Abstract   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   q  -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.
['Weigen Yan', 'Bo-Yin Yang', 'Yeong-Nan Yeh']
The behavior of Wiener indices and polynomials of graphs under five graph decorations