1. INTRODUCTION

In these notes we wish to discuss the topological approach to chemistry of conjugated

structures using the mathematical apparatus of graph theory (in the text the symbol

GT will be sometimes used instead of fully written: graph theory). The basic con­

cepts and definitions of GT will be given in the following chapter. Here, however,

we give some general ideas about the use of GT in chemistry.

There is hardly any concept in natural sciences which is closer to the notion of

graphs than the structural formulae of chemical compounds. In fact, there is no es­

sential difference between a graph and structural formula.127,1~8

A graph is, simply said, a mathematical structure which may be used to represent

the topology of a given molecule. Therefore, chemists can easily grasp the concepts

of GT. Moreover, chemists actually know and use a number of graph-theoretical theorems

without being aware of this fact in many cases. A classical example is provided by

the concept of alternant hydrocarbons introduced by Coulson and Rushbrooke2~ which is

for graph-theorists the two-colour problem.85,1~9 However, the language of GT is very

different from that of chemistry. Therefore, we offer a short glossary in Table 1.1

which should help the reader to follow more easily the text, because we shall freely

use and interchange the mathematical and chemical terminology throughout the text.

A. Graovac et al., Topological Approach to the Chemistry of Conjugated Molecules© Springer-Verlag Berlin · Heidelberg 1977

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TABLE 1.1

THE CORRESPONDENCE BETWEEN THE GRAPH-THEORETICAL AND CHEMICAL TERMINOLOGY

Graph-theoretical terminology

Molecular (chemical) graph

Vertex

Weighted edge

Edge

Weighted edge

Degree of a vertex

Tree

Cycle

Chain

Bipartite (bichromatic) graph

Non-bipartite graph

Adjacency matrix ~

Eigenvalue of ~

Eigenvector of ~

Characteristic polynomial

Chemical terminology

Structural formula

Atom

Atom of a specified element

Covalent chemical bond

Covalent chemical bond

between specified elements

Valency of an atom

Acyclic hydrocarbon

Ring

Linear polyene

Alternant hydrocarbon

Nonalternant hydrocarbon

Topological matrix

Molecular orbital energy level

Topological molecular orbital

Secular determinant

The advantage of using GT in chemical studies lies in the possibility to apply

directly its mathematical apparatus and proof techniques. Besides, a given problem

may be considered on a higher level of abstraction which enables a relatively simple

insight into the structural features of the molecule. This is a rather important ad­

vantage of GT because in many cases we wish to study directly the relations between

the particular structural features and a single physicochemical property of a molecule.

On the other hand, a purely numerical computerized study sometimes hides the importance

of a particular structural feature of a molecule which may account for a molecular

property of interest. In addition, the obtained graph-theoretical results have a gen­

eral validity and may be formulated as theorems and/or rules which can then be applied

to any similar group of molecules without any further numerical or conceptual work.

Finally, a graph-theoretic~l language is far more precise and contains a number of

terms which have no equivalent in chemistry.

The term molecular topology is appropriately used to describe the non-metric

properties of molecules. It should be noted that topology, a branch of mathematics,

investigates the nonmetric relationships of geometric (and more abstract) structures.

We define molecular topology as the totality of information contained in the molecular

graph. Let us emphasize here that the graph-theoretical methods should be expected

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primarily to be of use as a complementary approach where the topology and the combina­

torial nature of problem play an important role, in parallel to the application of the

group theory to problems where symmetry is an important feature of the system stud­

ied. 103a

Notes are composed as follows: first, the elements of GT are given and the

equivalence between GT and simple molecular orbital theory of conjugated molecules is

presented. Then, the pi-electron energy is derived in terms of topological parameters

of a molecule. Finally, resonance energies and substitution reactions of conjugated

structures are discussed by means' of topological theory.