The expected values and variances for Sombor indices in a general random chain

doi:10.1016/j.amc.2021.126521

Applied Mathematics and Computation

Volume 411, 15 December 2021, 126521

https://doi.org/10.1016/j.amc.2021.126521 Get rights and content

Abstract

This article is devoted to establish the distributions for Sombor indices in a general random chain, in which their explicit analytical expressions of the expected values and variances are obtained. As applications, these results for random hexagonal, random phenylene, random polyphenyl and random spiro chains are given. Finally, the distributions for the Sombor indices in these four random chains are asymptotic to normal distributions is shown.

Introduction

In this paper, we always suppose that a graph means a finite simple connected graph unless otherwise specified, and refer to Bondy and Murty [3] for the notations and terminologies freely used but not defined here.

Let $G = (V (G), E (G))$ be a graph, where $V (G)$ and $E (G)$ are the vertex set and the edge set of $G$ , respectively. It is a famous fact that chemical molecules can conveniently be described by a graph if we associate vertices with the atoms and edges with covalent bonds between atoms. Theoretical chemists focus on predicting physicochemical properties of chemical molecules by means of their structures. The topological indices play an important role in the prediction. For more details on topological indices, see [17], [25] and the references cited therein.

Inspired by Euclidean metrics, a class of novel topological indices, known as Sombor indices, is recently introduced by Gutman [11], including the (ordinary) Sombor index, the reduced Sombor index and the average Sombor index.

For a graph $G$ , the Sombor index $S O (G)$ , the reduced Sombor $S O_{r e d} (G)$ and the average Sombor index $S O_{a v r} (G)$ are defined as follows: $S O (G) = \sum_{u v \in E (G)} \sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)},$ $S O_{r e d} (G) = \sum_{u v \in E (G)} \sqrt{{(d_{G} (u) - 1)}^{2} + {(d_{G} (v) - 1)}^{2}},$ $S O_{a v r} (G) = \sum_{u v \in E (G)} \sqrt{{(d_{G} (u) - \bar{d})}^{2} + {(d_{G} (v) - \bar{d})}^{2}},$ where $\bar{d} = \frac{2 | E (G) |}{| V (G) |}$ is the average degree of $G$ .

In the following, we consider the general Sombor index, see [10], defined as $S O_{a} (G) = \sum_{u v \in E (G)} \sqrt{{(d_{G} (u) - a)}^{2} + {(d_{G} (v) - a)}^{2}},$ where $a$ is a constant or a parameter of $G$ . It is obvious that we can get the Sombor index, the reduced Sombor index and the average Sombor index by setting $a = 0, 1, \bar{d}$ , respectively.

Sombor indices have been the subject of intense scholarly debate because of their good chemical applicability. In [12], some basic properties of Sombor indices are presented. In [7], Sombor indices of chemical graphs is studied. In [19], the chemical graphs ordered by Sombor indices are given. For more results of Sombor indices, we refer the reader to see [[1], [2], [5], [8], [9], [13], [18], [23], [24]].

Currently, many researchers concentrate on the expected value of some indices. In [14], the expected values of the first Zagreb and Randić indices in random polyphenyl chains are given. In [21], [22], Raza determines the expected values for arithmetic, geometric indices, the sum-connectivity, harmonic, symmetric division, variable inverse sum degree and general Randic indices in random phenylene chains. In [26], [27], the expected values and variances for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index in a random polyphenyl chain are presented. In [20], the expected value for the reduced Sombor index in a random polyphenyl chain are given. In [10], the expected values for the Sombor indices in random hexagonal chains and random phenylene chains are obtained. Motivated by [10], [14], [20], [21], [22], [26], [27], we make researches on the expected values and variances for Sombor indices in a general random chain, defined in Definition 2.1.

This paper is organized as follows. In Section 2, we introduce the definition of general random chain and briefly recall some of the basic facts from probability theory. Section 3 is devoted to establish the distributions for Sombor indices in a general random chain. Based on the distributions, we obtain their explicit analytical expressions of the expected values and variances in Section 4. In Section 5, the asymptotic behaviors of the distributions for Sombor indices in random hexagonal, random phenylene, random polyphenyl and random spiro chains are presented.

Section snippets

Preliminaries

In this section, we start with some definitions of random hexagonal, random phenylene, random polyphenyl and random spiro chains.

The random hexagonal chain $R H C_{n} = R H C (n, p_{1}, p_{2}, p_{3})$ with $n$ hexagons is constructured by the following way:

(1)
$R H C_{1}$ is a hexagon and $R H C_{2}$ contains two hexagons, see Fig. 1.
(2)
For every $n > 2$ , $R H C_{n}$ is constructured by attaching one hexagon to $R H C_{n - 1}$ in three ways, resulted in $R H C_{n}^{1}, R H C_{n}^{2}, R H C_{n}^{3}$ with probability $p_{1}, p_{2}, p_{3}$ respectively, where $p_{1} + p_{2} + p_{3} = 1$ , see Fig. 2.

The random phenylene

The distribution for $S O_{a} (G_{n})$

This section devotes to establish the distrbution for $S O_{a} (G_{n})$ . As applications, the distributions for $S O_{a} (R H C_{n})$ , $S O_{a} (R P C_{n})$ , $S O_{a} (P P C_{n})$ and $S O_{a} (R S C_{n})$ are obtained.

Theorem 3.1

Let $G_{n} (n > 2)$ be a general random chain in Definition 2.1 with the link constants ${A_{i}}_{i = 1}^{t}$ for Sombor indices. Then $S O_{a} (G_{n}) = A^{T} X + S O_{a} (G_{2}),$ where $A^{T} = (A_{1}, A_{2}, \dots, A_{t})$ and $X \sim M (n - 2, p)$ , $p = (p_{1}, p_{2}, \dots, p_{t})$ .

Proof

By $(2.3)$ , $S O_{a} (G_{n})$ can be quantified as $S O_{a} (G_{n}) = (S O_{a} (G_{n}^{1}), S O_{a} (G_{n}^{2}), \dots, S O_{a} (G_{n}^{t})) Z_{n},$ and by (2.1), we have $S O_{a} (G_{n}^{i}) = S O_{a} (G_{n - 1}) + A_{i}, i = 1, 2, \dots, t .$

Associated (3.1)

The expected value and variance for $S O_{a} (G_{n})$

In this section, we present the explicit analytical expressions of the expected value and variance for $S O_{a} (G_{n})$ . In particular, the explicit analytical expressions of the expected values and variances for $S O_{a} (R H C_{n})$ , $S O_{a} (R P C_{n})$ , $S O_{a} (P P C_{n})$ and $S O_{a} (R S C_{n})$ are given.

Theorem 4.1

Let $G_{n} (n > 2)$ be a general random chain in Theorem 3.1. Then the expected value and variance for $S O_{a} (G_{n})$ are given by $E (S O_{a} (G_{n})) = (\sum_{i = 1}^{t} A_{i} p_{i}) (n - 2) + S O_{a} (G_{2}),$ $Var (S O_{a} (G_{n})) = (\sum_{i = 1}^{t} A_{i}^{2} p_{i} - {(\sum_{i = 1}^{t} A_{i} p_{i})}^{2}) (n - 2) .$

Proof

By Theorem 3.1, we have $S O_{a} (G_{n}) = A^{T} X + S O_{a} (G_{2}), X$

Asymptotic behaviors

In Section 3, the distrubutions for Sombor indices in random hexagonal, random phenylene, random polyphenyl and random spiro chains are presented, see Corollaries 3.3–3.6, respectively. In this section, we study the asymptotic behaviors for Sombor indices in these four random chains.

In probability theory and statistics, the binomial distribution can be approximated by normal distribution under certain conditions, known as De Moivre-Laplace Theorem.

Lemma 5.1

De Moivre-Laplace Theorem [6]

Let $X_{n} \sim B (n, p)$ . Then for each real number $x$ , we

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^☆: This work is supported by the National Nature Science Foundation of China (Grant No. 11971180, 12071156), the Guangdong Provincial Natural Science Foundation (Grant No. 2019A1515012052).

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The expected values and variances for Sombor indices in a general random chain☆

Abstract

Introduction

Section snippets

Preliminaries

The distribution for SOa(Gn)

The expected value and variance for SOa(Gn)

Asymptotic behaviors

De Moivre-Laplace Theorem [6]

Sombor index of polymers

MATCH Commun. Math. Comput. Chem.

Analysis of dendrimer generation by Sombor indices

J. Chem.

Graph theory

Graduate Texts in Mathematics

Statistics for Experimenters

Extremal values on the Sombor index of trees

MATCH Commun. Math. Comput. Chem.

Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance

Sombor index of chemical graphs

Appl. Math. Comput.

Sombor index of trees with at most three branch vertices

Appl. Math. Comput.

Molecular trees with extremal values of Sombor indices

Int. J. Quantum. Chem.

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

Int. J. Quantum. Chem.

Geometric approach to degree-based topological indices: Sombor indices

MATCH Commun. Math. Comput. Chem.

Some basic properties of Sombor indices

Open J. Discr. Appl. Math.

On Sombor index of graphs

MATCH Commun. Math. Comput. Chem.

The distribution for $S O_{a} (G_{n})$

The expected value and variance for $S O_{a} (G_{n})$