Information theory, distance matrix, and molecular branching

Abstract

Information theory was used in defining several measures of the topological properties of molecules, namely, information for adjacency, incidence, polynomial coefficients of the adjacency matrix, and for distances of molecular graphs. The latter was found to have a greater ability for discrimination between structural isomers than all known topological indices and to be a very appropriate measure of branching. All the quantities related to the distance matrix (two information measures, the Weiner number and the largest eigenvalue of the characteristic polynomial) were found to reflect in the same way the main features of molecular branching. On this basis, as well as on the basis of general expressions for information content and Wiener number derived for tree graphs, the essence of molecular branching was expressed in a number of rules. It was shown that these theoretical rules on branching are in agreement with the intuitive understanding of branching and are reflected in a large number of molecular properties.

Introduction of branching degrees of octane isomers

“…The sequence of isomers of increasing branching, as well as the values of branching degrees based on physicochemical properties of octanes, confirm the previous conclusions based on topological indices 1,3,4 that the most important structural feature regarding branching is the number of branches. In the case of the branching degrees of octane isomers derived here, it is overwhelmed only in the case of 234M5 < 3Et3M5.…”

Computer simulations of hyperbranched polymers: The influence of the Wiener index on the intrinsic viscosity and radius of gyration

“…W is a direct measure of molecular compactness. 20 Converse to DB, W is lower for more compact molecules at a given N with more sprawled out, linearlike structures having higher W values at a given N. Widmann and Davies 13 and Lyulin et al 14 creased. Lyulin et al also obtained a power law relationship where aϭ1.1 to 1.6 following the same trend of a with N. Since the ͓ ͔ values of Widmann and Davies are calculated directly from simulated R g 's through the assumption that ͓ ͔ϳR g 3 , it can be seen that their RMMC simulations indicate R g ϳW c , where cϭ0.7 to 0.9.…”

Information Theory of Networks

“…where k i is the occurrence of a distance possessing value i in the distance matrix of G. The motivation to introduce these measures was to find quantities which detect branching well, see [49]. In this context, branching of a graph correlates with the number of terminal vertices.…”