## Abstract

In the development of any scientific theory, the initial stage based on the accumulation of observational facts is necessarily followed by the formalization and generalization of the concepts involved. Graph theoretic molecular descriptors, which are referred as topological indices (TI) have been around for more than half century [1]. During this time many TIs have been defined [1], their mathematical properties have been scrutinized [2, 3], and more importantly they have proved to be useful in predicting molecular properties beyond any reasonable doubt [4]. Then, the field is mature enough to jump to the next stage of development. That is, the formalization and generalization of concepts that permits the elaboration of a physical theory for topological indices in molecular sciences. In 2001 one of the current authors (EE) proposed a graph theoretic scheme that permitted the generalization of several of the best known TIs [5]. In a subsequent series of papers we have shown several of the principal characteristics of this generalized scheme for graph theoretic indices in chemistry, hereafter named G-GTI [6-12].

Among the most relevant characteristics of G-GTI we can mention the following ones:

It groups the definition of many known TIs into one graph theoretic invariant, namely a quadratic form based on a generalized graph matrix and vectors [5, 10].

TIs are designed a-la-carte more than in ad hoc way to describe a particular experimental property, which improves significantly the predictability of the methods used [7-9].

The generalized graph matrix can be used to generalize quantum chemical concepts, such as the Hückel Molecular Orbital method or the Lennard-Jones potentials for united-atoms [6, 12].

The method permits the optimization of local vertex invariants (LOVIs) to account for atomic properties, which is an avenue to be explored for introducing heteroatoms in the G-GTI scheme.

The structural interpretation of the TIs can be carried out in a generalized way accounting for through-bonds and through-space inter-atomic interactions in molecules [5-12].

The aim of the current work is to review the G-GTI method with special emphasis in the methodological aspects of the method and its applications for predicting physico-chemical properties. Our objective is two-fold. First, that chemists interested in the prediction of molecular properties are aware of the generality and simplicity of the G-GTI approach in such a way that they can use it in modelling physical, chemical pharmacological, toxicological and environmental properties of organic molecules. Secondly, that graph theorists, both mathematicians and chemists, explore more deeply the possibilities of G-GTI for studying molecular graphs. We hope that this chapter be of interest for both communities.

Among the most relevant characteristics of G-GTI we can mention the following ones:

It groups the definition of many known TIs into one graph theoretic invariant, namely a quadratic form based on a generalized graph matrix and vectors [5, 10].

TIs are designed a-la-carte more than in ad hoc way to describe a particular experimental property, which improves significantly the predictability of the methods used [7-9].

The generalized graph matrix can be used to generalize quantum chemical concepts, such as the Hückel Molecular Orbital method or the Lennard-Jones potentials for united-atoms [6, 12].

The method permits the optimization of local vertex invariants (LOVIs) to account for atomic properties, which is an avenue to be explored for introducing heteroatoms in the G-GTI scheme.

The structural interpretation of the TIs can be carried out in a generalized way accounting for through-bonds and through-space inter-atomic interactions in molecules [5-12].

The aim of the current work is to review the G-GTI method with special emphasis in the methodological aspects of the method and its applications for predicting physico-chemical properties. Our objective is two-fold. First, that chemists interested in the prediction of molecular properties are aware of the generality and simplicity of the G-GTI approach in such a way that they can use it in modelling physical, chemical pharmacological, toxicological and environmental properties of organic molecules. Secondly, that graph theorists, both mathematicians and chemists, explore more deeply the possibilities of G-GTI for studying molecular graphs. We hope that this chapter be of interest for both communities.

Original language | English |
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Title of host publication | Theory and Applications II. Mathematical Chemistry Monographs No.9 |

Editors | I. Gutman, B. Furtula |

Pages | 217-230 |

Number of pages | 14 |

Publication status | Published - 2010 |

## Keywords

- G-GTI
- Generalized graph theoretic indices
- chemistry