Using the Wiener Index to Calculate the Matrix of Reaction Paths for Some Chemical Molecules

The Wiener index is one of the oldest distance-based graph invariants used in mathematical chemistry, obtained by summing the distances between all unordered pairs of vertices of a connected graph. This paper surveys twenty recent open-access works (2021–2026) on the Wiener index of molecular graphs and formalises a pendant-path reduction: a leaf path of vertices is removed from a chemical tree, the distance matrix is recomputed on the smaller graph, and the contribution of the removed path is restored through a closed-form correction that depends on the length of the path and the transmission of the anchor vertex in the reduced tree. For a representative ten-vertex tree, removing a leaf path of three vertices reduces the number of independent distance entries from to (about ) while exactly reproducing . A MATLAB implementation reproduces the Wiener index of nineteen molecular graphs (the -alkanes –, the cycles –, and the dendrimers and ) and verifies the reduction against known closed-form expressions. The proposed reduction is shown to be the multi-vertex generalisation of the linear-time leaf-deletion algorithm and a closed-form simplification of the cut method.