[1] Wiener index can be used for the representation of computer networks and enhancing lattice hardware security.
[4] The same quantity has also been studied in pure mathematics, under various names including the gross status,[5] the distance of a graph,[6] and the transmission.
[9] Thus, even though these two molecules have the same chemical formula, and the same numbers of carbon-carbon and carbon-hydrogen bonds, their different structures give rise to different Wiener indices.
[2] Later work on quantitative structure–activity relationships showed that it is also correlated with other quantities including the parameters of its critical point,[10] the density, surface tension, and viscosity of its liquid phase,[11] and the van der Waals surface area of the molecule.
[12] The Wiener index may be calculated directly using an algorithm for computing all pairwise distances in the graph.
When the graph is unweighted (so the length of a path is just its number of edges), these distances may be calculated by repeating a breadth-first search algorithm, once for each starting vertex.
Alternative but less efficient algorithms based on repeated matrix multiplication have also been developed within the chemical informatics literature.
[21] Benzenoids (graphs formed by gluing regular hexagons edge-to-edge) can be embedded isometrically into the Cartesian product of three trees, allowing their Wiener indices to be computed in linear time by using the product formula together with the linear time tree algorithm.
[22] Gutman & Yeh (1995) considered the problem of determining which numbers can be represented as the Wiener index of a graph.
[23] They showed that all but two positive integers have such a representation; the two exceptions are the numbers 2 and 5, which are not the Wiener index of any graph.