foreign research, domestic work is distinguished by a more pronounced applied nature, focus on solving problems of computer synthesis, and creating various data banks. Despite the high level of reports, the meeting noted an unacceptable lag in the training of specialists in mathematical chemistry. Only at Moscow and Novosibirsk universities are occasional courses given on individual issues. At the same time, it is time to seriously raise the question: what kind of mathematics should chemistry students study? Indeed, even in university mathematical programs of chemical departments such sections as the theory of groups, combinatorial methods, theory of graphs, and topology are practically not represented; in turn, university mathematicians do not study chemistry at all. In addition to the problem of training, the issue of scientific communications is urgent: an all-Union journal on mathematical chemistry is needed, published at least once a year. The journal "MATCH" (Mathematical Chemistry) has been published abroad for many years, and our publications are scattered across collections and a wide variety of periodicals.

Until recently, the Soviet reader could get acquainted with mathematical chemistry only from the book by V. I. Sokolov “Introduction to Theoretical Stereochemistry” (M.: Nauka, 1979) and the brochure by I. S. Dmitriev “Molecules without Chemical Bonds” (L.: Khimiya , 1977). Partially filling this gap, the Siberian branch of the Nauka publishing house published last year the book “Application of Graph Theory in Chemistry” (edited by N. S. Zefirov, S. I. Kuchanov). The book consists of three sections, with the first devoted to the use of graph theory in structural chemistry; the second part examines reaction graphs; the third shows how graphs can be used to facilitate the solution of many traditional problems in polymer chemical physics. Of course, this book is not yet a textbook (a significant part of the ideas discussed are original results of the authors); nevertheless, the first part of the collection can be fully recommended for an initial acquaintance with the subject.
      Another collection proceedings of the seminar of the Faculty of Chemistry of Moscow State University “Principles of symmetry and systematicity in chemistry” (edited by N. F. Stepanov) was published in 1987. The main topic of the collection is group-theoretic, graph-theoretic and system-theoretic methods in chemistry. The range of questions discussed is unconventional, and the answers to them are even less standard. The reader will learn, for example, about the reasons for the three-dimensionality of space, about the possible mechanism for the occurrence of dissymmetry in living nature, about the principles of designing the periodic system of molecules, about the planes of symmetry of chemical reactions, about the description of molecular forms without using geometric parameters, and much more. Unfortunately, the book can only be found in scientific libraries, since it has not gone on general sale.
      Since we are talking about the principles of symmetry and systematicity in science, it is impossible not to mention another unusual book “System. Symmetry. Harmony” (M.: Mysl, 1988). This book is dedicated to one of the variants of the so-called general theory of systems (GTS), proposed and developed by Yu.A. Urmantsev and which today has found the largest number of supporters among scientists of various specialties, both natural and humanities. The initial principles of Urmantsev's OTS are the concepts of system and chaos, polymorphism and isomorphism, symmetry and asymmetry, as well as harmony and disharmony.
      It seems that Urmantsev’s theory should attract the closest attention of chemists, if only because it traditionally elevates the chemical concepts of composition, isomerism, and dissymmetry to the rank of system-wide ones. In the book you can find striking symmetry analogues for example between isomers of leaves and molecular structures **. Of course, when reading the book, in some places a certain level of professional impartiality is necessary - for example, when it comes to chemical-musical parallels or the rationale for a mirror-symmetrical system of elements. Nevertheless, the book is permeated by the central idea of ​​finding a universal language expressing the unity of the universe, akin to which is perhaps the Castalian language of the “bead game” by Hermann Hess.
Speaking about the mathematical structures of modern chemistry, one cannot ignore the wonderful book by A. F. Bochkov and V. A. Smith “Organic Synthesis” (M.: Nauka, 1987). Although its authors are “pure” chemists, a number of ideas discussed in the book are very close to the problems raised above. Without dwelling on the brilliant form of presentation and depth of content of this book, after reading which you want to take up organic synthesis, we will emphasize only two points. Firstly, considering organic chemistry through the prism of its contribution to world science and culture, the authors draw a clear parallel between chemistry and mathematics as universal sciences that draw the objects and problems of their research from within themselves. In other words, to the traditional status of mathematics as the queen and servant of chemistry, we can add the peculiar hypostasis of its sister. Secondly, convincing the reader that organic synthesis is an exact science, the authors appeal to the accuracy and rigor of both structural chemistry itself and to the perfection of the logic of chemical ideas.
      If experimenters say so, is there any doubt that the hour of mathematical chemistry has come?

________________________
  * See "Chemistry and Life", 1988, No. 7, p. 22.
** See "Chemistry and Life", 1989, No. 2.