### Abstract

The Wiener index, or the Wiener number, also known as the "sum of distances" of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has been studied in depth. An index proposed by Schultz is shown to be related to the Wiener index for trees, and Ivan Gutman proposed a modification of the Schultz index with similar properties. We deduce a similar relationship between these three indices for catacondensed benzenoid hydrocarbons (graphs formed of concatenated hexagons, or hexagonal chains, or sometimes acenes). Indeed, we may define three families of generalized Wiener indices, which include the Schultz and Modified Schultz indices as special cases, such that similar explicit formulae for all generalized Wiener indices hold on hexagonal chains. We accomplish this by first giving a more refined proof of the formula for the standard Wiener index of a hexagonal chain, then extending it to the generalized Wiener indices via the notion of partial Wiener indices. Finally, we discuss possible extensions of the result.

Original language | English |
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Pages (from-to) | 426-435 |

Number of pages | 10 |

Journal | International Journal of Quantum Chemistry |

Volume | 106 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 Feb 1 |

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### Keywords

- Generalized polynomial
- Generalized wiener indices
- Hex chain
- Wiener indices
- Wiener polynomial

### ASJC Scopus subject areas

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

### Cite this

*International Journal of Quantum Chemistry*,

*106*(2), 426-435. https://doi.org/10.1002/qua.20732