Abstract Chemical graph theory (CGT) starts by defining matrices that represent
the molecular graph then proceed to extract numbering-independent matrix
invariants to be used as molecular descriptors in empirical quantitative structure to
activity (or property) relationships (QSAR/QSPR). Two proposed matrix representations of molecular structure are presented in this chapter as alternatives to
simple connectivity molecular graphs. Firstly, it is proposed to use a more
“nuanced” connectivity matrix by weighing the “ones” entered in a CGT molecular
graph matrix by the bond critical point electron densities associated with each bond
path to yield what we term the “electron density-weighted adjacency/connectivity
matrices (EDWAM/EDWCM)”. In a second approach, it is proposed to use the
localization and delocalization indices of the quantum theory of atoms in molecules
(QTAIM) to construct a richer representation of the molecular graph, a “fuzzy”
graph, whereby an edge exists between any two atoms (measured by the delocalization index between them) whether they share a bond path or not. Such a fuzzy
graph is represented by what we term “electron localization-delocalization matrix
(LDM)”. We show that the LDM representations of a series of molecules provide a
powerful tool for robust QSAR/QSPR modeling.
3.1 Introduction
A molecule can be abstracted as a network of points (vertices) connected by lines
(edges) and hence constituting a graph. Molecular graphs formed from a set of edges
each consisting of what chemists normally call a “chemical bond” can be—but
generally are not—complete. (A “complete graph” is one in which every pair of
vertices is connected by an edge, a trivial example being the graph of a diatomic
molecule). In contrast, a graph based on any pair-wise property such as inter-nuclear
distance, nuclear-nuclear repulsion, or a count of electrons delocalized between any
two pairs of atoms in the molecule necessarily constitutes a complete graph.
Molecular graphs, complete or incomplete, can be conveniently represented by
connectivity matrices as can be seen in the examples in Fig. 3.1 and in Refs. [1–9].
A complete graph where connectedness is indicated by 1 and disjointedness by 0
will have a non-zero entry for every non-diagonal element of the matrix while an
incomplete graph has finite entries only for connected vertices and zero elsewhere
in the matrix (Fig. 3.1).
A matrix representative of a complete graph with n vertices whereby connectivity is assigned “1” as in Fig. 3.1a is thus filled with ones except along the
diagonal and hence has n(n − 1)/2 edges, the number of non-diagonal elements of
its matrix representative. In practice, a complete graph such as the delocalization
matrix (DM), described below, may have zero (negligible) entries other than along
the diagonal when the delocalization index between a given pair of atoms in a
molecule has a magnitude below the precision to which the numerical entries are
reported.
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C.F. Matta et al.
the molecular graph then proceed to extract numbering-independent matrix
invariants to be used as molecular descriptors in empirical quantitative structure to
activity (or property) relationships (QSAR/QSPR). Two proposed matrix representations of molecular structure are presented in this chapter as alternatives to
simple connectivity molecular graphs. Firstly, it is proposed to use a more
“nuanced” connectivity matrix by weighing the “ones” entered in a CGT molecular
graph matrix by the bond critical point electron densities associated with each bond
path to yield what we term the “electron density-weighted adjacency/connectivity
matrices (EDWAM/EDWCM)”. In a second approach, it is proposed to use the
localization and delocalization indices of the quantum theory of atoms in molecules
(QTAIM) to construct a richer representation of the molecular graph, a “fuzzy”
graph, whereby an edge exists between any two atoms (measured by the delocalization index between them) whether they share a bond path or not. Such a fuzzy
graph is represented by what we term “electron localization-delocalization matrix
(LDM)”. We show that the LDM representations of a series of molecules provide a
powerful tool for robust QSAR/QSPR modeling.
3.1 Introduction
A molecule can be abstracted as a network of points (vertices) connected by lines
(edges) and hence constituting a graph. Molecular graphs formed from a set of edges
each consisting of what chemists normally call a “chemical bond” can be—but
generally are not—complete. (A “complete graph” is one in which every pair of
vertices is connected by an edge, a trivial example being the graph of a diatomic
molecule). In contrast, a graph based on any pair-wise property such as inter-nuclear
distance, nuclear-nuclear repulsion, or a count of electrons delocalized between any
two pairs of atoms in the molecule necessarily constitutes a complete graph.
Molecular graphs, complete or incomplete, can be conveniently represented by
connectivity matrices as can be seen in the examples in Fig. 3.1 and in Refs. [1–9].
A complete graph where connectedness is indicated by 1 and disjointedness by 0
will have a non-zero entry for every non-diagonal element of the matrix while an
incomplete graph has finite entries only for connected vertices and zero elsewhere
in the matrix (Fig. 3.1).
A matrix representative of a complete graph with n vertices whereby connectivity is assigned “1” as in Fig. 3.1a is thus filled with ones except along the
diagonal and hence has n(n − 1)/2 edges, the number of non-diagonal elements of
its matrix representative. In practice, a complete graph such as the delocalization
matrix (DM), described below, may have zero (negligible) entries other than along
the diagonal when the delocalization index between a given pair of atoms in a
molecule has a magnitude below the precision to which the numerical entries are
reported.
54
C.F. Matta et al.