### 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 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 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).

Original language | English |
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Pages (from-to) | 166-176 |

Number of pages | 11 |

Journal | International Journal of Quantum Chemistry |

Volume | 90 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 5 2002 |

### Keywords

- Cyclicity
- Distance matrix
- Distance-related indices
- Kirchhoff index
- Kirchhoff-sum index
- Quotient matrices
- Resistance-distance matrix
- Resistance-distance-related indices

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Condensed Matter Physics
- Physical and Theoretical Chemistry

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## Cite this

*International Journal of Quantum Chemistry*,

*90*(1), 166-176. https://doi.org/10.1002/qua.10057