## Abstract

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W^{λ} for λ> 0 and λ< 0 , and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A⊆ V(G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V(G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

Original language | English |
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Article number | 15 |

Journal | Acta Applicandae Mathematicae |

Volume | 171 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2021 |

## Keywords

- Caterpillar
- Complementary Wiener index
- Diameter
- Eccentric sequence
- Extremal tree structures
- Generalised Wiener index
- Harary index
- Hyper-Wiener index
- Wiener-type indices
- k-Steiner distance

## ASJC Scopus subject areas

- Applied Mathematics