Graphs preserving total distance upon vertex removal

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2018-07-01 DOI:10.1016/j.endm.2018.06.019

Snježana Majstorović, Martin Knor, Riste Škrekovski

引用次数: 1

Abstract

The total distance or Wiener index $W (G)$ of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Šoltés posed the problem of finding all graphs G such that the equality $W (G) = W (G - v)$ holds for all their vertices v. Up to now, the only known graph with this property is the cycle C₁₁. Our main object of study is a relaxed version of this problem: Find graphs for which total distance does not change when a particular vertex is removed. We show that there are infinitely many graphs that satisfy this property. This gives hope that Šoltes's problem may have also some solutions distinct from C₁₁.

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顶点移除后保持总距离的图

连通图G的总距离或维纳指数W(G)被定义为G中所有顶点对之间的距离和。1991年，Šoltés提出了一个问题，即找到所有图G，使得等式W(G)=W(G−v)对所有顶点v都成立。到目前为止，唯一已知的具有这个性质的图是循环C11。我们的主要研究对象是这个问题的一个宽松版本:找到当移除一个特定顶点时总距离不改变的图。我们证明有无穷多个图满足这个性质。这给了人们希望，Šoltes的问题可能也有一些不同于C11的解决方案。

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

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

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