Graph Theoretical Invariants Research Articles

A graph theoretical approach to structure-property and structure-activity correlations

The structure similarity and dissimilarity implied in many structure-property and structure-activity relationships has been examined from the graph theoretical point of view. The approach outlined is fundamentally different from generally used schemes in that, rather than seeking a new parametrization which will quantitatively fit observed data and trends,similarities among the skeletal forms and connectivities of the compounds of interest are studied quantitatively. The basis of the method is the assumption that skeletal forms of apparent similarity will yield similar enumerations for a number of graph theoretical invariants. In particular, allpaths within molecular skeletons are enumerated and sequences of path numbers (i.e., the number of paths of different length) are compared. The degree of similarity between molecules is proportional to the distance between points in the corresponding “structure space” obtained by interpreting the entries in molecular path sequences as coordinates inn-dimensional space. As anexample of the use of the concept of structural similarity, structure-activity data relating cerebral dopamine agonist properties for a series of N-substituted 2-aminotetralins are considered. The analysis suggests that the method may find wide application in the field of structure-activity correlations and structure-property studies. The data could be mass spectra, the “fingerprint” regions of infrared spectra, optical rotation and circular dichroism measurements, or any of many not fullyunderstood complex experimental findings suspected of having an inherent structural basis.

On a graph theoretical basis for ordering of structures

Algebraic aspects of ordering of structures are examined. By using paths as the graph theoretical invariants, sequences which enumerate paths of differnet length and characterize individual molecules are derived. A rule for ordering such sequences is proposed. The partial ordering thus derived can be used for prediction of the relative magnitudes of various molecular properties of isomers.

Graph theoretical analysis of diamagnetic susceptibilities of alkanes and alkenes

An alternative description of the additivity of diamagnetic susceptibilities of alkanes by Hameka is given. The analysis is based on an examination of molecular graphs and enumeration of selected graph theoretical invariants (subgraphs). The generality of the apporach is illustrated by extending the analysis to alkenes.

The graph-like state of matter. Part 8.—LCGI schemes and the statistical analysis of experimental data

All additivity schemes can be expressed in terms of an LCGI (linear combination of graphtheoretical invariants) and they are exact in a mathematically almost trivial way. With the aid of the combinatorial method of Mobius inversion, the LCGI formulation tackles efficiently the confirmation (or rejection) of theories from experimental data by exact calculation. Optimisation methods are out of place for this purpose. However, optimisation over a truncated LCGI series does remain the most consistent approach to data reduction and prediction. These conclusions are illustrated by reference to standard molar enthalpy data for alkanes. Our example provides a paradigm for the collection and statistical analysis of molecular data.

The graph-like state of matter. Part 2.—LCGI schemes for the thermodynamics of alkanes and the theory of inductive inference

The theory of graphs, a rapidly developing mathematical discipline, is immensely useful in chemistry. As first appreciated by Cayley (A. Cayley, Collected Mathematical Papers, Cambridge University Press, 1889–97), a molecule becomes a graph, when we regard atoms as points and bonds as lines. Theories based on this notion define a state of matter. Formally, they almost always equate measurable quantities to linear combinations of graph-theoretical invariants (LCGI).Using quite elementary concepts and terms of graph theory, a single systematic definition of about 50 words, contains as special cases, practically all that is useful in previously proposed additivity scheme for predicting standard thermodynamic data. For example, the bond additivity schemes, the schemes for alkanes by Allen, by Laidler and by Tatevskii are included as individual members of the set of schemes so defined, which constitute a canonical hierarchy of LCGI schemes.In this way, estimation of standard enthalpies of alkanes can be rationalised, and that of standard entropies substantially improved. This is exemplified with the aid of recent improvements in computer optimisation procedures. The theory of inductive inference, especially as developed by Carnap (R. Carnap, The Continuum of Inductive Methods, University of Chicago Press, 1952), is useful not merely in suggesting methods of economical and systematic construction of inductive schemes, but in assessing the validity of their postulational basis in the light of experimental data. The customary calculation from standard enthalpies of substantial destabilisation energies in the form of “steric corrections” is criticised as being biassed. It is doubted whether quantum theory had so far contributed significantly to results of analyzing enthalpy data on alkanes.

16. Gases and solids

An alternative description of the additivity of diamagnetic susceptibilities of alkanes by Hameka is given. The analysis is based on an examination of molecular graphs and enumeration of selected graph theoretical invariants (subgraphs). The generality of the apporach is illustrated by extending the analysis to alkenes.