Mean Wiener numbers and other mean extensions for alkane trees

Area of Science:

• Computational chemistry
• Graph theory
• Structure-property relationships

Background:

• The Wiener number, a topological index, quantifies molecular size and shape.
• Understanding the properties of alkane isomers is crucial in various chemical applications.
• The vast number of alkane isomers for larger carbon counts presents a computational challenge.

Purpose of the Study:

• To compute and analyze the average Wiener number and other graphical extension measures for alkane structural isomers.
• To investigate the asymptotic behavior of these measures for large numbers of carbon atoms (N).
• To relate computed graphical extensions to the geometric properties of random alkane mixtures.

Main Methods:

• Systematic computation of Wiener numbers for alkane isomers up to N=90.
• Application of graph theory to represent alkane structures and calculate intersite distances.
• Fitting computed data to various proposed asymptotic functional forms.
• Development of a heuristic argument connecting graphical and geometric extensions.

Main Results:

• Average Wiener numbers and other graphical extension measures were calculated for N up to 90.
• The study identified and tested several asymptotic forms for these measures.
• A correlation was established between the average graphical extension and the geometric extension of alkane mixtures.

Conclusions:

• The Wiener number and related topological indices provide valuable insights into the average properties of large sets of alkane isomers.
• Asymptotic analysis helps predict the behavior of these measures for very large molecules.
• The findings contribute to understanding the relationship between molecular structure and macroscopic properties in complex mixtures.