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

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.

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