Corroborating a Modification of the Wiener Index

Sa？etak In a recent work [Chem. Phys. Lett. 333 (2001) 319-321] Nikoli？, Trinajsti？, and Randi？ put forward a novel modification mW of the Wiener index. We now show that mW possesses the basic properties required by a topological index to be acceptable as a measure of the extent of branching of the carbon-atom skeleton of the respective molecule (and therefore to be a structure-descriptor, potentially applicable in QSPR and QSAR studies). In particular, if Tn is any n-vertex tree, different from the n-vertex path Pn and the n-vertex star Sn, then mW(Pn) < mW(Tn) < mW(Sn). We also show how the concept of the modified Wiener index can be extended to weighted molecular graphs