The behavior of Wiener indices and polynomials of graphs under five graph decorations

The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A qq-analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial, extends this concept by trying to capture the complete distribution of distances in the graph.Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.