Application of read-across methods as a framework for the estimation of emissions from chemical processes

Sudhakar Takkellapati; Michael A. Gonzalez; Sudhakar Takkellapati; Michael A. Gonzalez

doi:10.3934/ctr.2023018

Clean Technologies and Recycling

2023, Volume 3, Issue 4: 283-300. doi: 10.3934/ctr.2023018

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Research article Topical Sections

Application of read-across methods as a framework for the estimation of emissions from chemical processes

US Environmental Protection Agency, Office of Research and Development, Center for Environmental Solutions and Emergency Response, Land Remediation and Technology Division, Environmental Decision Analytics Branch, 26 W. Martin Luther King Dr., Cincinnati, OH 45268, USA

Received: 13 November 2023 Revised: 19 December 2023 Accepted: 27 December 2023 Published: 28 December 2023

The read-across method is a popular data gap filling technique with developed application for multiple purposes, including regulatory. Within the US Environmental Protection Agency's (US EPA) New Chemicals Program under Toxic Substances Control Act (TSCA), read-across has been widely used, as well as within technical guidance published by the Organization for Economic Co-operation and Development, the European Chemicals Agency, and the European Center for Ecotoxicology and Toxicology of Chemicals for filling chemical toxicity data gaps. Under the TSCA New Chemicals Review Program, US EPA is tasked with reviewing proposed new chemical applications prior to commencing commercial manufacturing within or importing into the United States. The primary goal of this review is to identify any unreasonable human health and environmental risks, arising from environmental releases/emissions during manufacturing and the resulting exposure from these environmental releases. The authors propose the application of read-across techniques for the development and use of a framework for estimating the emissions arising during the chemical manufacturing process. This methodology is to utilize available emissions data from a structurally similar analogue chemical or a group of structurally similar chemicals in a chemical family taking into consideration their physicochemical properties under specified chemical process unit operations and conditions. This framework is also designed to apply existing knowledge of read-across principles previously utilized in toxicity estimation for an analogue or category of chemicals and introduced and extended with a concurrent case study.

Keywords:

Citation: Sudhakar Takkellapati, Michael A. Gonzalez. Application of read-across methods as a framework for the estimation of emissions from chemical processes[J]. Clean Technologies and Recycling, 2023, 3(4): 283-300. doi: 10.3934/ctr.2023018

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Abstract

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,

$W(G) = \sum\limits_{\{u, v \}\subseteq V(G)}d(u, v).$

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

$S(G) = \sum\limits_{\{u, v \}\subseteq V(G)}(d_u+d_v)d(u, v).$

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

$Gut(G) = \sum\limits_{\{u, v \}\subseteq V(G)}d_ud_vd(u, v).$

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

$Gut(T) = 4W(T)-(n-1)(2n-1).$

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

$SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d^2_u+d^2_v}.$

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

$SW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v).$

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

$DWW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}f(d_u, d_v)d(u, v),$

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

$f(d_u, d_v)\geq 0\; \; \text{and}\; \; f(d_u, d_v) = f(d_v, d_u).$

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

$SW(G) = (n-1)\left(\frac{\sqrt{2}(n^2-5n+9)}{3}+\sqrt{5}(n-2)\right).$

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

$SW(G) = \sqrt{2}rW(G).$

Moreover, if $G = K_n$ , then

$SW(G) = \frac{\sqrt{2}n(n-1)^2}{2}.$

If $G = C_n$ , then

$SW(G) =\left\{ \begin{array}{l}\frac{\sqrt{2}n^3}{4} , \qquad \qquad \mathit{\text{if n is even;}}\\ \frac{\sqrt{2}n(n^2-1)}{4}, \ \quad \mathit{\text{if n is odd.}} \end{array} \right.$

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

$SW(G) = n_1n_2\left[\sqrt{n_1^2+n_2^2}+\sqrt{2}(n_1+n_2)-2\sqrt{2}\right].$

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

$SW(G) = (n-1)(\sqrt{n^2-2n+2}+\sqrt{2}n-2\sqrt{2}).$

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

$\begin{align*} SW(G) = & \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ = & \sqrt{5}(1+2+\cdots+n-2)+\sqrt{2}(n-1)\\ & +2\sqrt{2}(1+2+\cdots+n-3)\\ & +\sqrt{5}(n-2)+\cdots+2\sqrt{2}+2\sqrt{5}+\sqrt{5}\\ = & \frac{\sqrt{5}(n-1)(n-2)}{2}+\sqrt{2}(n-1)\\ & +2\sqrt{2}\left(1+3+\cdots+\frac{(n-3)(n-2)}{2}\right)\\ & +\sqrt{5}(1+2+\cdots+n-2)\\ = & \sqrt{2}(n-1)+\sqrt{5}(n-1)(n-2)\\ & +\frac{2\sqrt{2}(n-1)(n-2)(n-3)}{6}\\ = & (n-1)\left(\frac{\sqrt{2}(n^2-5n+9)}{3}+\sqrt{5}(n-2)\right). \end{align*}$

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

$SW(G) = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{r^2+r^2}d(u, v) = \sqrt{2}rW(G).$

In particular, if $G = K_n$ , then

$\begin{align*} SW(G) & = \sqrt{2}(n-1)W(G)\\& = \sqrt{2}(n-1)\frac{n(n-1)}{2}\\ & = \frac{\sqrt{2}n(n-1)^2}{2}. \end{align*}$

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

$SW(G) = \left\{ \begin{array}{l}\frac{\sqrt{2}n^3}{4} , \qquad \qquad \text{if } ~~n ~~\text{ is even;}\\ \frac{\sqrt{2}n(n^2-1)}{4}, \ \quad \text{if } ~~n ~~ \text{is odd.} \end{array} \right.$

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

$\begin{align*} SW(G) = & \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ = & n_1n_2\sqrt{n_1^2+n_2^2}+2\binom{n_1}{2}\sqrt{n_2^2+n_2^2}\\ & +2\binom{n_2}{2}\sqrt{n_1^2+n_1^2}\\ = & n_1n_2\left[\sqrt{n_1^2+n_2^2}+\sqrt{2}(n_1+n_2)-2\sqrt{2}\right]. \end{align*}$

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

$SW(K_{1, \, n-1}) = (n-1)(\sqrt{n^2-2n+2}+\sqrt{2}n-2\sqrt{2}).$

This completes the proof. □

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

$\sqrt{2}\delta W(G)\leq SW(G)\leq \sqrt{2}\Delta W(G)$

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

$\sqrt{2} W(G)\leq SW(G)\leq \sqrt{2}(n-1) W(G).$

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

$\begin{align} \frac{1}{\sqrt{2}}S(G)\leq SW(G) \leq S(G)-(2-\sqrt{2})\delta W(G) \end{align}$

(2.1)

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

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \geq \sum\limits_{\{u, v\}\subseteq V(G)}\frac{1}{\sqrt{2}}(d_u+d_v)d(u, v)\\ & = \frac{1}{\sqrt{2}}S(G) \end{align*}$

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

$\sqrt{d^2_u+d^2_v}\leq d_u+(\sqrt{2}-1)d_v$

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

$\begin{align*} \begin{aligned} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} [d_u+(\sqrt{2}-1)d_v]d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} (d_u+d_v)d(u, v)-\sum\limits_{\{u, v\}\subseteq V(G)} (2-\sqrt{2})\delta d(u, v)\\ & \leq S(G)-(2-\sqrt{2})\delta W(G)\end{aligned} \end{align*}$

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

$\frac{\sqrt{2}}{\Delta}Gut(G)\leq SW(G) \leq \frac{\sqrt{2}}{\delta}Gut(G)$

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

Proof. Note that

$\sqrt{d^2_u+d^2_v} = d_ud_v\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}},$

then we have

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \leq \sum\limits_{\{u, v\}\subseteq V(G)} d_ud_v\sqrt{\frac{1}{\delta^2}+\frac{1}{\delta^2}}d(u, v)\\ & = \frac{\sqrt{2}}{\delta}\sum\limits_{\{u, v\}\subseteq V(G)} d_ud_vd(u, v)\\ & = \frac{\sqrt{2}}{\delta}Gut(G) \end{align*}$

and

$\begin{align*} SW(G) & = \sum\limits_{\{u, v\}\subseteq V(G)}\sqrt{d^2_u+d^2_v}d(u, v)\\ & \geq \sum\limits_{\{u, v\}\subseteq V(G)} d_ud_v\sqrt{\frac{1}{\Delta^2}+\frac{1}{\Delta^2}}d(u, v)\\ & = \frac{\sqrt{2}}{\Delta}\sum\limits_{\{u, v\}\subseteq V(G)} d_ud_vd(u, v)\\ & = \frac{\sqrt{2}}{\Delta}Gut(G). \end{align*}$

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

$DWW(B_{k+1}) = \sum\limits_{l = 1}^{k+1}A_l,$

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

$n_1 = 1\; \; \mathit{\text{and}}\; \; n_{j+1} = d_1d_2\cdots d_j$

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

$\begin{align*} \begin{aligned} A_1 = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d_1, d_j+1 )+kn_{k+1}f(d_1, 1), \\ A_l = & \Big[2\binom{d_{l-1}}{2}+4\binom{d_{l-1}}{1}\binom{d_{l-1}d_{l-2}-d_{l-1}}{1}+\cdots\\ & +2(l-1)\binom{d_{l-1}\cdots d_{2}}{1}\binom{n_l-d_{l-1}\cdots d_{2}}{1}\Big]f(d_l+1, d_l+1)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d_l+1, d_j+1)\\&+(k-l+1)n_{k+1}f(d_l+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d_l+1, d_j+1)\\ & +(k-l+3)n_{k+1}f(d_l+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d_l+1, d_j+1)\\ & +(k-l+5)n_{k+1}f(d_l+1, 1)\Big]+\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d_l+1, d_j+1)\\ & +(k+l-1)n_{k+1}f(d_l+1, 1)\Big], \\ A_{k+1} = & f(1, 1)\Big[2\binom{d_{k}}{2}+4\binom{d_{k}}{1}\binom{d_{k}d_{k-1}-d_{k}}{1}+\cdots\\ & +2k\binom{d_{k}\cdots d_{2}}{1}\binom{n_{k+1}-d_{k}\cdots d_{2}}{1}\Big].\end{aligned} \end{align*}$

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

$DWW(B_{k+1}) = \sum\limits_{i = 1}^{k+1}A_i.$

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

$\begin{align*} A_1 = & n_2f(d_1, d_2+1)+2n_3f(d_1, d_3+1)+\cdots\\ & +(k-1)n_kf(d_1, d_k+1)+kn_{k+1}f(d_1, 1)\\ = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d_1, d_j+1)+kn_{k+1}f(d_1, 1), \\ A_2 = & 2 \binom{d_1}{2}f(d_2+1, d_2+1)+n_2\Big[d_2f(d_2+1, d_3+1)\\ & +2d_2d_3f(d_2+1, d_4+1) + \cdots\\ &+ d_2d_3\cdots d_{k-1}(k-2)f(d_2+1, d_k+1)\\ & +d_2d_3\cdots d_{k}(k-1)f(d_2+1, 1)\\ & +(n_2-1)(3d_2f(d_2+1, d_3+1)\\ & +4d_2d_3f(d_2+1, d_4+1)+\cdots\\ & +d_2d_3\cdots d_{k-1}kf(d_2+1, d_k+1)\\ &+d_2d_3\cdots d_{k}(k+1)f(d_2+1, 1))\Big]\\ = & d_1(d_1-1)f(d_2+1, d_2+1)\\ & +n_2\Big[n_3/d_1f(d_2+1, d_3+1) \\ & +2n_4/d_1f(d_2+1, d_4+1) + \cdots\\ &+ n_k/d_1(k-2)f(d_2+1, d_k+1)\\ & +n_{k+1}/d_1(k-1)f(d_2+1, 1)\\ & +(n_2-1)(3n_3/d_1f(d_2+1, d_3+1)\\ & +4n_4/d_1f(d_2+1, d_4+1)+\cdots\\ & +kn_k/d_1f(d_2+1, d_k+1)\\ & +n_{k+1}/d_1(k+1)f(d_2+1, 1))\Big]\\ = & d_1(d_1-1)f(d_2+1, d_2+1)\\ &+ n_2/d_1\Big[\sum\limits_{j = 3}^{k}(j-2)n_jf(d_2+1, d_j+1)\\ & +(k-1)n_{k+1}f(d_2+1, 1)\\ & +(n_2-1)(\sum\limits_{j = 3}^{k}jn_jf(d_2+1, d_j+1)\\ & +(k+1)n_{k+1}f(d_2+1, 1))\Big]\\ = &d_1(d_1-1)f(d_2+1, d_2+1)\\ & + \sum\limits_{j = 3}^{k}(j-2)n_jf(d_2+1, d_j+1)\\ & +(k-1)n_{k+1}f(d_2+1, 1)\\ & +(n_2-1)\Big[\sum\limits_{j = 3}^{k}jn_jf(d_2+1, d_j+1)\\ & +(k+1)n_{k+1}f(d_2+1, 1)\Big], \\ A_3 = & \Big[2\binom{d_2}{2}+4\binom{d_2}{1}\binom{n_3-d_2}{1}\Big]f(d_3+1, d_3+1)\\ & +n_3\Big[d_3f(d_3+1, d_4+1)\\ & +2d_3d_4f(d_3+1, d_5+1)+\cdots\\ & +(k-3)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +(k-2)d_3d_4\cdots d_k f(d_3+1, 1)\\ & +(d_2-1)(3d_3f(d_3+1, d_4+1)\\ & +4d_3d_4 f(d_3+1, d_5+1)+\cdots\\ & +(k-1)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +kd_3d_4\cdots d_k f(d_3+1, 1))\\ & +(n_3-d_2)(5d_3f(d_3+1, d_4+1)\\ & +6d_3d_4f(d_3+1, d_5+1)+\cdots\\ & +(k+1)d_3d_4\cdots d_{k-1}f(d_3+1, d_{k}+1)\\ & +(k+2)d_3d_4\cdots d_k f(d_3+1, 1))\Big]\\ = & \Big[2\binom{d_2}{2}+4\binom{d_2}{1}\binom{n_3-d_2}{1}\Big])f(d_3+1, d_3+1)\\ & +\sum\limits_{j = 4}^{k}n_j(j-3)f(d_3+1, d_j+1)\\ & +(k-2)n_{k+1}f(d_3+1, 1)\\ & +(d_2-1)\Big[\sum\limits_{j = 4}^{k}n_j(j-1)f(d_3+1, d_j+1)\\ & +kn_{k+1}f(d_3+1, 1)\Big]\\&+(n_3-d_2)\Big[\sum\limits_{j = 4}^{k}n_j(j+1)f(d_3+1, d_j+1)\\ & +n_{k+1}(k+2)f(d_3+1, 1)\Big]. \end{align*}$

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

$\begin{align*} A_l = & \Big[2\binom{d_{l-1}}{2}+4\binom{d_{l-1}}{1}\binom{d_{l-1}d_{l-2}-d_{l-1}}{1}+\cdots\\ & +2(l-1)\binom{d_{l-1}\cdots d_{2}}{1}\binom{n_l-d_{l-1}\cdots d_{2}}{1}\Big]f(d_l+1, d_l+1)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d_l+1, d_j+1)\\&+(k-l+1)n_{k+1}f(d_l+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d_l+1, d_j+1)\\ & +(k-l+3)n_{k+1}f(d_l+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d_l+1, d_j+1)\\ & +(k-l+5)n_{k+1}f(d_l+1, 1)\Big]+\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d_l+1, d_j+1)\\ & +(k+l-1)n_{k+1}f(d_l+1, 1)\Big]. \end{align*}$

In particular, we have

$\begin{align*} A_{k+1} = & f(1, 1)\Big[2\binom{d_{k}}{2}+4\binom{d_{k}}{1}\binom{d_{k}d_{k-1}-d_{k}}{1}+\cdots\\ & +2k\binom{d_{k}\cdots d_{2}}{1}\binom{n_{k+1}-d_{k}\cdots d_{2}}{1}\Big]. \end{align*}$

This completes the proof. □

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

$DWW(B_{k, d}) = \sum\limits_{l = 1}^{k}A_l,$

where

$\begin{align*} A_1 = & \sum\limits_{j = 2}^{k-1}d(d+1)^{j-2}(j-1)f(d, d+1)+(k-1)d(d+1)^{k-2}f(d, 1), \\ A_l = & \Big[2\binom{d+1}{2}+4\binom{d+1}{1}\binom{(d+1)^2-(d+1)}{1}+\cdots\\ & +2(l-1)\binom{d^{l-2}}{1}\binom{n_l-d^{l-2}}{1}\Big]f(d+1, d+1)\\ & +\sum\limits_{j = l+1}^{k-1}n_j(j-l)f(d+1, d+1)+(k-l)n_{k}f(d+1, 1)\\ & +(d_{l-1}-1)\Big[\sum\limits_{j = l+1}^{k-1}f(d+1, d+1)n_j(j-l+2)\\ & +(k-l+2)n_{k}f(d+1, 1)\Big]\\ & +(d_{l-1}d_{l-2}-d_{l-1})\Big[\sum\limits_{j = l+1}^{k-1}n_j(j-l+4)f(d+1, d+1)\\ & +(k-l+4)n_{k}f(d+1, 1)\Big] +\cdots\\ & +(n_l-d_{l-1}\cdots d_2)\Big[\sum\limits_{j = l+1}^{k-1}n_j(j+l-2)f(d+1, d+1)\\ & +(k+l-2)n_{k}f(d+1, 1)\Big], \\ A_{k} = & f(1, 1)\Big[2\binom{d}{2}+4\binom{d}{1}\binom{d(d-1)}{1}+\cdots\\ & +2(k-1)\binom{d^{k-2}}{1}\binom{(d-1)d^{k-2}}{1}\Big]. \end{align*}$

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

$DWW(T_{k, d}) = \sum\limits_{l = 1}^{k+1}A_l,$

where

$\begin{align*} A_1 = & \sum\limits_{j = 2}^{k}n_j(j-1)f(d, d)+kd(d-1)^{k-1}f(d, 1), \\ A_l = & \Big[2\binom{d-1}{2}+4\binom{d-1}{1}\binom{(d-1)(d-2)}{1}+\cdots\\ & +2(l-1)\binom{(d-1)^{l-2}}{1}\binom{n_l-(d-1)^{l-2}}{1}\Big]f(d, d)\\ & +\sum\limits_{j = l+1}^{k}n_j(j-l)f(d, d)+(k-l+1)n_{k+1}f(d, 1)\\ & +(d-2)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+2)f(d, d)\\ & +(k-l+3)n_{k+1}f(d, 1)\Big]\\ & +(d-1)(d-2)\Big[\sum\limits_{j = l+1}^{k}n_j(j-l+4)f(d, d)\\ & +(k-l+5)n_{k+1}f(d, 1)\Big] +\cdots\\ & +(n_l-(d-1)^{l-2})\Big[\sum\limits_{j = l+1}^{k}n_j(j+l-2)f(d, d)\\ & +(k+l-1)n_{k+1}f(d, 1)\Big], \\ A_{k+1} = & f(1, 1)\Big[2\binom{d-1}{2}+4\binom{d-1}{1}\binom{(d-1)(d-2)}{1}+\cdots\\ & +2k\binom{(d-1)^{k-1}}{1}\binom{(d-1)^{k}}{1}\Big]. \end{align*}$

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:

$AcenFac = 0.00198\times SW+0.008141.$

Table 1. Experimental values of AcenFac and SW index for octane isomers.

Molecule	AcenFac	SW
Octane	0.397898	202.8093
2-methyl-heptane	0.377916	191.2453
3-methyl-heptane	0.371002	182.2057
4-methyl-heptane	0.371504	179.1925
3-ethyl-hexane	0.362472	170.5225
2, 2-dimethyl-hexane	0.339426	170.4970
2, 3-dimethyl-hexane	0.348247	166.8181
2, 4-dimethyl-hexane	0.344223	169.9447
2, 5-dimethyl-hexane	0.35683	179.0977
3, 3-dimethyl-hexane	0.322596	158.2653
3, 4-dimethyl-hexane	0.340345	160.7917
2-methyl-3-ethyl-pentane	0.332433	157.9633
3-methyl-3-ethyl-pentane	0.306899	149.3210
2, 2, 3-trimethyl-pentane	0.300816	148.2544
2, 2, 4-trimethyl-pentane	0.30537	157.5862
2, 3, 3-trimethyl-pentane	0.293177	145.1517
2, 3, 4-trimethyl-pentane	0.317422	153.9314
2, 2, 3, 3-tetramethylbutane	0.255294	135.0271

| Show Table

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Figure 1. Scatter plot between acentric factor of Octane isomers and their SW index.

DownLoad: Full-Size Img PowerPoint

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

$DWR(G) = \sum\limits_{\{u, v \}\subseteq V(G)}f(d_u, d_v)r(u, v),$

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.

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