The Wiener index is one of the most widely studied parameters in chemical graph theory. It is defined as the sum of the lengths of the shortest paths between all unordered pairs of vertices in a given graph. In 1991, Šoltés posed the following problem regarding the Wiener index: Find all graphs such that its Wiener index is preserved upon removal of any vertex. The problem is far from being solved, and to this day, only one graph with such property is known: the cycle graph on 11 vertices. In this paper, we solve a relaxed version of the problem, proposed by Knor et al. in 2018. For a given k, the problem is to find (infinitely many) graphs having exactly k vertices such that the Wiener index remains the same after removing any of them. We call these vertices good vertices, and we show that there are infinitely many cactus graphs with exactly k cycles of length at least 7 that contain exactly 2k good vertices and infinitely many cactus graphs with exactly k cycles of length \(c \in \{5,6\}\) that contain exactly k good vertices. On the other hand, we prove that G has no good vertex if the length of the longest cycle in G is at most 4.

该Wiener指数是化学图论研究最广泛的参数之一。它定义为给定图中所有无序顶点对之间的最短路径长度的总和。1991年，Šoltés提出了有关Wiener指数的以下问题：查找所有图形，以便在删除任何顶点时保留其Wiener指数。问题远未解决，到目前为止，只有一个具有这种特性的图是已知的：11个顶点上的循环图。在本文中，我们解决了由Knor等人提出的问题的宽松版本。在2018年。对于给定的k，问题是要找到（无限多个）正好为k的图去除顶点后，维纳指数保持不变的顶点。我们称这些顶点为好顶点，并且我们证明有无限多个仙人掌图，其长度恰好为k个周期，至少7个包含正好2个k良好顶点，以及无限多个仙人掌图，恰恰是长度为k个周期，\ {5,6 \} \）恰好包含k个良好顶点。另一方面，如果G中最长循环的长度最多为4 ，则我们证明G没有良好的顶点。