Resistance-distance matrix: A computational algorithm and its application

• Published in: International journal of quantum chemistry Vol. 90; no. 1; pp. 166 - 176
• Main Authors: Babić, D., Klein, D. J., Lukovits, I., Nikolić, S., Trinajstić, N.
• Format: Journal Article
• Language: English
• Published: New York Wiley Subscription Services, Inc., A Wiley Company 2002
• Subjects: cyclicity; distance matrix; distance-related indices; Kirchhoff index; Kirchhoff-sum index; quotient matrices; resistance-distance matrix; resistance-distance-related indices
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/qua.10057

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 JΩ, 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 C70 isomers. Among the considered indices only the Kirchhoff index correctly ranks according to their cyclicity, the Schlegel graphs for platonic solids, C60 isomers, and C70 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