| Issue |
RAIRO-Oper. Res.
Volume 56, Number 4, July-August 2022
|
|
|---|---|---|
| Page(s) | 2495 - 2511 | |
| DOI | https://doi.org/10.1051/ro/2022118 | |
| Published online | 09 August 2022 | |
Comparative results between the number of subtrees and Wiener index of graphs
1
College of Mathematics, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, P.R. China
2
MIIT Key Laboratory of Mathematical Modelling and High Performance, Computing of Air Vehicles, Nanjing 210016, P.R. China
* Corresponding author: kexxu1221@126.com
Received:
18
November
2021
Accepted:
11
July
2022
For a graph G, we denote by N(G) the number of non-empty subtrees of G. If G is connected, its Wiener index W(G) is the sum of distances between all unordered pairs of vertices of G. In this paper we establish some comparative results between N and W. It is shown that N(G) > W(G) if G is a graph of order n ≥ 7 and diameter 2 or 3. Also some graphs are constructed with large diameters and N > W. Moreover, for a tree T ≇ Sn of order n, we prove that W(T) > N(T) if T is a starlike tree with maximum degree 3 or a tree with exactly two vertices of maximum degrees 3 one of which has two leaf neighbors, or a broom with klog2 n leaves. And a method is provided for constructing the graphs with N < W. Finally several related open problems are proposed to the comparison between N and W.
Mathematics Subject Classification: 05C09 / 05C30 / 05C35
Key words: Number of subtrees / Wiener index / graph with diameter 2 or 3 / tree
© The authors. Published by EDP Sciences, ROADEF, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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