The statistical analysis for Sombor indices in a random polygonal chain
networks

Jia-Bao Liu1,∗, Ya-Qian Zheng1,∗, Xin-Bei Peng1

1School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601,
P.R. China

††footnotetext: E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>* Corresponding author.

Abstract. The Sombor indices, a new category of degree-based topological
molecular descriptors, have been widely investigated due to their excellent
chemical applicability. This paper aims to establish Sombor indices
distributions in random polygonal chain networks and to achieve expressions of
the expected values and variances. The expected values and variances of the
Sombor indices for polyonino, pentachain, polyphenyl, and cyclooctane chains
are obtained. Since the end connection of a random chain network follows a
binomial distribution, the Sombor indices of any chain network follow the
normal distribution when the number of polygons connected by the chain,
indicated by $n$, approaches infinity. Keywords: Degree distribution;
Polygonal chains; Expected value; Variance; Sombor indices.

##  1 Introduction

Let $\mathbb{G}=\left(V\left(\mathbb{G}\right),\
E\left(\mathbb{G}\right)\right)$ denote a graph with the edge set
$E\left(\mathbb{G}\right)$ and the vertex set $V\left(\mathbb{G}\right)$. The
degree of vertex $u$ is represented by $d_{\mathbb{G}}\left(u\right)$. If
$u,v\in V\left(\mathbb{G}\right)$ are adjacent, the edge connecting them is
labeled by $uv$. Please refer to[1] for concepts or notations in graph theory
that are not addressed in this paper. A multitude of degree-based topological
indices play a very significant role in the fields of mathematics and
chemistry. For further information on topological indices, refer to[2].
Gutman[3] has proposed a new group of topological indices called the Sombor
indices, which includes the (ordinary) Sombor index, the reduced Sombor index,
and the average Sombor index, inspired by the Euclidean metric.

For a graph $\mathbb{G}$, the formulas of the Sombor index
$SO\left(\mathbb{G}\right)$, the reduced Sombor index
$SO_{red}\left(\mathbb{G}\right)$ and the average Sombor index
$SO_{avr}\left(\mathbb{G}\right)$ are given as follows:

$\displaystyle SO\left(\mathbb{G}\right)=\sum_{uv\in
E\left(\mathbb{G}\right)}\sqrt{d^{2}_{\mathbb{G}}\left(u\right)+d^{2}_{\mathbb{G}}\left(v\right)}\hfill\
\ ,$ $\displaystyle SO_{red}\left(\mathbb{G}\right)=\sum_{uv\in
E\left(\mathbb{G}\right)}\sqrt{\left(d_{\mathbb{G}}\left(u\right)-1\right)^{2}+\left(d_{\mathbb{G}}\left(v\right)-1\right)^{2}}\hfill\
\ ,$ $\displaystyle SO_{red}\left(\mathbb{G}\right)=\sum_{uv\in
E\left(\mathbb{G}\right)}\sqrt{\left(d_{\mathbb{G}}\left(u\right)-\frac{2m}{n}\right)^{2}+\left(d_{\mathbb{G}}\left(v\right)-\frac{2m}{n}\right)^{2}}\hfill\
\ ,$

where $\frac{2m}{n}$ denotes the average degree of graph $\mathbb{G}$ and $m$,
$n$ represent the set of edges and the set of vertices, respectively.

It is obvious that we can obtain the general equation for the Sombor
indices[3], defined as

$\displaystyle SO_{a}\left(\mathbb{G}\right)=\sum_{uv\in
E\left(\mathbb{G}\right)}\sqrt{\left(d_{\mathbb{G}}\left(u\right)-a\right)^{2}+\left(d_{\mathbb{G}}\left(v\right)-a\right)^{2}}\hfill\
\ .$

Figure 1: A $l$-polygonal chain $PC_{n}$ with $n$ polygons, where $O_{n}$ is
the $n$-th polygon. Figure 2: $k$ sorts of permutations in an $l$-polygonal
chain.

An $l$-cycles (polygon) $PC_{n}$ can be seen as a new structure. Between a new
polygon with $l$-cycles and a chain of polygons with $n-1$ $l$-cycles, with
line segments acting as bridges connecting the endpoints of the segments. When
$1\leqslant i\leqslant n$, $O_{i}$ is known as the $i$-th polygon of the
$PC_{n}$; The polygonal chain is unique when $n=1,\ 2$ (see Fig $1$); When
$n\geqslant 3$, obtain $PC^{i}_{n}$ by joining the vertex of $O_{n}$ with the
vertex $V_{n-1}$ of $O_{n-1}$, where $1\leqslant i\leqslant n$.

For an $l$-cycle random polygon chain $PC_{n}(n;\ p_{1},\ \ldots,\ p_{k})$.
The transition from $PC_{n-1}$ to $PC_{n}$ is assumed to be a stable
stochastic process, indicated by $p_{i}$, $p_{i}$ is a constant, independently
of $n$, where $1\leqslant i\leqslant k$ (see Fig 2). In other words, the
stated process is viewed as a zeroth-order Markovprocess.

Topological indices and graph invariants based on the distances between graph
vertices are widely used to characterize molecular graphs, establish
relationships between the structure and properties of molecules, predict the
biological activity of compounds, and enable their chemical applications. The
Sombor index is one of the more novel topological indices available and is
highly correlated with many physical and chemical indices of molecular
compounds. The Sombor index, a new category of degree-based topological
molecular descriptors, has received a lot of attention derived from its
excellent chemistry applicability. Roberto et al[5], were concerned the
results of the Sombor index of chemical graphs. In[6], the Sombor index of the
chemistry tree was given, and the boiling temperatures of benzenoid
hydrocarbons and the (reduced) Sombor index were confirmed to be substantially
linked. Existing bounds and extremal results related to the Sombor index and
its variants were collected by Liu and You[7]. In[8], the authors
systematically described the general properties of the Sombor indices. For
additional outstanding Sombor indices results, we recommend that the reader
refer to[10, 9, 11, 12, 13]. Meanwhile, more scientific studies were focused
on the exploration of the mathematical expected values or variances of some
degree-based topological indices. In[14], a simple mathematical formula for
the expected values of the Wiener index were established in a random
cyclooctane chain. In random hydrocarbon chains, Raza et al[15] have
discovered the expected values for the several chemistry indices. The expected
values of the basic Randic index for conjugated hydrocarbons were reported by
the[16]. Zhang and Li[17] discussed the expected values of the four types of
degree-based topological indices in a random polyphenylene chain. In[18], the
expected values of the Kirchhoff indices in the random polyphenyl and spiro
chains were obtained. Many studies have revealed the expected values and
variances of degree-based topological indices for a wide range of substances
across this time period, and the reader can refer to[19, 20, 21, 22].

As a result of the preceding research, we offered precise analytical formulas
for the Sombor indices’s expected values and variances in this study.

The structure of the paper is detailed following. In Section $2$, we define a
general random polygonal chain and introduce some basic probability theory
notions. The distributions of the Sombor indices in a general random chain are
established in Section $3$. Based on these distributions, we obtain
expressions for their expected values and variances in Section $4$. As an
application of Sections $3$ and $4$, in Section $5$, the Sombor indices for
the polyonino, pentachain, polyphenyl, and cyclooctane chains, as well as
their expected values and variances of the Sombor indices are calculated. The
asymptotic behavior of the distributions of the Sombor indices for all the
polygonal chains given in this paper is illustrated in Section $6$.

##  2 Preliminaries

For a random polygonal chain $PC_{n}$, the $SO_{a}(PC_{n})$ are random
variables. First, we introduce some of the probability theories that will used
in next section.

######  Definition 1.

The random chain $G_{n}=G(n;\ p_{1},\ p_{2},\ \ldots,\ p_{k})$ with $n$
identical graphs, a graph $\mathbb{G}$ is given, created by the following
ways:

$(1)$ $\mathbb{G}_{2}$ is made up for two polygons, see Fig 1.

$(2)$ For each $n>2,\ \mathbb{G}_{n}$ is created by attaching one $O_{n}$ to
$\mathbb{G}_{n-1}$ in certain ways, resulting in $\mathbb{G}^{1}_{n},\
\mathbb{G}^{2}_{n},\ \ldots,\ \mathbb{G}^{k}_{n}$ with probability $p_{1},\
p_{2},\ \ldots,\ p_{k}$, respectively, where $\sum_{i=1}^{k}p_{i}=1$.

Some of the fundamental concepts of probability theory are illustrated below.

The polynomial distribution is a binomial distribution extension with n
independent replicate experiments, each with k possible results. It is usually
denoted by $M(n,\mathbf{p})$, where $n$ represents the number of experiments
and the vector $\mathbf{p}$ is the probability of occurrence of the event.

$\\{\mathbf{p}|\mathbf{p}=(p_{1},\ p_{2},\ \ldots,\
p_{k})^{T}\in\mathbb{R}^{t},i=1,\ 2,\ \ldots,\ k,\ p_{i}\geqslant 0\ and\
\sum_{i=1}^{k}p_{i}=1\\}.$

The sample space of the multinomial distribution is

$\mathbb{S}=\\{\mathbf{X}|\mathbf{X}=(x_{1},\ x_{2},\ \ldots,\
x_{k})^{T}\in\mathbb{Z}^{t},\ i=1,\ 2,\ldots,\ k,\ x_{i}\geqslant 0\ and\
\sum_{i=1}^{k}x_{i}=n\\}.$

The probability function of the multinomial distribution is

$f(x)=\frac{n!}{\prod_{i=1}^{k}x_{i}!}\prod_{i=1}^{k}p_{i}^{x_{i}},\
x\in\mathbb{S},$

Assume that the random vector $\mathbf{X}$ follows the polynomial
distribution, which is denoted by $\mathbf{X}\thicksim M(n,\mathbf{p})$. The
following conclusion about $X$ can be obtained.

$\mathbb{E}(\mathbf{X})=n\mathbf{p},\
\mathbb{V}\mathbf{a}\mathbf{r}(\mathbf{X})=n(\mathbf{d}\mathbf{i}\mathbf{a}\mathbf{g}(\mathbf{p})-\mathbf{p}\mathbf{p}^{T}).$
(2.1)

The bernoulli distribution $B(p_{1})$, binomial distribution$M(n,\ p_{1})$,
and category distribution $C(p)$ are obtained by setting $n=1$ and $k=1$,
$n>1$ and $k=2$, and $n=1$ and $k>2$ in that order.

The multinomial distribution’s two essential properties are described below.

######  Proposition 2.1.

(Addition Rule[23]). Let $\mathbf{X}_{i}\thicksim M(n_{i},\mathbf{p})$, where
$i=1,\ 2,\ldots,\ k$, for each $X_{i}$ is an independent vector of each other.
Then

$\sum_{i=1}^{k}\mathbf{X}_{i}\thicksim M(\sum_{i=1}^{k}n_{i},\ \mathbf{p}).$

######  Proposition 2.2.

(Marginal Distribution[23]). Let $\mathbf{X}=(\mathbf{X}_{1},\
\mathbf{X}_{2},\ \ldots,\ \mathbf{X}_{k})\thicksim M(n,\ \mathbf{p})$. Then
$X_{i}\thicksim B(n,\ p_{i})$ for each $i=1,\ 2,\ \ldots,\ k$.

For further information on general probability distributions, please refer
to[24]. The Lemma 2.3 below is commonly applied.

######  Lemma 2.3.

Let $X$ be a random variable and $a,\ b\in\mathbb{R}$. Then

$\mathbb{E}(\mathbf{A}\mathbf{X}+\mathbf{b})=\mathbf{A}\mathbb{E}(\mathbf{X})+\mathbb{E}(\mathbf{b}),\
\mathbb{V}\mathbf{a}\mathbf{r}(\mathbf{A}\mathbf{X}+\mathbf{b})=\mathbf{A}\mathbb{V}\mathbf{a}\mathbf{r}(\mathbf{X})\mathbf{A}^{T}.$

A general method of calculating the expected values and variances for Sombor
indices in polygonal chain are given.

######  Theorem 2.4.

Let $PC_{n}$ be a random polygonal chain of length $n$ and
$\mathbf{X}\thicksim B(n-2,\ p_{1})$, where $n>2$. Then

$\displaystyle SO(PC_{n})$ $\displaystyle=$ $\displaystyle
A\mathbf{X}+B(n-2)+C,$ $\displaystyle\mathbb{E}(SO_{a}(PC_{n}))$
$\displaystyle=$ $\displaystyle(p_{1}A+B)(n-2)+C,$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}(SO_{a}(PC_{n}))$ $\displaystyle=$
$\displaystyle A^{2}(n-2)p_{1}(1-p_{1}),$

$where$

$(A,B,C)=\Bigl{(}A_{1}-A_{2},\ A_{2},\ -2A_{2}+SO_{a}(PC_{n})\Bigr{)}.$

###### Proof.

In order to quantify the random variable $SO_{a}(PC_{n})$, we defined a family
of $3$-dimensional random vectors ${\mathbb{Z}_{k}}$ as follows:

$\displaystyle\mathbb{Z}_{k}=\begin{cases}(1,0,0),&PC_{n}=PC^{1}_{n};\\\
(0,1,0),&PC_{n}=PC^{2}_{n};\\\ (0,0,1),&PC_{n}=PC^{3}_{n}.\\\ \end{cases}$

By the definition of the random polygonal chain, we can check that
$\mathbb{Z}_{k}$ follows the categprical distribution $C(1,\ p_{1},\ p_{2},\
p_{3})$ and $\mathbb{Z}_{3},\ \mathbb{Z}_{4},\ \ldots,\ \mathbb{Z}_{n}$ are
independent.

For each $k=3,\ 4,\ \ldots,\ n$, $SO_{a}(PC_{k})$ can be quantified as

$SO_{a}(PC_{k})=\Bigl{(}SO_{a}(PC^{1}_{k}),\ SO_{a}(PC^{2}_{k}),\
SO_{a}(PC^{3}_{k})\Bigr{)}\mathbb{Z}_{k}.$ (2.2)

By the definition of Sombor indices, for each $k=3,\ 4,\ \ldots,\ n$, and
$i=1,\ 2,\ 3$, we have

$SO_{a}(PC^{i}_{k})-SO_{a}(PC_{k-1})=SO_{a}(PC^{i}_{3})-SO_{a}(PC_{2}),$

we denoted $A_{i}=SO_{a}(PC^{i}_{3})-SO_{a}(PC_{2}),\ i=1,\ 2,\ 3$. Then

$SO_{a}(PC^{i}_{k})=SO_{a}(PC_{k-1})+A_{i},\ i=1,\ 2,\ 3.$ (2.3)

Associated (2.2) with (2.3), the $SO_{a}(PC_{k})$ satisfies the following
recursive relation

$SO_{a}(PC_{k})=SO_{a}(PC_{k-1})+(A_{1},\ A_{2},\ A_{3})\mathbb{Z}_{k},$ (2.4)
$SO_{a}(PC_{n})=(A_{1},A_{2},A_{3})\mathbf{X}+SO_{a}(PC_{2}),\
\mathbf{X}=(X_{1},\ X_{2},\ X_{3})^{T}=\sum_{k=3}^{n}\mathbb{Z}_{k},$ (2.5)

where $\mathbf{X}$ follows the Multinomial distribution $M(n-2,\ p_{1},\
p_{2},\ p_{3})$ by Proposition 2.1.

Thus, we obtain

$\displaystyle A_{1}$ $\displaystyle=SO_{a}(PC^{1}_{3})-SO_{a}(PC_{2}),$
$\displaystyle A_{2}$ $\displaystyle=SO_{a}(PC^{2}_{3})-SO_{a}(PC_{2})=A_{3}.$

By (2.5), we have

$\displaystyle SO_{a}(PC_{n})$ $\displaystyle=$
$\displaystyle(A_{1},A_{2},A_{2})\mathbf{X}+SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle(A_{1}-A_{2})(1,0,0)\mathbf{X}+A_{2}(1,1,1)\mathbf{X}+TI(PC_{2})$
$\displaystyle=$ $\displaystyle(A_{1}-A_{2})X_{1}+A_{2}(n-2)+SO_{a}(PC_{2}),$

where $X_{1}$ follows the binomial distribution $B(n-2,\ p_{1})$ by
Proposition 2.2.

Put $A=A_{1}-A_{2},\ B=A_{2},\ C=SO_{a}(PC_{2})$. Applying Lemma 2.3, we
obtain

$\mathbb{E}\bigl{(}SO_{a}(PC_{n})\bigr{)}=(p_{1}A+B)(n-2)+C,\
\mathbb{V}\mathbf{a}\mathbf{r}\bigl{(}SO_{a}(PC_{n})\bigr{)}=A^{2}(n-2)p_{1}(1-p_{1}).$

∎

##  3 The distribution for $SO_{a}(\mathbb{G}_{n})$

We discover that $SO_{a}(PC_{n})$ are regarded as random variables. Then, the
distributions of $SO_{a}(PC_{n})$ of a $(2k+1)$-polygonal chain and
$SO_{a}(PC_{n})$ of a $2k$-polygonal chain are presented in this section.

######  Theorem 3.1.

Let $\mathbb{G}_{n}(n>2)$ be an random polygonal chain also with connect
constants $\\{A_{i}\\}^{k}_{i=1}$ for Sombor indices, by Definition $2.1$.
Then

$SO_{a}(\mathbb{G}_{n})=\mathbf{A}^{T}X+SO_{a}(\mathbb{G}_{2}),$

$where$ $\mathbf{A}^{T}=(A_{1},\ A_{2},\ \ldots,\ A_{k})$ $and$ $\mathbf{X}$
follow the $M(n-2,\ \mathbf{p}),\ \mathbf{p}=(p_{1},\ p_{2},\ \ldots,\
p_{k})$.

###### Proof.

Associated $(2.2),\ (2.3),\ (2,4)$ with $(2.5)$ can be obtained. ∎

######  Corollary 3.2.

Let $PC_{n}$ be a $(2k+1)$-polygonal chain of length $n$ and $X\sim B(n-2,\
p_{1})$, where $k\geqslant 2$ and $n>2$. Then $SO_{a}(PC_{n})=AX+Bn+C$, where

$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\sqrt{2}\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert+2v_{1},$ $\displaystyle B$
$\displaystyle=$ $\displaystyle\sqrt{2}(2k-3)\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert+v_{1},$ $\displaystyle C$
$\displaystyle=$ $\displaystyle 4\sqrt{2}\left\lvert
2-a\right\rvert-\sqrt{2}\left\lvert 3-a\right\rvert+8v_{1}.$

Specifically, we have

$\displaystyle SO(PC_{n})$ $\displaystyle=$
$\displaystyle(6\sqrt{2}+2\sqrt{13})X+(4\sqrt{2}k-3\sqrt{2}+\sqrt{13})n+5\sqrt{2}+8\sqrt{13},$
$\displaystyle SO_{red}(PC_{n})$ $\displaystyle=$
$\displaystyle(4\sqrt{2}+2\sqrt{5})X+(2\sqrt{2}k-\sqrt{2}+\sqrt{5})n+3\sqrt{2}+8\sqrt{5},$
$\displaystyle SO_{avr}(PC_{n})$ $\displaystyle=$ $\displaystyle
A_{1}X+B_{1}n+C_{1},$

where

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}(2k+3)+2\mu_{1}}{n(2k+1)},$ $\displaystyle B_{1}$
$\displaystyle=$
$\displaystyle\frac{6\sqrt{2}nk-7\sqrt{2}n+4\sqrt{2}k-8\sqrt{2}+\mu_{1}}{n(2k+1)},$
$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{-4\sqrt{2}nk+9\sqrt{2}n+11\sqrt{2}+8\mu_{1}}{n(2k+1)},$
$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13},$
$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle\sqrt{4n^{2}k^{2}-4n^{2}k-8nk+5n^{2}+12n+8}.$

###### Proof.

According to the concept of Sombor indices, we can obtain the Sombor indices
of a random $(2k+1)$-polygonal chain. Specific proofs are as follows.

$\displaystyle SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle(4k-2)\sqrt{2}\left\lvert 2-a\right\rvert+\sqrt{2}\left\lvert
3-a\right\rvert+4v_{1},$ $\displaystyle SO_{a}(PC^{1}_{3})$ $\displaystyle=$
$\displaystyle(6k-4)\sqrt{2}\left\lvert 2-a\right\rvert+3\sqrt{2}\left\lvert
3-a\right\rvert+6v_{1},$ $\displaystyle SO_{a}(PC^{2}_{3})$ $\displaystyle=$
$\displaystyle(6k-5)\sqrt{2}\left\lvert 2-a\right\rvert+2\sqrt{2}\left\lvert
3-a\right\rvert+8v_{1},$ $\displaystyle\vdots$ $\displaystyle
SO_{a}(PC^{k}_{3})$ $\displaystyle=$ $\displaystyle(6k-5)\sqrt{2}\left\lvert
2-a\right\rvert+2\sqrt{2}\left\lvert 3-a\right\rvert+8v_{1},$

then relevant variables of $PC_{n}$ are derived by

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
SO_{a}(PC^{1}_{3})-SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle(2k-2)\sqrt{2}\left\lvert 2-a\right\rvert+2\sqrt{2}\left\lvert
3-a\right\rvert+2v_{1},$ $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle
SO_{a}(PC^{2}_{3})-SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle(2k-3)\sqrt{2}\left\lvert 2-a\right\rvert+\sqrt{2}\left\lvert
3-a\right\rvert+4v_{1},$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

Applying (2.3), one obtains

$SO_{a}(PC_{n})=(A_{1}-A_{2})X+A_{2}n-2A_{2}+SO_{a}(PC_{2}),\ X\sim B(n-2,\
p_{1}).$

First part can be obtained by right away. If $\left\lvert
V(PC_{n})\right\rvert=n(2k+1)$ and $\left\lvert
E(PC_{n})\right\rvert=2n(k+1)-1$, then the average degree of $PC_{n}$ is
$\frac{4n(k+1)+2}{n(2k+1)}$. The proof is completed by putting $a=0,\ 1,\
\frac{4n(k+1)+2}{n(2k+1)}$ into the equation, respectively.

∎

######  Corollary 3.3.

Let $PC_{n}$ be a random $2k$-polygonal chain of length $n$ and $X\sim B(n-2,\
p_{1})$, where $k\geqslant 2$ and $n>2$. Then $SO_{a}(PC_{n})=AX+Bn+C$, where

$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\sqrt{2}\left\lvert
2-a\right\rvert-2v_{1},$ $\displaystyle B$ $\displaystyle=$
$\displaystyle\sqrt{2}((2k-3)\left\lvert 2-a\right\rvert+\left\lvert
3-a\right\rvert)+4v_{1},$ $\displaystyle C$ $\displaystyle=$
$\displaystyle\sqrt{2}\left\lvert 2-a\right\rvert-\sqrt{2}\left\lvert
3-a\right\rvert-4v_{1}.$

In particular, we have

$\displaystyle SO(PC_{n})$ $\displaystyle=$
$\displaystyle(2\sqrt{2}-2\sqrt{13})X+(4\sqrt{2}k+4\sqrt{13}-3\sqrt{2})n-\sqrt{2}-4\sqrt{13},$
$\displaystyle SO_{red}(PC_{n})$ $\displaystyle=$
$\displaystyle(\sqrt{2}-2\sqrt{5})X+(2k-1)\sqrt{2}n+4\sqrt{5}n-\sqrt{2}k-4\sqrt{5},$
$\displaystyle SO_{avr}(PC_{n})$ $\displaystyle=$ $\displaystyle
A_{2}X+B_{2}n+C_{2},$

where

$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{2}n-2\mu_{2}-2\sqrt{2}}{nk},$ $\displaystyle B_{2}$
$\displaystyle=$
$\displaystyle\frac{3\sqrt{2}nk-2\sqrt{2}k-4\sqrt{2}n+4\sqrt{2}+4\mu_{2}}{nk},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{-\sqrt{2}nk+2\sqrt{2}n-2\sqrt{2}-4\mu_{1}}{nk},$
$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13},$
$\displaystyle\mu_{2}$ $\displaystyle=$
$\displaystyle\sqrt{n^{2}k^{2}-2n^{2}k+2nk+2n^{2}-4n+2}.$

###### Proof.

According to the definition of Sombor indices, we can obtain the Sombor
indices of a random $2k$-polygonal chain.

$\displaystyle SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(4k-5\left\lvert 2-a\right\rvert)+\left\lvert
3-a\right\rvert\Bigr{)}+4v_{1},$ $\displaystyle SO_{a}(PC^{1}_{3})$
$\displaystyle=$ $\displaystyle\sqrt{2}\Bigl{(}(6k-7\left\lvert
2-a\right\rvert)+3\left\lvert 3-a\right\rvert\Bigr{)}+6v_{1},$ $\displaystyle
SO_{a}(PC^{2}_{3})$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(6k-8\left\lvert 2-a\right\rvert)+2\left\lvert
3-a\right\rvert\Bigr{)}+8v_{1},$ $\displaystyle\vdots$ $\displaystyle
SO_{a}(PC^{k}_{3})$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(6k-8\left\lvert 2-a\right\rvert)+2\left\lvert
3-a\right\rvert\Bigr{)}+8v_{1},$

then relevant variables of $PC_{n}$ are given by

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
SO_{a}(PC^{1}_{3})-SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(2k-2\left\lvert 2-a\right\rvert)+\left\lvert
3-a\right\rvert\Bigr{)}+2v_{1},$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle SO_{a}(PC^{2}_{3})-SO_{a}(PC_{2})$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(2k-3\left\lvert 2-a\right\rvert)+\left\lvert
3-a\right\rvert\Bigr{)}+4v_{1}.$

Appiying (2.3), we obtain

$SO_{a}(PC_{n})=(A_{1}-A_{2})X+A_{2}n-2A_{2}+SO_{a}(PC_{2}),\ X\sim B(n-2,\
p_{1}).$

Setting $A=A_{1}-A_{2},\ B=A_{2}$ and $C=-2A_{2}+SO_{a}(PC_{2})$, we promptly
get the first component. Setting $A=A1-A2,B=A2$, and
$C=-2A_{2}+SO_{a}(PC_{2})$, we promptly get the first component. Since
$\left\lvert V(PC_{n})\right\rvert=2nk$ and $\left\lvert
E(PC_{n})\right\rvert=2nk+n-1$, then the $\frac{2m}{n}$ of $PC_{n}$ is
$\frac{2nk+n-1}{nk}$. The proof is performed directly by setting $a=0,\ 1,\
\frac{2nk+n-1}{nk}$. ∎

##  4 The expected values and variances for $SO_{a}(\mathbb{G}_{n})$

The exact analytical equations of $\mathbb{E}(SO_{a}(PC_{n}))$ and
$\mathbb{V}\mathbf{a}\mathbf{r}(SO_{a}(PC_{n}))$ of a $(2k+1)$-polygonal chain
and a $2k$-polygonal chain are discussed in this section, respectively.

######  Theorem 4.1.

Let $\mathbb{G}_{n}$ be a random polygonal chain. The expected values and
variances of $SO_{n}(\mathbb{G}_{n})$ are calculated by

$\displaystyle\mathbb{E}(\mathbb{G}_{n})$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{k}A_{i}p_{i}\right)(n-2)+SO_{a}(G_{2}),$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}(\mathbb{G}_{n})$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{k}A^{2}_{i}p_{i}-(\sum_{i=1}^{k}A_{i}p_{i})^{2}\right).$

###### Proof.

$SO_{a}(\mathbb{G}_{n})\mathbf{A}^{T}+SO_{a}(\mathbb{G}_{2}),\ \mathbf{X}\sim
M(n-2,\mathbf{p}),\ \mathbf{p}=(p_{1},\ p_{2},\ \ldots,\ p_{k})^{T}.$

By Lemma 2.3, we obtain

$\displaystyle\mathbb{E}\Bigl{(}SO_{a}(\mathbb{G}_{n})\Bigl{)}$
$\displaystyle=$ $\displaystyle
E(\mathbf{A}^{T}\mathbf{X}+SO_{a}(\mathbb{G}_{2}))$ $\displaystyle=$
$\displaystyle\mathbf{A}^{T}E(\mathbf{X})+SO_{a}(\mathbb{G}_{2})$
$\displaystyle=$
$\displaystyle(\sum_{i=1}^{k}A_{i}p_{i})(n-2)+SO_{a}(\mathbb{G}_{2}),$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(\mathbb{G}_{n})\Bigl{)}$
$\displaystyle=$ $\displaystyle
Var(\mathbf{A}^{T}\mathbf{X}+SO_{a}(\mathbb{G}_{2}))$ $\displaystyle=$
$\displaystyle\mathbf{A}^{T}Var(\mathbf{X})\mathbf{A}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sum_{i=1}^{k}A_{i}p_{i}-(\sum_{i=1}^{k}A_{i}p_{i})^{2}\Bigr{)}(n-2).$

∎

######  Corollary 4.2.

Let $PC_{n}$ be a random $(2k+1)$-polygonal chain of length $n$, where
$k\geqslant 2$ and $n>2$. Then

$\mathbb{E}\Bigl{(}SO_{a}(PC_{n})\Bigl{)}=Mn+N,\
\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PC_{n})\Bigl{)}=Pn+Q,$

where

$\displaystyle M$ $\displaystyle=$ $\displaystyle\sqrt{2}\left\lvert
2-a\right\rvert(p_{1}+2k-3)+\sqrt{2}(1+p_{1})\left\lvert
3-a\right\rvert+(4-2p_{1})v_{1},$ $\displaystyle N$ $\displaystyle=$
$\displaystyle(-2\sqrt{2}p_{1}-4\sqrt{2}k+10\sqrt{2})\left\lvert
2-a\right\rvert-3\sqrt{2}\left\lvert 3-a\right\rvert+4p_{1}v_{1},$
$\displaystyle A^{2}$ $\displaystyle=$ $\displaystyle 3v_{1}^{2}+2\left\lvert
2-a\right\rvert\left\lvert 3-a\right\rvert-2\sqrt{2}(\left\lvert
2-a\right\rvert+\left\lvert 3-a\right\rvert),$ $\displaystyle P$
$\displaystyle=$ $\displaystyle A^{2}p_{1}(1-p_{1}),$ $\displaystyle Q$
$\displaystyle=$ $\displaystyle-2A^{2}p_{1}(1-p_{1}),$ $\displaystyle v_{1}$
$\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13}.$

In particular, we have

$\displaystyle\begin{aligned}
\mathbb{E}\Bigl{(}SO(PC_{n})\Bigr{)}&=\Bigl{(}(5\sqrt{2}-2\sqrt{13})p_{1}+4\sqrt{2}k-3\sqrt{2}+4\sqrt{13}\Bigr{)}n\\\
&+(4\sqrt{13}-4\sqrt{2})p_{1}-8\sqrt{2}k+11\sqrt{2},\end{aligned}$ (4.6)
$\displaystyle\begin{aligned}
\mathbb{E}\Bigl{(}SO_{red}(PC_{n})\Bigr{)}&=\Bigl{(}(3\sqrt{2}-2\sqrt{5})p_{1}+2\sqrt{2}k+4\sqrt{5}-\sqrt{2}\Bigr{)}n\\\
&+(4\sqrt{5}-2\sqrt{2})p_{1}-4\sqrt{2}k+4\sqrt{2},\end{aligned}$ (4.7)
$\displaystyle\mathbb{E}\Bigl{(}SO_{avr}(PC_{n})\Bigl{)}=M_{1}n+N_{1},$ (4.8)

where

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}np_{1}(2k+1)+\sqrt{2}n(6k-7)+4\sqrt{2}(k-2)+(4-2p_{1})\mu_{1}}{n(2k+1)},$
$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\frac{-4\sqrt{2}np_{1}-14\sqrt{2}nk+(4\mu_{1}-4\sqrt{2})p_{1}-8\sqrt{2}+23\sqrt{2}n+26\sqrt{2}}{n(2k+1)},$
$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13},$
$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle\sqrt{4n^{2}k^{2}-4n^{2}k-8nk+5n^{2}+12n+8}.$

$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO(PC_{n})\Bigl{)}$
$\displaystyle=$ $\displaystyle(16\sqrt{26}+84)p_{1}(1-p_{1})(n-2),$ (4.9)
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\bigl{(}SO_{red}(PC_{n})\bigr{)}$
$\displaystyle=$ $\displaystyle(8\sqrt{10}+28)(n-2)p_{1}(1-p_{1}),$ (4.10)
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\bigl{(}SO_{avr}(PC_{n})\bigr{)}=\widetilde{\sigma}(n-2)p_{1}(1-p_{1}),$
(4.11)

where

$\displaystyle\widetilde{\sigma}=\frac{16(\sqrt{2}+1)\sqrt{4n^{2}k^{2}-4n^{2}k-8nk+5n^{2}+12n+8}}{n^{2}(2k+1)^{2}}-\frac{64(4n^{2}k^{2}-4n^{2}k+2n+1)}{n^{2}(2k+1)^{2}}+84.$

######  Corollary 4.3.

Let $PC_{n}$ be a random $(2k)$-polygonal chain of length $n$, where
$k\geqslant 2$ and $n>2$. Then

$\mathbb{E}\Bigl{(}SO_{a}(PC_{n})\Bigl{)}=Mn+N,\
\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PC_{n})\Bigl{)}=Pn+Q,$

where

$\displaystyle M$ $\displaystyle=$
$\displaystyle\sqrt{2}\Bigl{(}(p_{1}+2k-3)\left\lvert
2-a\right\rvert+\left\lvert 3-a\right\rvert\Bigr{)}+(4-2p_{1})v_{1},$
$\displaystyle N$ $\displaystyle=$ $\displaystyle\sqrt{2}\left\lvert
2-a\right\rvert\left(-2p_{1}-4k+7\right)-3\sqrt{2}\left\lvert
3-a\right\rvert-2\sqrt{2}(4-2p_{1})v_{1},$ $\displaystyle P$ $\displaystyle=$
$\displaystyle\Bigl{(}16a^{2}-72a+84-8\sqrt{2}v_{1}\left\lvert
2-a\right\rvert\Bigl{)}p_{1}(1-p_{1}),$ $\displaystyle Q$ $\displaystyle=$
$\displaystyle-2P.$

In particular, we have

$\displaystyle\begin{aligned}
\mathbb{E}\Bigl{(}SO(PC_{n})\Bigl{)}&=\Bigl{(}2p_{1}(\sqrt{2}-\sqrt{13})+4\sqrt{2}k-4\sqrt{13}-3\sqrt{2}\Bigl{)}n\\\
&+4p_{1}(\sqrt{26}-\sqrt{2})+8\sqrt{2}k+5\sqrt{2}-8\sqrt{26},\end{aligned}$
(4.12) $\displaystyle\begin{aligned}
\mathbb{E}\Bigl{(}SO_{red}(PC_{n})\Bigl{)}&=\Bigl{(}\sqrt{2}(p_{1}+2k-1)+(4-2p_{1})\sqrt{5}\Bigl{)}n\\\
&-2\sqrt{2}p_{1}+4\sqrt{10}p_{1}-4\sqrt{2}k-8\sqrt{10}+\sqrt{2},\end{aligned}$
(4.13) $\displaystyle\begin{aligned}
\mathbb{E}\Bigl{(}SO_{avr}(PC_{n})\Bigl{)}&=\frac{-2\sqrt{2}np_{1}+2\sqrt{2}p_{1}+8\sqrt{2}n-6\sqrt{2}}{k}\\\
&+\frac{4np_{1}-8n-4}{nk}\mu_{2},\end{aligned}$ (4.14)
$\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO(PC_{n})\Bigl{)}=(84-16\sqrt{26})p_{1}(1-p_{1})(n-2),$
(4.15)

$\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{red}(PC_{n})\Bigl{)}=(28-16\sqrt{10})(n-2)p_{1}(1-p_{1}),$
(4.16) $\displaystyle\begin{aligned}
\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{avr}(PC_{n})\Bigl{)}&=\frac{-80n^{2}k^{2}-8n^{2}k+8nk+16n^{2}-32n+16}{n^{2}k^{2}}\\\
&+\frac{-8\sqrt{2}(n-1)\mu_{2}}{n^{2}k^{2}},\end{aligned}$ (4.17)

where

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13},$
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\sqrt{n^{2}k^{2}-2n^{2}k+2nk+2n^{2}-4n+2}.$

##  5 Application of Corollaries 4.2 and 4.3

The expected values and variances of topological indices of random polygonal
chains have received much attention in scientific research, such as random
polygonal chains, random pentagonal chains, random polyphenyl chains, and
random cyclooctane chains. This section studied the Sombor indices
distribution of these polygonal chains as well as their expected values and
variances. According to the common research methods presented in 4.2 and 4.3,
there was the following analysis.

A random polyonino chain is a random modified domino chain with $n$
squares[25], represented by $GPC_{n}(n;\ p_{1},\ 1-p_{1})$. Due to the
numerous intriguing combinatorial subjects that result from them, for example,
the dominance problems[26, 27], car dominoes, enumeration problems[28, 29]
with perfect matching, and so on, generalized domino graphs have aroused the
curiosity about certain mathematicians.

In a random generalized poly-energy chain, there are two sorts of local
permutations, due to the definition of a random $l$-poly energy chain (see
Fig. 3). Based on the calculation of the Sombor indices for an random
polygonal chain presented in Section 3, the generalized formula of the Sombor
indices of $GPC_{n}$ is obtained by

$\displaystyle SO_{a}(GPC_{n})$ $\displaystyle=$
$\displaystyle(2\sqrt{2}\left\lvert 2-a\right\rvert-2v_{1})X$ $\displaystyle+$
$\displaystyle\Bigl{(}\sqrt{2}(\left\lvert 2-a\right\rvert+\left\lvert
3-a\right\rvert)+4v_{1}\Bigr{)}n$ $\displaystyle+$
$\displaystyle\sqrt{2}\left\lvert 2-a\right\rvert-\sqrt{2}\left\lvert
3-a\right\rvert-4v_{1},$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

Figure 3: Two sorts of local permutations in a random polyonino chain. Figure
4: Two sorts of local permutations in a random pentachain.

######  Theorem 5.1.

The expected value and variance of the Sombor indices for $GPC_{n}$ are given
by

$\displaystyle\mathbb{E}\Bigl{(}SO_{a}(GPC_{n})\Bigr{)}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sqrt{2}(1+p_{1})\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert+(4-2p_{1})v_{1}\Bigr{)}n$
$\displaystyle+$ $\displaystyle\sqrt{2}(-2p_{1}-1)\left\lvert
2-a\right\rvert-3\sqrt{2}\left\lvert
3-a\right\rvert-2\sqrt{2}(4-2p_{1})v_{1},$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(GPC_{n})\Bigr{)}=\Bigl{(}16a^{2}-72a+84-8\sqrt{2}v_{1}\left\lvert
2-a\right\rvert\Bigr{)}(n-2)p_{1}(1-p_{1}),$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

A random pentachain $P(n;\ p_{1},\ 1-p_{1})$ of length $n$ with $n$
pentagons[30, 31], labeled it by $P_{n}$.

According to the concept of a random $l$-polygon chain, a random pentachain
has two sorts of local permutations (see Fig 4). Depending on the calculation
of the Sombor indices for an random polygonal chain presented in Section 3,
the generalized formula of the Sombor indices of $P_{n}$ is obtained by

$\displaystyle SO_{a}(P_{n})$ $\displaystyle=$
$\displaystyle\Bigl{(}2\sqrt{2}\left\lvert 2-a\right\rvert+\sqrt{2}\left\lvert
3-a\right\rvert+2v_{1}\Bigr{)}X$ $\displaystyle+$
$\displaystyle\Bigl{(}\sqrt{2}\left\lvert 2-a\right\rvert+\sqrt{2}\left\lvert
3-a\right\rvert+v_{1}\Bigr{)}n$ $\displaystyle+$ $\displaystyle
4\sqrt{2}\left\lvert 2-a\right\rvert-\sqrt{2}\left\lvert
3-a\right\rvert+8v_{1},$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

######  Theorem 5.2.

The expected value and variance of the Sombor indices for $P_{n}$ are given by

$\displaystyle\mathbb{E}\Bigl{(}SO_{a}(P_{n})\Bigr{)}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sqrt{2}((1+p_{1})\left\lvert
2-a\right\rvert+\left\lvert 3-a\right\rvert)+(4-2p_{1})v_{1}\Bigr{)}n$
$\displaystyle+$ $\displaystyle(2\sqrt{2}-2\sqrt{2}p_{1})\left\lvert
2-a\right\rvert-3\sqrt{2}\left\lvert 3-a\right\rvert+4p_{1}v_{1},$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}(SO_{a}(P_{n})\Bigr{)}=A^{2}(n-2)p_{1}(1-p_{1}),$

where

$\displaystyle A^{2}$ $\displaystyle=$ $\displaystyle 3v_{1}^{2}+2\left\lvert
2-a\right\rvert\left\lvert 3-a\right\rvert-2\sqrt{2}\Bigl{(}\left\lvert
2-a\right\rvert+\left\lvert 3-a\right\rvert\Bigr{)},$ $\displaystyle v_{1}$
$\displaystyle=$ $\displaystyle\sqrt{2a^{2}-10a+13}.$

Random polyphenyl chain of length $n$, with $n$ hexagons, labeled by
$PPC_{n}(n;\ p_{1},\ p_{2},\ 1-p_{1}-p_{2})$. Polygonal chains are nothing
more than an unbranched heavy hydrocarbons graph with modifications that are
widely seen in chemical synthesis, medicinal synthesis, and heat exchangers,
and they have long aroused the curiosity of chemists[32, 33]. There are three
types of configurations according to the random polygonal chain concept, as
shown in Fig 5. Based on the calculation of the Sombor indices for an random
polygonal chain presented in Section 3, the generalized formula of the Sombor
indices of $PPC_{n}$ is obtained by

$\displaystyle SO_{a}(PPC_{n})$ $\displaystyle=$
$\displaystyle\Bigl{(}2\sqrt{2}\left\lvert 2-a\right\rvert-2v_{1}\Bigr{)}X$
$\displaystyle+$ $\displaystyle\Bigl{(}3\sqrt{2}\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert+4v_{1}\Bigr{)}n$
$\displaystyle+$ $\displaystyle\sqrt{2}\left\lvert
2-a\right\rvert-\sqrt{2}\left\lvert 3-a\right\rvert-4v_{1},$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

######  Theorem 5.3.

The expected value and variance of the Sombor indices for $PPC_{n}$ are
derived by

$\displaystyle\mathbb{E}\Bigl{(}SO_{a}(PPC_{n})\Bigr{)}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sqrt{2}((3+p_{1})\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert)+(4-2p_{1})v_{1}\Bigr{)}n$
$\displaystyle-$ $\displaystyle 5\left\lvert
2-a\right\rvert-3\sqrt{2}\left\lvert
3-a\right\rvert-2\sqrt{2}(4-2p_{1})v_{1},$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PPC_{n})\Bigr{)}=\Bigl{(}16a^{2}-72a+84-8\sqrt{2}v_{1}\left\lvert
2-a\right\rvert\Bigr{)}(n-2)p_{1}(1-p_{1}).$

Figure 5: Three sorts of permutations in a random polyphenyl chain. Figure 6:
Four sorts of permutations in a random cyclooctane chain.

The random $8$-polygonal chain $COC_{n}(n;\ p_{1},\ p_{2},\ p_{3},\
1-p_{1}-p_{2}-p_{3})$ is a random cyclooctanechain with $n$ octagons. Aromatic
hydrocarbons and their derivatives have constantly attracted the interest of
chemists[34, 35]. According to the definition of a random $l$-polygon chain,
there are four sorts of permutations in random cyclooctane chains (see Fig 6).
According to the calculation of the Sombor indices for a random polygonal
chain presented in Section 3, the generalized formula of the Sombor indices of
$COC_{n}$ is obtained by

$\displaystyle SO_{a}(COC_{n})$ $\displaystyle=$
$\displaystyle(2\sqrt{2}\left\lvert
2-a\right\rvert-2v_{1})X+(5\sqrt{2}\left\lvert
2-a\right\rvert+\sqrt{2}\left\lvert 3-a\right\rvert+4v_{1})n$ $\displaystyle+$
$\displaystyle\sqrt{2}\left\lvert 2-a\right\rvert-\sqrt{2}\left\lvert
3-a\right\rvert-4v_{1},$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

######  Theorem 5.4.

The expected value and variance of the Sombor indices for $COC_{n}$ are
obtained by

$\displaystyle\mathbb{E}\Bigl{(}SO_{a}(COC_{n})\Bigr{)}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sqrt{2}((5+p_{1})\left\lvert
2-a\right\rvert+\sqrt{2}(1+p_{1})\left\lvert
3-a\right\rvert)+(4-2p_{1})v_{1}\Bigr{)}n$ $\displaystyle-$
$\displaystyle(2\sqrt{2p_{1}+6})\left\lvert
2-a\right\rvert-3\sqrt{2}\left\lvert 3-a\right\rvert-4p_{1}v_{1},$
$\displaystyle\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(COC_{n})\Bigr{)}=\Bigl{(}3v_{1}^{2}+2\left\lvert
2-a\right\rvert*\left\lvert 3-a\right\rvert-2\sqrt{2}(\left\lvert
2-a\right\rvert*\left\lvert 3-a\right\rvert)\Bigr{)}(n-2)p_{1}(1-p_{1}),$

where $v_{1}=\sqrt{2a^{2}-10a+13}$.

##  6 Asymptotic behaviors for $SO_{a}(G_{n})$

The distribution of the Sombor indices of random polygonal chains is presented
in Section 3, see Corollaries $4.2-4.3$, respectively. In addition, the
preceding section’s applications to four specific polygonal chains were
integrated. Therefore, in this part, asymptotic behavior of the Sombor indices
within those random chains.

Under specific conditions, the normal distribution is a good approximation of
the binomial distribution and can be used to calculate the probability of the
binomial distribution. Since the probability thus obtained is only an
approximation to the true probability value of the binomial distribution, this
application of the normal distribution is known as the normal approximation to
the binomial distribution.

######  Theorem 6.1.

Let $X$ obeys bernoulli distribution. For $n\to\infty$, $X$ asymptotically
obeys normal distributions. One has

$\displaystyle\lim_{n\to\infty}\sup_{a\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{X_{n}-np}{np(1-p)}\leqslant
a)-\int_{-\infty}^{a}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0.$

Theorem 6.1 leads to the following conclusion.

######  Proposition 6.2.

Let $PC_{n}$, $GPC_{n}$, $P_{n}$, $PPC_{n}$, and $COC_{n}$ be a random
polygonal chain, polyonino chain, pentachain, polyphenyl, and cyclooctane
chain, respectively.

For each $x\in\mathbb{R},\ n\to\infty$, we have

$\displaystyle\lim_{n\to\infty}\sup_{x\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{SO_{a}(PC_{n})-\mathbb{E}\Bigl{(}SO_{a}(PC_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PC_{n})\Bigr{)}}}\leqslant
x)-\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0,$
(6.18)
$\displaystyle\lim_{n\to\infty}\sup_{x\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{SO_{a}(GPC_{n})-\mathbb{E}\Bigl{(}SO_{a}(GPC_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(GPC_{n})\Bigr{)}}}\leqslant
x)-\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0,$
(6.19)
$\displaystyle\lim_{n\to\infty}\sup_{x\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{SO_{a}(P_{n})-\mathbb{E}\Bigl{(}SO_{a}(P_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(P_{n})\Bigr{)}}}\leqslant
x)-\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0,$
(6.20)
$\displaystyle\lim_{n\to\infty}\sup_{x\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{SO_{a}(PPC_{n})-\mathbb{E}\Bigl{(}SO_{a}(PPC_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PPC_{n})\Bigr{)}}}\leqslant
x)-\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0,$
(6.21)
$\displaystyle\lim_{n\to\infty}\sup_{x\in\mathbb{R}}\left\lvert\mathbb{P}(\frac{SO_{a}(COC_{n})-\mathbb{E}\Bigl{(}SO_{a}(COC_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(COC_{n})\Bigr{)}}}\leqslant
x)-\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{t^{2}}{2}}\,dt\right\rvert=0.$
(6.22)

###### Proof.

By Corollary 4.2 and Corollary 4.3, we have

$\frac{SO_{a}(PC_{n})-\mathbb{E}\Bigl{(}SO_{a}(PC_{n})\Bigr{)}}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}\Bigl{(}SO_{a}(PC_{n})\Bigr{)}}}=\frac{X-\mathbb{E}(X)}{\sqrt{\mathbb{V}\mathbf{a}\mathbf{r}(X)}},\
X\sim B(n-2,\ p_{1}).$

By Theorem 6.1 and $(4.6-4.19),(6.20)$ can be proved.

Similarly, $(6.21-6.24)$ are obtained by such proof process. ∎

According to Proposition 6.2, the distributions of the Sombor indices of the
random polygonal chain networks are consistent with the asymptotic normal
distributions, and when $n>30$, $p_{1}$ is a constant between $0$ and $1$. The
normal distributions can be regarded as an approximation to the distributions
of the Sombor indices of the random chain networks in this paper,and please
refer to Table 1 for the specific values.

Table 1: Expected values and variances of Normal distribution for Sombor indices. $Indices$ | $Normal\ \ parameters$
---|---
$SO(PC_{n})_{2k+1}$ | $\mu=\Bigl{(}(5\sqrt{2}-2\sqrt{13})p_{1}+4\sqrt{2}k-3\sqrt{2}+4\sqrt{13}\Bigr{)}n+(4\sqrt{13}-4\sqrt{2})p_{1}-8\sqrt{2}k+11\sqrt{2}$
| $\sigma^{2}=(16\sqrt{26}+84)p_{1}(1-p_{1})(n-2)$
$SO_{red}(PC_{n})_{2k+1}$ | $\mu=\Bigl{(}(3\sqrt{2}-2\sqrt{5})p_{1}+2\sqrt{2}k+4\sqrt{5}-\sqrt{2}\Bigr{)}n+(4\sqrt{5}-2\sqrt{2})p_{1}-4\sqrt{2}k+4\sqrt{2}$
| $\sigma^{2}=(8\sqrt{10}+28)(n-2)p_{1}(1-p_{1})$
$SO_{avr}(PC_{n})_{2k+1}$ | $\mu=M_{1}n+N_{1}$
| $\sigma^{2}=\widetilde{\sigma}(n-2)p_{1}(1-p_{1})$
$SO(PC_{n})_{2k}$ | $\mu=\Bigl{(}(2p_{1}(\sqrt{2}-\sqrt{13})+4\sqrt{2}k-4\sqrt{13}-3\sqrt{2}\Bigr{)}n+4p_{1}(\sqrt{26}-\sqrt{2})+8\sqrt{2}k+5\sqrt{2}-8\sqrt{26}$
| $\sigma^{2}=(84-16\sqrt{26})p_{1}(1-p_{1})(n-2)$
$SO_{red}(PC_{n})_{2k}$ | $\mu=\Bigl{(}\sqrt{2}(p_{1}+2k-1)+(4-2p_{1})\sqrt{5}\Bigr{)}n-2\sqrt{2}p_{1}+4\sqrt{10}p_{1}-4\sqrt{2}k-8\sqrt{10}+\sqrt{2}$
| $\sigma^{2}=(28-16\sqrt{10})(n-2)p_{1}(1-p_{1})$
$SO_{avr}(PC_{n})_{2k}$ | $\mu=\frac{-2\sqrt{2}np_{1}+2\sqrt{2}p_{1}+8\sqrt{2}n-6\sqrt{2}}{k}+\frac{4np_{1}-8n-4}{nk}\mu_{2}$
|
$\sigma^{2}=\frac{-80n^{2}k^{2}-8n^{2}k+8nk+16n^{2}-32n+16}{n^{2}k^{2}}+\frac{-8\sqrt{2}(n-1)\mu_{2}}{n^{2}k^{2}}$

##  7 Conclusion

In this paper, a method for calculating the distributions of the Sombor
indices of a random polygonal chain has been established. The expected values
and variances of the Sombor indices of a random polygonal chain have been
calculated. As an application, we also have obtained the expected values and
variances of the Sombor indices for polyonino chain, pentachain, polyphenyl
chain, and cyclooctane chain. Based on the central limit theorem, it is also
discovered from a probabilistic perspective that since the end connections of
random chains obey a binomial distribution, when the number $n$ of polygons
connected by any chain tends to infinity, is the Sombor indices of any chain
at that point from a normal distribution. Sombor indices can help to predict
the physicochemical properties of many molecules. We have compared the Sombor
indices with some existing topological indices and will find that sometimes
the Sombor indices show better predictive power than existing indices, which
also provide new ideas in our future research discussions.

## Declartion of competing interest

We confirm that these results have neither published elsewhere nor are under
consideration. The authors have no conflict of interest to disclose.

## Data availability statements

The data that support the finding of this study are availability within the
article. All relevant data are also availability from the corresponding author
upon reasonable request.

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