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On the Djoković-Winkler relation and its closure in subdivisions of fullerenes, triangulations, and chordal graphs

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 Djoković-Winkler relation $Θ$ and its transitive closure $Θ^\ast$. In this paper we study the behavior of $Θ$ and $Θ^\ast$ with respect to full subdivisions. We apply our results to describe $Θ^\ast$ in full subdivisions of fullerenes, plane triangulations, and chordal graphs.