Wiener index and graphs, almost half of whose vertices satisfy Šoltés property

Abstract

The Wiener index $W\left(G\right)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, Šoltés formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equality $W\left(G\right)=W(G-v)$ holds. The cycle $C_{11}$ is the only known graph with this property. In this paper we consider the following relaxation of the original problem: find a graph with a large proportion of vertices such that removing any one of them does not change the Wiener index of a graph. As the main result, we build an infinite series of graphs with the proportion of such vertices tending to $\frac{1}{2}$.

Introduction

Let $G=(V,E)$ be a simple connected graph. The Wiener index of a graph $G$ is defined as the sum of distances between all pairs of vertices of $G$:

$$W\left(G\right)=\frac{1}{2}\sum _{v,u\in V,v\ne u}{d}_G(u,v),$$

where $d_G(x,y)$ is a usual graph distance, i.e. the length of a shortest path from $x$ to $y$ in $G$. This index was introduced in $1947$ by Wiener [9] for applications in chemistry, where he used similar expression to determine the boiling point of the paraffin. In $1971$, Hosoya [4] first defined this index in terms of graph theory. After that, the index for different families of graphs was investigated because of pure mathematical interest. Very recently, Egorov and Vesnin [3] investigated correlation of hyperbolic volumes of fullerenes (fullerene is a molecule consisting entirely of carbon atoms) with Wiener index and other related indexes. We also refer an interested reader to survey papers by Dobrynin, Entringer and Gutman [2] and Knor, Škrekovski and Tepeh [7].

Let us define the transmission of a vertex $v\in V$ in $G$ as

$$t_G\left(v\right)=\sum _{u\in V\setminus \left\{v\right\}}{d}_G(v,u).$$

Clearly, the Wiener index of a graph $G$ may be expressed in terms of vertices transmissions:

$$W\left(G\right)=\frac{1}{2}\sum _{v\in V}{t}_G\left(v\right).$$

Let us denote by $G-v$ the graph obtained by removing the vertex $v$ from $G$. A vertex $v\in V$ is called good if $G-v$ is connected and $W\left(G\right)=W(G-v)$. In 1991, Šoltés [8] posed the following problem.

Problem 1

Find all graphs $G$ such that the equality $W\left(G\right)=W(G-v)$ holds for all their vertices $v$.

Today the only known graph with this property is $C_{11}$. In other words, Šoltés asked about graphs such that all their vertices are good. After thirty years of study of this problem, which led to lots of elegant ideas, the problem is far from being solved. However, there are some partial results. In $2018$, Knor, Majstorović and Škrekovski in [6] constructed an infinite series of graphs with one good vertex. The same authors also showed [5] that for $k\ge 3$ there are infinitely many graphs with one good vertex of degree $k$. In [1], for any $k\in \mathbb{N}$, Bok, Jedličková and Maxová found an infinite series of graphs with exactly $k$ good vertices.

For a graph $G=(V,E)$ and vertices $v,u,w\in V$, let us define functions ${\Delta }_v\left(G\right)=W\left(G\right)-W(G-v)$ and ${\delta }_G^v(u,w)=d_G(u,w)-d_{G-v}(u,w)$. From the definitions we directly have the following

Proposition 1

Given a graph $G=(V,E)$, $v\in V$. If $G-v$ is connected then the following equality holds:

$${\Delta }_v\left(G\right)=t_G\left(v\right)+\frac{1}{2}\sum _{u,w\in V\setminus \left\{v\right\}}{\delta }_G^v(u,w).$$

Instead of finding graphs with a fixed number of good vertices, one may try to find graphs with a fixed proportion of such vertices. So we denote a proportion of good vertices in $G$ as $\frac{|\{v\in V|{\Delta }_v\left(G\right)=0\}|}{\left|V\right|}$. Now we are ready to formulate the following relaxation of Problem 1.

Problem 2

For a fixed $\alpha \in (0,1]$ construct an infinite series $S$ of graphs such that for all $G=(V,E)$ from $S$ the following inequality takes place:

$$\frac{|\{v\in V|{\Delta }_v\left(G\right)=0\}|}{\left|V\right|}\ge \alpha .$$

Clearly, a solution to this problem for $\alpha =1$ would give an infinite series of solutions to Problem 1. Let us note that the construction from [1] after a slight modification yields an infinite series of graphs with the proportion of good vertices tending to $\frac{1}{3}$ (we will discuss it in more detail in Section 3).

In this work, we provide two constructions based on different ideas in order to improve this constant. In Section 2, we find an infinite series of graphs with the proportion of good vertices tending to $\frac{2}{5}$ as the number of vertices tends to infinity. In Section 3, we find one more series with the proportion tending to $\frac{1}{2}$. In Section 4, we discuss results and open problems.