On the Djokovi\'c-Winkler relation and its closure in subdivisions of fullerenes, triangulations, and chordal graphs

Abstract

It was recently pointed out that certain SiO$_2$ layer structures and SiO$_2$ nanotubes can be described as full subdivisions aka subdivision graphs of partial cubes. A key tool for analyzing distance-based topological indices in molecular graphs is the Djokovi\'c-Winkler relation $\Theta$ and its transitive closure $\Theta^\ast$. In this paper we study the behavior of $\Theta$ and $\Theta^\ast$ with respect to full subdivisions. We apply our results to describe $\Theta^\ast$ in full subdivisions of fullerenes, plane triangulations, and chordal graphs.