Šoltés problem for the Kirchhoff index of a graph

The Kirchhoff index $Kf\left(G\right)$ of a connected graph $G$ is defined as the sum of resistance distances between all pairs of vertices in $G$. We say that $v\in V\left(G\right)$ is a good vertex if the Kirchhoff index remains unchanged when $v$ is removed, i.e. $Kf\left(G\right)=Kf(G-v)$. 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 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. Due to the challenge of finding more examples of such graphs, we shift our focus to 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 $\beta$-Kirchhoff Šoltés problem, which seeks to find an infinite family of graphs in which the proportion of good vertices is at least $\beta$, with $\beta \in (0,1]$ 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 $\gamma \in (0,1]$ 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 $\beta$ of good vertices, $1/7\le \beta <1/5$ tends to a certain irrational number. Furthermore, we prove the existence of infinitely many graphs with half good vertices, and for each $s\in N$, 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 $C_5$.