发表机构
Chemnitz University of Technology; Josip Juraj Strossmayer University of Osijek; University of Zagreb（Chemnitz 工业大学; 约瑟普·尤拉伊·施特罗斯马耶尔奥西耶克大学; 萨格勒布大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究基尔霍夫Šoltés问题，证明C5是基尔霍夫Šoltés图，并构造多个无限图族，其好顶点比例满足不同渐近条件，为寻找其他解提供关键线索。
AI 中文摘要
我们称$v\in V(G)$为\textit{好顶点}，如果移除$v$后基尔霍夫指数保持不变，即$Kf(G)=Kf(G-v)$。1991年，Šoltés研究了图的维纳指数，并提出了识别移除任意顶点后维纳指数保持不变的图的问题。在本文中，我们探索类似的概念：识别\textit{基尔霍夫Šoltés图}，即所有顶点都是好顶点的图。我们证明$C_5$是一个基尔霍夫Šoltés图。我们考虑基尔霍夫Šoltés问题的几个放宽版本，其主要目标是识别包含至少一个好顶点的图。其中之一是\textit{$\beta$-基尔霍夫Šoltés问题}，它寻求找到一个无限图族，其中好顶点的比例至少为$\beta$，其中$\beta \in (0,1]$是一个指定的有理数。另一个涉及构造无限图族，使得好顶点的比例随着图的阶数增长而增加并渐近趋近于给定的实数$\gamma\in (0,1]$。我们证明这两个放宽版本都有无限多个解。特别地，我们证明存在无限多个图，其好顶点的比例$\beta$满足$1/7\leq \beta<1/5$且趋于某个无理数。此外，我们证明存在无限多个具有一半好顶点的图，并且对于每个$s\in\mathbb{N}$，我们构造一个无限图族，其好顶点的比例趋于$\frac{s+1}{2s+1}$。这些发现可能对解决原始问题（即确定是否存在除$C_5$之外的额外解）至关重要。
英文摘要
We say that $v\in V(G)$ is a \textit{good vertex} if the Kirchhoff index remains unchanged when $v$ is removed, i.e. $Kf(G)=Kf(G-v)$. In 1991, Šoltés studied the Wiener index of a graph and posed the problem of identifying graphs for which the removal of an arbitrary vertex preserves the Wiener index. In this paper, we explore a similar concept: identifying \textit{Kirchhoff Šoltés graphs}, i.e. graphs in which all vertices are good vertices. We show that the cycle $C_5$ is a Kirchhoff Šoltés graph. We consider several relaxed versions of the Kirchhoff Šoltés problem, where the primary objective is to identify graphs containing at least one good vertex. One of them is the \textit{$β$-Kirchhoff Šoltés problem}, which seeks to find an infinite family of graphs in which the proportion of good vertices is at least $β$, with $β\in (0,1]$ being a specified rational number. Another one involves constructing infinite families of graphs where the proportion of good vertices increases and asymptotically approaches a given real number $γ\in (0,1]$ as the order of the graph grows. We demonstrate that both relaxed versions have infinitely many solutions. In particular, we prove the existence of infinitely many graphs for which the proportion $β$ of good vertices, $1/7\leq β<1/5$ tends to a certain irrational number. Furthermore, we prove the existence of infinitely many graphs with half good vertices, and for each $s\in\mathbb{N}$, we construct an infinite family of graphs whose proportion of good vertices tends to $\frac{s+1}{2s+1}$. These findings could be pivotal in addressing the original problem of determining whether there are additional solutions beyond $C_5$.
Journal refApplied Mathematics and Computation, Volume 510, 2026
