Resistance‐distance matrix: A computational algorithm and its application

Abstract

Abstract The distance matrix D , the resistance‐distance matrix Ω , the related quotient matrices D / Ω and Ω / D and the corresponding distance‐related and resistance‐distance‐related descriptors: the Wiener index W, the Balaban indices J and J Ω , the Kirchhoff index Kf, the Wiener‐sum index WS, and Kirchhoff‐sum index KfS are presented. A simple algorithm for computing the resistance‐distance matrix is outlined. The distance‐related and the resistance‐distance‐related indices are used to study cyclicity in four classes of polycyclic graphs: five‐vertex graphs containing a five‐cycle and Schlegel graphs representing platonic solids, buckminsterfullerene isomers and C 70 isomers. Among the considered indices only the Kirchhoff index correctly ranks according to their cyclicity, the Schlegel graphs for platonic solids, C 60 isomers, and C 70 isomers. The Kirchhoff index further produces the reverse order of five‐vertex graphs containing a five‐cycle (which could be simply altered to the correct order by adding a minus sign to the Kirchhoff indices for these graphs). © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001