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The present book is an attempt to outline some, certainly not all, mathematical aspects of modern organic chemistry. We have focused our attention on topological, graph-theoretical and group-theoretical features of organic chemistry, Parts A, B and C. The book is directed to all those chemists who use, or who intend to use mathe­ matics in their work, and especially to graduate students. The level of our exposition is adjusted to the mathematical background of graduate students of chemistry and only some knowledge of elementary algebra and calculus is required from the readers of the book. Some less well-known. but still elementary mathematical facts are collected in Appendices 1-4. This, however, does not mean that the mathematical rigor and numerous tedious, but necessary technical details have been avoided. The authors' intention was to show the reader not only how the results of mathematical chemistry look, but also how they can be obtained. In accordance with this, Part 0 of the book contains a few selected advanced topics which should give the reader the flavour of the contemporary research in mathe­ matical organic chemistry. One of the authors (I.G.) was an Alexander von Humboldt fellow in 1985 when the main part of the book was written. He gratefully acknowledges the financial support of the Alexander von Humboldt Foundation which enabled his stay at the Max-Planck-Institut fUr Strahlenchemie in M iilheim and the writing of this book.

### Introduction

Abstract
Mathematical chemistry nowadays presents a variety of approaches to understanding the mathematical structures which lie behind existing chemical concepts, to establishing and investigating novel mathematical models of chemical phenomena, and applying mathematical ideas and techniques in chemistry. Throughout the entire history of chemistry certain scientists, usually not numerous, were inclined to contemplate connections between mathematics and chemistry and the possibility of using mathematics for deducing known and predicting new chemical facts. Extensive use of mathematical methods is traditional in various branches of physical chemistry, especially in thermodynamics (partial derivatives, proper and improper differentials, path integrals etc.) and chemical kinetics (coupled non-linear differential equations). A real need for mathematics in chemistry appeared, however, only after the discovery, made by physicists in the first three decades of our century, that the basic properties of atoms and molecules can be explained and predicted by means of quantum theory. The awareness that chemistry cannot be understood without a knowledge of quantum physics, including its sophisticated mathematical apparatus, was the actual driving force which led to the introduction of mathematics and mathematical thinking into (or at least not very far from) chemical laboratories.
Ivan Gutman, Oskar E. Polansky

### Chapter 1. Topological Aspects in Chemistry

Abstract
Since the time of Van’t Hoff, chemists are used to thinking about molecules as geometric objects in which atoms have a certain spatial arrangement. The geometric parameters of molecules (interatomic distances, bond angles, dihedral angles, etc.) can be measured with a rather high degree of accuracy and are indeed known in a considerable number of cases.
Ivan Gutman, Oskar E. Polansky

### Chapter 2. Molecular Topology

Abstract
A precise, but formal and not easily understandable definition of molecular topology will be given in Sect. 3 of this chapter. For most of the applications, however, the present simple description will fully suffice. Suppose a molecule M is composed of n atoms A1, A2, ... , A n . Suppose also that for any two atoms A i and A j we can decide which of the following two statements is correct:
(a)
In the molecule M there is a chemical bond between the atoms A i and A j .

(b)
In the molecule M the atoms A i and A j are not chemically bonded.

Ivan Gutman, Oskar E. Polansky

### Chapter 3. Chemical Graphs

Abstract
The concept of a molecular graph was introduced in Sect. 2.1. Elements of the mathematical apparatus of graph theory will be given in Chap. 4. The purpose of the present chapter is to review the manifold types of molecular graphs which occur in mathematical investigations in organic chemistry.
Ivan Gutman, Oskar E. Polansky

### Chapter 4. Fundamentals of Graph Theory

Abstract
It is not necessary to persuade the reader that graphs are one of the basic mathematical objects with which the present book is concerned. Chap. 2 gave a general conceptual basis for the use of graphs for representing the topology of a molecule. In Chap. 3 we got acquainted with a variety of types of molecular graphs. The present chapter will, finally, provide a precise mathematical characterization of a graph. We shall list here a number of additional graph-theoretical definitions and mention a few basic properties of graphs.
Ivan Gutman, Oskar E. Polansky

### Chapter 5. Graph Theory and Molecular Orbitals

Abstract
In the present chapter, as well as throughout the entire book, we assume that the reader knows the basic facts about the Hückel molecular orbital (HMO) theory [35, 51, 62]. Hence HMO theory is an approximate quantum-mechanical approach to the description of the π-electrons in unsaturated conjugated molecules. The wave function for a π-electron is presented in the LCAO form
$${\psi_i} = \sum\limits_{{j = 1}}^n {{c_{{ij}}}} \left| {{p_j} >} \right.$$
(1)
where {p j > symbolizes a p π -vorbital located on the j-th atom of the conjugated molecule, and the summation goes over all n atoms which participate in the conjugation.
Ivan Gutman, Oskar E. Polansky

### Chapter 6. Special Molecular Graphs

Abstract
In graph theory a connected acyclic graph is called a tree. Hence we may say that the topology of acyclic molecules is represented by trees; the essential topological properties of acyclic molecules coincide with those of trees. In the following we shall get acquainted with the basic properties of trees.
Ivan Gutman, Oskar E. Polansky

### Chapter 7. Fundamentals of Group Theory

Abstract
Groups are sets of elements amended with a combination law that satisfies certain conditions (axioms).
Ivan Gutman, Oskar E. Polansky

### Chapter 8. Symmetry Groups

Abstract
In 3-dimensional space only the few following types of symmetry operations are feasible in order to transform a symmetrical molecule into itself:
• E... the identity operation which keeps all the points unchanged.
• C n ... a rotation about an axis of symmetry by the angle 2π/n where n denotes any natural number; C n is called an n-fold axis of rotation. C n and C n n-1 differ only in the direction of the rotation. The value n = 1 indicates that no symmetry axis exists, i.e. C1 = E; the values n ≧ 7 are only occasionally realized except C which coincides with the molecular axis in collinear molecules. Among several axes of symmetry that one which has the largest value of n is called the principal axis. The rotations obey the following relations:
$$C_n^kC_n^l = C_n^{{k + 1}}; C_{{km}}^k = {C_m}; C_n^n = E$$
(1)
.
Ivan Gutman, Oskar E. Polansky

### Chapter 9. Automorphism Groups

Abstract
The notion of graph automorphism has already been introduced in Section 4.1. An automorphism may be understood as a bijective (that is one-to-one) mapping of the vertex set V(G) of the graph onto itself which preserves the edge relation ℰ(G) of the graph. Evidently, only those vertices can be mapped onto each other which are equivalent, i.e. they are indistinguishable apart from their labels. A subset of V(G) formed by all mutually equivalent vertices is called an orbit of the graph vertices.
Ivan Gutman, Oskar E. Polansky

### Chapter 10. Some Interrelations between Symmetry and Automorphism Groups

Abstract
Apart from the identity operation E, the elements of a space group are geometrically well-defined subspaces of the 3-dimensional space, i.e. points (C), straight lines (C n , S n ) and planes (σ h , σ v , σ d ). Thus the operations associated with these elements are realized in each point of the space. On the other hand, each point of the space and, thus, each center of the molecule is transformed by such an operation, provided the point does not belong to the symmetry element considered. As a final consequence of these circumstances, the molecule treated by means of symmetry groups is considered to be rigid. This means that each atom of the molecule is associated with a triple of coordinates characterizing its mean position with absolute precision. Vibrations of the atoms about their mean positions are discussed in terms of elongation vectors (see paragraph 8.4.3). This concept begins to break down when an internal degree of freedom (e.g. the torsion of a methyl group) becomes fully excited. As the shall show later, automorphism groups of the molecular graph are well-suited to treat the symmetry in non-rigid molecules.
Ivan Gutman, Oskar E. Polansky

### Chapter 11. Topological Indices

Abstract
Whereas the topology of a molecule, represented by the molecular graph is an essentially non-numerical mathematical object, various measurable properties of molecules are usually expressed by means of numbers. In order to link molecular topology to any real molecular property one must first convert the information contained in the molecular graph into a numerical characteristic. Every number which is uniquely determined by a graph is called a graph invariant. Those invariants of molecular graphs which are used for structure-property or structure-activity correlations are usually called topological indices (of the corresponding molecule).
Ivan Gutman, Oskar E. Polansky

### Chapter 12. Thermodynamic Stability of Conjugated Molecules

Abstract
We already know that the molecular orbital energy levels of the π-electrons in conjugated molecules are (within the HMO approximation) related to the eigenvalues of the molecular graph via
$${E_j} = \alpha + {\lambda_j}\beta$$
(1)
.
Ivan Gutman, Oskar E. Polansky

### Chapter 13. Topological Effect on Molecular Orbitals

Abstract
Chemical experience is gained from the study of real existing molecules, which are thought to be composed of distinct particles of matter, namely atomic nuclei and electrons. Abstracting the molecular structure as far as possible one arrives at molecular topology. In the course of abstraction one and the same molecular topology may be obtained from very different molecules (see Fig. 2.1). Thus the pronounced differences in the chemical properties should be an exclusive consequence of the properties of the atoms used in the various realizations of a given topology. In view of this, one may wonder whether topology plays any role at all. The successful application of topological indices in various correlations (see Chap. 11) may be evaluated as some positive evidence, but not as a rigorously proved answer: neither the physical meaning of the topological indices nor the physical interrelations causing the correlations are sufficiently understood. Contrary to that, the topological effect on molecular orbitais (TEMO) [162] provides solid evidence that topology determines at least a frame within which series of physically and chemically diverse species may be realized.
Ivan Gutman, Oskar E. Polansky