The Szeged Index and the Wiener Index of Partial Cubes with Applications to Chemical Graphs

TL;DR

This paper generalizes methods for calculating the Szeged and Wiener indices of partial cubes, enabling efficient computation for certain chemical graphs like $C_4C_8$ systems, with potential applications in chemical graph theory.

Contribution

It introduces a reduction technique for computing indices of partial cubes via quotient graphs, extending previous results and enabling linear-time calculations for specific chemical graph families.

Findings

Szeged and Wiener indices of partial cubes can be computed via quotient graphs.

Quotient graphs of partial cubes are themselves partial cubes.

Indices for $C_4C_8$ systems can be computed in linear time.

Abstract

In this paper we study the Szeged index of partial cubes and hence generalize the result proved by V. Chepoi and S. Klav\v{z}ar, who calculated this index for benzenoid systems. It is proved that the problem of calculating the Szeged index of a partial cube can be reduced to the problem of calculating the Szeged indices of weighted quotient graphs with respect to a partition coarser than $\Theta$-partition. Similar result for the Wiener index was recently proved by S. Klav\v{z}ar and M. J. Nadjafi-Arani. Furthermore, we show that such quotient graphs of partial cubes are again partial cubes. Since the results can be used to efficiently calculate the Wiener index and the Szeged index for specific families of chemical graphs, we consider $C_4C_8$ systems and show that the two indices of these graphs can be computed in linear time.