Degree-weighted Wiener index of a graph

1.   Introduction

In theoretical chemistry, the topological index of a graph, also called molecular structure descriptor, is a real number related to a structural graph of a molecule, and is often used to predict the physico-chemical properties and biological activities of molecules. A large number of molecular structure descriptors have been conceived and several of them have found applications in quantitative structure-activity and structure-property relationships (QSAR/QSPR) studies. In particular, degree-based topological indices and distance-based topological indices are the most important molecular structure descriptors that play an important role in QSAR/QSPR.

Throughout in this paper, $G$ is a simple connected undirected graph with the vertex set $V(G)$ and edge set $E(G)$. For $u, v \in V(G)$, $d_v$ is the degree of vertex $v$ in $G$ and $d(u, v)$ is the distance between vertices $u$ and $v$ in $G$. As a molecular descriptor, the Wiener index, introduced by Wiener ^[1] in 1947, is considered as one of the most used topological indexes with high correlation with many physical and chemical indices of molecular compounds. The Wiener index equals the sum of distances between all pairs of vertices of a graph $G$, that is,

In 1989, the Schultz index ^[2] of a chemical graph $G$ was put forward as a topological index of alkanes. It is defined as

The proposal of this index has opened up the research on the degree-distance-type index. Plavšić et al. ^[3] showed that the Wiener index and the Schultz index are highly intercorrelated topological indices. For arbitrary catacondensed benzenoid graphs, Dobrynin ^[4] proved that the Schultz index has the same discriminating power with the Wiener index. So, it is both significant and interesting to study the Schultz index for some given class of graphs (or network), no matter whether they are molecular graphs or not.

In 1994, Gutman ^[5] proposed the Schultz index of the second kind, often called the Gutman index, and defined it as

Bounds of this index have been extensively studied using mathematical methods; see ^[6]. Moreover, for a tree $T$ on $n$ vertices, the Gutman index and Wiener index are closely related by

In 2021, from a geometric perspective (degree radius), Gutman ^[7] introduced a novel degree-based topological index called the Sombor index, which is defined as

Note that the Sombor index is the sum of Euclidean distances of the degrees of the two vertices of each edge in the graph. This index is widely studied in mathematics and chemistry; see ^[8].

Inspired by the above research, we propose a new topological index called the Sombor-Wiener (SW) index, and define it as

The new index can be regarded as the sum of the product of degree radius and distance between any two vertices in the graph, which is a novel version of the distance-based topological index.

Naturally, we define a general topological index $DWW(G)$ of a graph $G$ contributed by the degree weights of all vertices as

where $f(d_u, d_v)$ is a real function of $d_u$ and $d_v$ with

Clearly, the general topological index, called the degree-weighted Wiener index, is the generalization of the Schultz index, the Gutman index, and the SW index.

In this paper, we study the basic properties of the SW index, and the linear regression analysis of the SW index, with respect to acentric factor of octane isomers. In addition, we give the calculation formula of degree-weighted Wiener index for generalized Bethe trees. Our results generalize some known formulae on the Schultz index and Gutman index.

2.   Basic properties of the SW index

Theorem 2.1. Let $G$ be a connected graph with $n$ vertices.

(i) If $G = P_n$, then

(ii) If $G$ is $r$-regular, then

Moreover, if $G = K_n$, then

If $G = C_n$, then

(iii) If $G = K_{n_1, \, n_2}$, then

In particular, if $G = K_{1, \, n-1}$, then

Proof. (i) If $G = P_n$, then

(ii) If $G$ is $r$-regular, then

In particular, if $G = K_n$, then

If $G$ is a cycle $C_n$, from ^[9], we have

(iii) If $G = K_{n_1, \, n_2}$, then

Let $n_1 = 1$ and $n_2 = n-1$, then

This completes the proof. □

Theorem 2.2. Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$, then

with equality if, and only if, $G$ is regular.

Proof. By definition of $SW(G)$, we have the proof. □

Corollary 2.3. Let $G$ be a connected graph with $n$ vertices, then

Theorem 2.4. Let $G$ be a connected graph with the minimum degree $\delta$, then

with equality (left and right) if, and only if, $G$ is regular.

Proof. First, we prove the left-hand side of (2.1). By Cauchy-Schwarz's inequality, we have

with equality if, and only if, $d_u = d_v$ for $u, v\in V(G)$, that is, $G$ is regular.

Second, we prove the righthand side of (2.1). For any $u, v \in V(G)$ ($d_u\geq d_v$), we have

with equality if, and only if, $d_u = d_v$. Thus,

with equality if, and only if, $G$ is regular.

This completes the proof. □

Theorem 2.5. Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$, then

with equality (left and right) if, and only if, $G$ is regular.

Proof. Note that

then we have

and

This completes the proof. □

3.   Degree-weighted Wiener index of generalized Bethe trees

The generalized Bethe tree is an important graph structure that has wide applications in many fields. The investigation on topological indices of generalized Bethe trees and dendrimer trees frequently appeared in various journals. A Bethe tree $B_{k, d}$ is a rooted tree at $k$ levels whose root is on level $1$ and has degree equal to $d$, the vertices of levels from $2$ to $k-1$ have degrees equal to $d+1$, and the vertices on the level $k$ have degree equal to $1$; see ^[10]. In 2007, Rojo ^[11] generalized the notion of a Bethe tree as follows: A generalized Bethe tree $B_{k}$ is a rooted tree whose vertices at the same level have equal degrees. Moreover, a regular dendrimer tree $T_{k, d}$ is a generalized Bethe tree of $k+1$ levels with each non-pendent vertex having degree $d$.

Theorem 3.1. Let $B_{k+1}$ be a generalized Bethe tree of $k+1$ levels. If $d_1$ denotes the degree of rooted vertex and $d_i+1$ denotes the degree of vertices on the $i$-th level of $B_{k+1}$ for $i < 1\leq k$, then

where $n_j$ is the number of vertices on the $j$-th level of $B_{k+1}$, and

for $1\leq j\leq k$, and

Proof. Let $A_i$ be the value of degree-weighted Wiener index of vertices on the $i$-th level of $B_{k+1}$, then

By definition of $B_{k+1}$, we have

By calculating similarly to the above, for any $2\leq l \leq k$, we have

In particular, we have

This completes the proof. □

Corollary 3.2. The degree-weighted Wiener index of a Bethe tree $B_{k, d}$ is

where

Corollary 3.3. The degree-weighted Wiener index of a regular dendrimer tree $T_{k, d}$ is

where

4.   Applications of SW indices to the acentric factor of octane isomers

In this section, the chemical applicability of the SW index is investigated. The acentric factor (AcenFac) is a measure of the non-sphericity of molecules. We consider the correlation between acentric factors of octane isomers and the respective SW indices. The experimental values of acentric factors of octane isomers were taken from http://www.moleculardescriptors.eu/dataset/dataset.htm.

Using the data from Table 1, we find the correlation of AcenFac with the value of SW index for octane isomers; see Figure 1. The following equations give the regression models for the SW index:

Thus, the SW index can also help to predict the properties of octane isomers.

5.   Conclusions

In this paper, we propose the SW index, and establish some mathematical relations between the Harary-Sombor index and other classic topological indices. Morover, we obtain the calculation formula of degree-weighted Wiener index for generalized Bethe trees. In addition, some numerical results are discussed. We calculate the SW index of octane isomers. The regression models show that the AcenFac and SW index of octane isomers are highly correlated.

In 1993, Klein and Randić ^[12] introduced the notion of resistance distance. Naturally, from the perspective of distance, we similarly propose the degree-weighted resistance-distance index of a graph $G$ and define it as

where $r(u, v)$ is the resistance distance between $u$ and $v$. It would be interesting to explore chemical and mathematical properties and possible predictive potential of this index.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are thankful to the anonymous referees for their helpful comments that improved the quality of the manuscript. This work was funded by the National Natural Science Foundation of China under Grant No. 12261074.

Conflict of interest

The authors declare no conflicts of interest to this work.