Orientations of graphs with maximum Wiener index

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-03-03 DOI:10.1002/jgt.23090

Zhenzhen Li, Baoyindureng Wu

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引用次数: 0

Abstract

In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree $T$ achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph $Θ_{a, b, c}$ with $a \geq b \geq c$ and $b > 1$ , the maximum Wiener index is achieved by the one in which the union of the paths between $u_{1}$ and $u_{2}$ forms a directed cycle of length $a + b + 2$ , where $u_{1}$ and $u_{2}$ are the vertex of degree 3. We confirm the validity of the conjecture.

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具有最大维纳指数的图形方向

本文研究了树和θ图的方向的维纳指数。如果树的一个方向上不存在边改变方向两次的子路径，则称为无之字形方向。Knor、Škrekovski 和 Tepeh 猜想，达到最大维纳指数的树的每个方向都是无之字形。我们通过构建一个反例推翻了这一猜想。Knor、Škrekovski 和 Tepeh 猜想，在有和的θ图的所有方向中，达到最大维纳指数的方向是和之间的路径联合形成长度为和的有向循环的方向，其中和是阶数为 3 的顶点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .