Chemical Graph Theory

Nenad Trinajstic

• Publisher: CRC Press
• Year: 2018
• eBook ISBN: 9781351461566
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

New Edition! Completely Revised and Updated Chemical Graph Theory, 2nd Edition is a completely revised and updated edition of a highly regarded book that has been widely used since its publication in 1983. This unique book offers a basic introduction to the handling of molecular graphs - mathematical diagrams representing molecular structures. Using mathematics well within the vocabulary of most chemists, this volume elucidates the structural aspects of chemical graph theory: (1) the relationship between chemical and graph-theoretical terminology, elements of graph theory, and graph-theoretical matrices; (2) the topological aspects of the Hückel theory, resonance theory, and theories of aromaticity; and (3) the applications of chemical graph theory to structure-property and structure-activity relationships and to isomer enumeration. An extensive bibliography covering the most relevant advances in theory and applications is one of the book's most valuable features. This volume is intended to introduce the entire chemistry community to the applications of graph theory and will be of particular interest to theoretical organic and inorganic chemists, physical scientists, computational chemists, and those already involved in mathematical chemistry.

Chapter 1

INTRODUCTION

Graph theory is a branch of (discrete) mathematics that deals with the way objects are connected.¹ Thus, the connectivity in a system is a fundamental quality of graph theory. The principal concept in graph theory is that of a graph. For a mathematician, a graph is the application of a set on itself, i.e., a collection of elements of the set and the binary relations between these elements.² Graphs are one-dimensional objects,³ but they can be embedded or realized in spaces of higher dimensions.

Graph theory is related to topology (in fact graph theory is one-dimensional topology³), matrix theory, group theory, set theory, probability, combinatorics, and numerical analysis. It has been used in such diverse fields as economics⁴ and theoretical physics,⁵ psychology⁶ and nuclear physics,⁷ biomathematics⁸ and linguistics,⁹ sociology¹⁰ and zoology,¹¹ technology¹² and anthropology,¹³ computer science¹⁴ and geography,¹⁵ biology¹⁶ and engineering,² etc.

Chemical graph theory is a branch of mathematical chemistry that is concerned with analyses of all consequences of a connectivity in a chemical graph. Chemical graph serves as a convenient model for any real or abstract chemical system (molecule or reaction scheme in a chemical transformation).^17,18 In other words, chemical graph theory is concerned with all aspects of the application of graph theory to chemistry. By the use of the term chemical it is emphasized that one is allowed in chemical graph theory, unlike in graph theory, to rely on the intuitive understanding of many concepts and theorems rather than on formal mathematical proofs.

The recent years have witnessed a remarkable growth of chemical graph theory.¹⁹, ²⁰, ²¹, ²², ²³, ²⁴, ²⁵, ²⁶, ²⁷, ²⁸, ²⁹, ³⁰, ³¹, ³², ³³, ³⁴, ³⁵, ³⁶, ³⁷, ³⁸, ³⁹, ⁴⁰, ⁴¹, ⁴², ⁴³ There are many reasons for the increasing popularity of graph theory in chemistry.¹⁷, ¹⁸, ¹⁹, ²⁰, ^21,25, ²⁶, ²⁷, ^{28,30,34,37,38,41,42,44}, ⁴⁵, ⁴⁶ First, there is hardly any concept in the natural sciences which is closer to the notion of graph than the structural (constitutional) formula of a chemical compound,⁴⁷ because a graph is, simply said, a mathematical model⁴⁸ which may be used directly to represent a molecule when the only property considered is the internal connectivity, i.e., whether or not a chemical bond joins two atoms in a molecule. Since almost all discussions in chemistry are carried out by means of graphic display of compounds and reactions, chemists manipulate graphs on a daily basis, albeit many a chemist is not aware of this fact. Thus, it appears that the natural language of chemistry by which chemists communicate is provided by (chemical) graph theory. Second, graph theory provides simple rules by which chemists may obtain many qualitative predictions about the structure and reactivity of various compounds. All these predictions can be reached using nothing more than pencil and paper. Furthermore, the obtained results have in many cases a general validity and may be formulated as theorems and/or rules which can then be applied to a variety of similar problems without any further numerical or conceptual work. Third, graph theory may be used as a foundation for the representation, classification, and categorization of a very large number of chemical systems...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Chapter 1 Introduction
• Chapter 2 Elements of Graph Theory
• Chapter 3 Chemical Graphs
• Chapter 4 Graph-Theoretical Matrices
• Chapter 5 The Characteristic Polynomial of a Graph
• Chapter 6 Topological Aspects of Hückel Theory
• Chapter 7 Topological Resonance Energy
• Chapter 8 Enumeration of Kekulé Valence Structures
• Chapter 9 The Conjugated-Circuit Model
• Chapter 10 Topological Indices
• Chapter 11 Isomer Enumeration
• Index