The edge-Wiener index is conceived in analogous to the traditional Wiener index and it is defined as the sum of distances between all pairs of edges of a graph G. In the recent years, it has received considerable attention for determining the variations of its computation. Motivated by the method of computation of the traditional Wiener index based on canonical metric representation, we present the techniques to compute the edge-Wiener and vertex-edge-Wiener indices of G by dissecting the original graph G into smaller strength-weighted quotient graphs with respect to Djoković-Winkler relation. These techniques have been applied to compute the exact analytic expressions for the edge-Wiener and vertex-edge-Wiener indices of coronoid systems, carbon nanocones and SiO 2 nanostructures. In addition, we have reduced these techniques to the subdivision of partial cubes and applied to the circumcoronene series of benzenoid systems.

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The advent of two‐dimensional transition metal dichalcogenides has triggered an interest in exploring a new class of high‐performance materials with intriguing physicochemical attributes. Molybdenum and tungsten disulfides have attracted significant attention due to surface excitons and trions and the large spin‐orbit effects of these compounds. Moreover, WS2is especially intriguing due to large relativistic effects, which result in bound excitons at the edge and biexciton formation in the bilayers. Hence, we present a relativistic topological model for the characterization of these two types of metal disulfides. We have employed relativistically weighted graph‐theoretical methods for obtaining structural descriptors of these compounds through the inclusion of different shapes on the boundaries and employing the topological cut techniques.

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