Reverse Wiener Indices

By subtracting from the graph diameter all topological distances one obtains a new symmetrical matrix, reverse Wiener RW, with zeroes on the main diagonal, whose sums over rows or columns give rise to new integer-number graph invariants σi whose half-sum is a novel topological index (TI), the reverse Wiener index Λ. Analytical forms for values of σi and Λ of several classes of graphs are presented. Relationships with other TIs are discussed. Unlike distance sums, σi values increase from the periphery towards the center of the graph, and they are equal to the graph vertex degrees when the diameter of the graph is equal to 2. Structural descriptors computed from the reverse Wiener matrix were tested in a large number of quantitative structure-property relationship models, demonstrating the usefulness of the new molecular matrix