

































 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  

(ISSN: 2992-4421 )                                                                                                                                    Jyothi.Mj 1 * 

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*On SiO4 Molecular Topological Characterization of Chemical Structures* 

 

 

On SiO4 Molecular Topological Characterization of Chemical Structures 

Jyothi. Mj*  ,  k. shivashankara**   

*Department of Mathematics, Maharanis Science College for Women, Mysore 570005, 

India   

**Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore  

570005, India  

 

Abstract 

In this paper, our aim is to study valency-based molecular invariants for SiO4 in a chain 

network. We compute the harmonic polynomial, atom bond connectivity polynomial, forgotten 

polynomial, geometric arithmetic polynomial, Randic polynomial, reciprocal Randic polynomial, 

symmetric division polynomial, inverse symmetric division polynomial, sigma polynomial, 

Sombor polynomial, and their degree-base topological indices for SiO4 embedded in a silicate 

chain network for various conditions. Physio-chemical properties of chemical compounds, such 

as formation enthalpies, boiling points, chromatographic retention times, vapour pressure, and 

surface areas, can be determined using our investigated results, such as the H-index, ABC-index, 

F-index, GA-index, R-index, RR-index, SDD-index, ISDDindex, S-index, and SO-index. We also 

create graphical representations of the results that describe the dependence of topological 

indices on polynomial structure parameters. 

 

 

Keywords: SiO4 in a chain, ABC polynomial and ABC index, geometric arithmetic 

polynomial, Randic index and reciprocal randic polynomial, sigma and Sombor index 

2020  Mathematics Subject Classification: 05C07, 05C09, 05C31, 05C76, 05 C 99 

  

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Introduction 

One of the standard procedures used in the study of structure-property relations is the 

use of structure descriptors. The ability to correlate and predict physical, chemical, and 

biological activity (property) from a molecular structure is a challenging problem in 

theoretical and computational chemistry [1, 2]. A topological index is a number that 

describes the graph’s topology. It is one of the best quantification methods because it 

can be computed quickly for a large number of molecules and can be obtained directly 

from molecular structures. Wiener, a chemist, used a topological index for the first time 

in 1947 while studying the relationship between molecular structure and the physical 

and chemical properties of certain hydrocarbon compounds [3, 17]. Liu et al. discussed 

several aspects of graph theory in 

[5]-[12]. 

Mathematical chemistry describes how to use polynomials and functions to offer 

instructions concealed in the symmetry of molecular graphs, and graph theory has 

many applications in modern chemistry, particularly organic chemistry. The atoms and 

bonds of a molecular structure are represented by vertices and edges, respectively, in 

chemical graph theory. Many applications of topological indices are employed in 

theoretical chemistry, [13, 14], particularly QSPR/QSAR research. Many famous 

researchers have studied topological indices to get information about different families 

of graphs [4, 15]. In qualitative structure-property relationships (QSPR) and qualitative 

structure-activity relationships (QSAR), topological indices are used directly as simple 

numerical descriptors in comparison with physical, biological, or chemical 

characteristics of molecules, which is a benefit. Many researchers have worked on 

various chemical compounds and computed topological descriptors of various 

molecular graphs during the last few decades [18]. 

In chemical graph theory, a molecular graph is a simple connected graph that contains 

chemical atoms and bonds, which are often referred to as vertices and edges, 

respectively, and there must be a linkage between the vertices set VG and edges set EG.If 

two atoms have an atom-bond, then it is denoted by e ∼ f, the valency of every atom of G 

is actually the total number of atoms connected to f of G and it is denoted by df, [16, 19]. 

Several polynomials closely related to degree-based indices are also introduced. 

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In 2012, Zhang introduced harmonic index [31]. The harmonic polynomial 

corresponding to harmonic index is defined as 

�(�, �) = ∑ �

�

�����
��∈�(�) & �(�) = ∑

�

�����
��∈�(�)                      (1) 

In 1998, Estrada et.al introduced atom bond connectivity index [21]. The ABC 

polynomial corresponding to the ABC indices is expressed as 

���(�, �) =  � �

�

��+��
−2

��+��

��∈�(�)

&         ���(�) =  �

�

��+��
−2

��+��
      (2)

��∈�(�)

 

In 2015, Formula and Gutmann introduced Forgotten Topological index or F-index [22]. 

The forgotten polynomial and index are defined as 

�(�, �) =  � �[ (��)��[(��)�]

��∈�(�)

&         �(�) =  � [ (��)� + [(��)�]      (3)

��∈�(�)

 

The first GA-index was proposed by Vukicevic [23]. The geometric arithmetic 

polynomial and index are defined as 

��(�, �) =  � �

���+��
�

��+��

��∈�(�)

&        ��(�) =  �
���+��

�

��+��

                (4)

��∈�(�)

 

The Randic polynomial and index, [24] are defined as 

�(�, �) =  � �

�

���+��

��∈�(�)

&        �(�) =  �
1

���+��

                          (5)

��∈�(�)

 

The reciprocal Randic polynomial and index [25] are defined as 

��(�, �) =  � �
���   ��

��∈�(�)

&        ��(�) =  � ���   ��
                      (6)

��∈�(�)

 

The symmetric division degree polynomial and index [26] are defined as 

���(�, �) =  � �

[ (��)��[(��)�]

(��)(��)

��∈�(�)

&       ���(�) =  �
[ (��)� + [(��)�]

(��)(��)
   (7)

��∈�(�)

 

The inverse symmetric Division Degree polynomial and index [27] are defined as 

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����(�, �) =  � �

(��)(��)

[ (��)��[(��)�]

��∈�(�)

&      ����(�) =  �
(��)(��)

[ (��)� + [(��)�]
   (8)

��∈�(�)

 

Sigma polynomial and index [28] are defined as 

�(�, �) =  � � (�����)�

��∈�(�)

&         �(�) =  �  (��−��)�                                  (9)

��∈�(�)

 

The concept of Sombor index was recently introduced by Gutman [29]. The Sombor 

polynomial and index are defined as 

�(�, �) =  � �
�[ (��)��[(��)�]

��∈�(�)

&         ��(�) =  � �[ (��)� + [(��)�]          (10)

��∈�(�)

 

In this study, the atom-bond partition set of SiO4 in a chain network, which is 

partitioned according to the valencies of their Si and O2 atoms, is used to generate the 

ten polynomials mentioned above and their corresponding indices. 

1 Chain of SiO4 

A SiO4 tetrahedron, the fundamental building block of silicates, is created by fusing 

metal oxides or mixing metal carbonates with sand. The SiO4 tetrahedron is present in 

almost all silicates. As shown in Figure 1, a tetrahedron SiO4 is a pyramid with a 

triangular base (a single tetrahedron SiO4), and the silicon atom Si is bonded with 

evenly spaced oxygen atoms. The resulting SiO4, a silicate tetrahedron that connects 

with other SiO4 horizontally, forms a single chain. Similar to this, when two SiO4 

molecules join corner to corner, each one shares its O2 atoms with the other, as shown in 

Figure 1. These two molecules of SiO4 can be joined with two other molecules once this 

sharing process is finished. We now have a silicate chain, SCqp, where p and q stand for 

the total number of SiO4 atoms in one silicate chain and the number of silicate chains 

that were formed, respectively. The pq number of SiO4 tetrahedrons used in the chain of 

SiO4 SCqp is shown in Figure 1. 

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1.1 Result and discussion 

Here, we have observed that there are three types of atom bonds on the bases of the 

valency of each atom of SCqp in a chain of 

types of atoms, vi and vj, with valencies of

respectively. Three different types of atom

based on the valencies (3 and 6) of atoms. Table 1 provides the division of the set of 

atom bonds based on valency. 

 

Table 1: Atom

Type of atom-bond 

Number of atom bonds 

Theorem 2.1. For p >1 and p 

(3�� + 3� − 4)�
�

� + (3�� − 6�

 

 

 

Figure 1: 1 

e, we have observed that there are three types of atom bonds on the bases of the 

in a chain of SiO4 SCqp. As a result, there are two different 

, with valencies of  and dvi = 3 and 

Three different types of atom-bonds (3 ∼ 3), (3 ∼ 6), and (6 ∼ 6) in SC

based on the valencies (3 and 6) of atoms. Table 1 provides the division of the set of 

 

Table 1: Atom-bond partition of SCqp, for p = q 

3 = de ∼ df = 3 3 = de ∼ df = 6 6 = de ∼ d

3p + 2 3(pq + q) − 4 3(pq − 2q

and p = q, the harmonic polynomial ofSCqp, is (

+ 2)�
�

� 

 

e, we have observed that there are three types of atom bonds on the bases of the 

. As a result, there are two different 

= 3 and dvj = 6, 

∼ 3), (3 ∼ 6), and (6 ∼ 6) in SCqp are 

based on the valencies (3 and 6) of atoms. Table 1 provides the division of the set of 

df = 6 

q) + 2 

(3� + 2)�
�

� +

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Proof.  

Using Table”1” enter the following formula harmonic polynomial (1), we get 

��SC�
�

, �� =  � �
�

���

����~����

+ � �
�

���

����~����

+ � �
�

���

����~����

 

This gives 

��SC�
�

, �� = (3� + 2)�
�

� + (3�� + 3� − 4)�
�

� + (3�� − 6� + 2)�
�

� 

 

By taking the first derivative of the polynomial in Theorem 2.1 at y = 1, we get the 

harmonic index of Silicate Network SC�
�

as follows: 

Corollary 2.2. For p >1 and p = q, the harmonic index of SC�
�

is 
����������

��
 

Theorem 2.3. For p >1 and p = q, the ABS polynomial ofSC�
�

 is (3� + 2)�
�

� + 3�� + 3� −

4)�
�

�� + (3�� − 6� + 2)�
�

�� 

Proof. Using Table”1” enter the following formula ABC polynomial (2), we get 

����SC�
�

, �� = � �
�

�����

(�)(�)

����~����

+ � �
�

�����

(�)(�)

����~����

+ � �
�

�����

(�)(�)

����~����

 

This gives 

����SC�
�

, �� = (3� + 2)�
�

� + (3�� + 3� − 4)�
�

�� + (3�� − 6� + 2)�
�

�� 

 

By taking the first derivative of the polynomial in Theorem 2.3 at y = 1, we get the 

ABC index of chain ���� (SC�
�

) of as follows: 

Corollary 2.4. For p >1 and p = q, the ABC index of
  ����������

��
 

Theorem 2.5. For p >1 and p = q, the forgotten topological polynomial of SC�
�

is (3p + 2) 

y18+ (3p2 + 3p − 4) y45+ (3p2 − 6p + 2) y72. 

 

 

 

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Proof. Using Table”1” enter the following formula forgotten topological polynomial (3), 

weget 

��SC�
�

, �� =  � �[�����]

����~����

+ � �[�����]

����~����

+ � �[�����]

����~����

 

This gives 

��SC�
�

, �� = (3� + 2)��� + (3�� + 3� − 4)��� + (3�� − 6� + 2)��� 

 

By taking the first derivative of the polynomial in Theorem 2.5 at y = 1, we get the 

forgotten index of chain of SiO4 (SCpp) as follows: 

Corollary 2.6. For p >1 and p = q, the forgotten topological index of SC�
�

is 351p2 −243p. 

Theorem 2.7. For p >1 and p = q, the geometric arithmetic polynomial of  SC�
�(3� +

2)�
√�

� + (3�� + 3� − 4)�
�

� + (3�� − 6� + 2)�
�

√� 

Proof. Using Table”1” enter the following formula geometric arithmetic polynomial (4), 

we get 

���SC�
�

, �� = � �
√����

���

����~����

+ � �
√����

���

��~�

+ � �
√����

���

����~����

 

This gives 

���SC�
�

, �� = (3� + 2)�
√�

� + (3�� + 3� − 4)�
�

� + (3�� − 6� + 2)�
�

√� 

 

By taking the first derivative of the polynomial in Theorem 2.7 at y = 1, we get the 

geometric arithmetic index of chain of SiO4 (SC�
�

) as follows: 

Corollary 2.8. For p >1 and p = q, the geometric arithmetic index ofSC�
�

 �� (3� + 2)�
√�

� +

(3�� + 3� − 4)�
�

� + (3�� − 6� + 2)�
�

√� 

Theorem 2.9. For p >1 and p = q, the randic polynomial of  SC�
�

 �� (3� + 2)�
�

� +

(3�� + 3� − 4)�
�

√�
� + (3�� − 6� + 2)�

�

� 

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Proof. Using Table”1” enter the following formula randic polynomial polynomial (5), we 

get 

��SC�
�

, �� =  � �
�

�(�)(�)

����~����

+ � �
�

�(�)(�)

����~����

+ � �
�

�(�)(�)

����~����

 

This gives 

��SC�
�

, �� = (3� + 2)�
�

� + (3�� + 3� − 4)�
�

√�
� + (3�� − 6� + 2)�

�

� 

 

By taking the first derivative of the polynomial in Theorem 2.9 at y = 1, we get the 

randic polynomial index of chain of SiO4 (SC�
�

) as follows: 

Corollary 2.10. For p >1 and p = q, the randic polynomial index ofSC�
�

 is 
√���������.√�� ����.√�

�
 

Theorem 2.11. For p >1 and p = q, the reciprocal randic polynomial ofSC�
�

 is(3� +

2)�� + 

(3�� + 3� − 4)��√� + (3�� − 6� + 2)�√�. 

Proof. Using Table “1” enter the following formula reciprocal randic polynomial 

polynomial 

(6), we get 

���SC�
�

, �� =  � ��(�)(�)

����~����

+ � ��(�)(�)

��~�

+ � ��(�)(�)

����~����

 

This gives 

���SC�
�

, �� =  (3� + 2)�� + (3�� + 3� − 4)��√� + (3�� − 6� + 2)�√�. 

 

By taking the first derivative of the polynomial in Theorem 2.11 at y = 1, we get the 

reciprocal randic polynomial index of chain of SiO4 (SC�
�

) as follows: 

Corollary 2.12. For p >1 and p = q, the reciprocal randic polynomial index of 3(3� + 2) +

3√2(3�� + 3� − 4) + √6(3�� − 6� + 2). 

Theorem 2.13. For p>1 and p=q. the symmetric degree polynomial of SC�
�

is(3�� − 3� +

4)�� + (3�� + 3� − 4)�
��

�  

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Proof. Using Table enter the following formula symmetric division degree polynomial 

(7), we get 

����SC�
�

, �� =  � �
[(�)��(�)�]

(�)(�)

����~����

+ � �
[(�)��(�)�]

(�)(�)

����~����

+ � �
[(�)��(�)�]

(�)(�)

��~�

 

This gives 

����SC�
�

, �� = (3� + 2)�� + (3�� + 3� − 4)�
��

� + (3�� − 6� + 2)�� 

= (3�� − 3� + 4)�� + (3�� + 3� − 4)�
��

�  

By taking the first derivative of the polynomial in Theorem 2.19 at y = 1, we get the 

symmetric division degree index of chain of SiO4 (SC�
�

 as follows: 

Corollary 2.14. For p >1 and p = q, the symmetric division degree index of SC�
�

is2(3�� −

3� + 4) +
������������

�
 

Theorem 2.15. For p >1 and p = q, the inverse symmetric division polynomial of SC�
�

 is 

(3�� − 3� + 4)�
�

� + (3�� + 3� − 4)�
�

�� 

Proof. Using Table “1” enter the following formula inverse symmetric division degree 

polynomial (8), we get 

�����SC�
�

, �� =  � �
[(�) �(�) ]

(�)��(�)�

����~����

+ � �
[(�)��(�)�]

(�)(�)

��~�

+ � �
[(�)��(�)�]

(�)(�)

����~����

 

This gives 

����SC�
�

, �� = (3� + 2)�
�

� + (3�� + 3� − 4)�
�

�� + (3�� − 6� + 2)�
�

� 

= (3�� − 3� + 2)�
�

� + (3�� + 3� − 4)�
�

�� 

 

By taking the first derivative of the polynomial in Theorem 2.15 at y = 1, we get the 

inverse symmetric division degree index of chain of SiO4 SC�
�

as follows: 

Corollary 2.16. For p >1 and p = q, the inverse symmetric division degree index of SC�
�

is  

�����������

��
. 

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Theorem 2.17. For p >1 and p 

3�� − 3� + 4. 

Proof. Using Table “1” enter the following formula sigma polynomial (7), we get

�(G, �)�SC�
�

, �� =  

�

This gives 

����SC�
�

, �� = (

= (

By taking the first derivative of the polynomial in Theorem 2.19 at 

sigma index of chain of

Corollary 2.18. For p >1 and p 

Theorem2.19. For p >1 and√ p 

(3�� + 3� − 4)��√� + (3�� − 6

Proof. Using Table “1” enter the following formula somber polynomial (7), we get

���SC�
�

, �� =  �

����~����

This gives 

���SC�
�

, �� = (3� + 2

By taking the first derivative of the polynomial in Theore

somber index of chain of

Corollary 2.20. For p >1 and p = q, the somber index of

3� − 4)√5. 

and p = q, the sigma polynomial of   SC�
�

 ��(3�� + 3�

1” enter the following formula sigma polynomial (7), we get

� � �(���)�

����~����

+ � �(���)�

����~����

+ � �(

��~�

� (3� + 2) + (3�� + 3� − 4)�� + (3�� − 6� + 2

(3�� + 3� − 4)�� + 3�� − 3� + 4 

By taking the first derivative of the polynomial in Theorem 2.19 at y 

) as follows: 

and p = q, the sigma index of SC�
�p is 9(3p2 + 3p − 4).

p = q, the somber polynomial of SC�
�

is (3� + 2)�

6� + 2)��√� 

1” enter the following formula somber polynomial (7), we get

��(�)��(�)�

�

+ � ��(�)��(�)�

����~����

+ � ��

��~�

2)��√� + (3�� + 3� − 4)��√� + (3�� − 6� + 2

By taking the first derivative of the polynomial in Theorem 2.19 at y 

) as follows: 

q, the somber index ofSC�
� �� 9(2�� − 3� + 2)√

� − 4)�� +

1” enter the following formula sigma polynomial (7), we get 

(���)�
 

2) 

 

y = 1, we get the 

. 

)��√� +

1” enter the following formula somber polynomial (7), we get 

�(�)��(�)�
 

2)��√� 

 

y = 1, we get the 

)√2 + 3(3�� +

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1.2 Results for p<q and p is odd 

Here, in chain of SiO4 (SC�
�

), we observed for p < q and p is odd, atom-bonds on the bases of 

valency of every atom of SC�
�changed. So, on the base of valency, Table 2 provides the 

partition of the set of atom-bonds. 

Table 2: Atom-bond partition of SC�
�

, for p is odd and p < q 

Type of atom-bond 3 = de ∼ df = 3 3 = de ∼ df = 6 6 = de ∼ df = 6 

Number of atom bonds 3(p + 1) 3pq + p + 2q − 5 3pq − 2(2p + q − 1) 

Theorem 2.21. Let p be odd and p < q. Then the harmonic polynomial ofSC�
�

 �� 3(� + 1)�
�

� +

(3�� + � + 2� − 5)�
�

� + (3�� − 2(2� + � − 1)�
�

� 

Proof. Using the atom-bond partition from Table 2, in the formula of harmonic polynomial 

(1), we get 

��SC�
�

, �� =  � �
�

���

����~����

+ � �
�

���

��~�

+ � �
�

���

����~����

 

This gives 

��SC�
�, �� = 3(� + 1)�

�

� + (3�� + � + 2� − 5)�
�

� + (3�� − 2(2� + � − 1)�
�

� 

 

By taking the first derivative of the polynomial in Theorem 2.21 at y = 1, we get the 

harmonic index of Silicate Network SC�
�

as follows: 

Corollary 2.22. Let p be odd and p < q. Then the harmonic index 

of SC�
�

 �� � + 
�������������

��
+ 1. 

Theorem 2.23. Let p be odd and p < q. Then the ABC polynomial ofSC�
�

 �� 3(� + 1)�
�

�(3�� +

� + 2� − 5)�
�

�� + (3�� − 2(2� + � − 1)�
�

�� 

 

Proof. Using the atom-bond partition from Table 2, in the formula of ABC polynomial (2) , 

we get 

����SC�
�, �� =  � �

�
�����

(�)(�)

����~����

+ � �
�

�����

(�)(�)

����~����

+ � �
�

�����

(�)(�)

����~����

 

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This gives 

����SC�
�, �� = 3(� + 1)�

By taking the first derivative of the polynomial in Theorem 2.23 at 

index of chain of SiO4 (SC�
�) as follows:

Corollary 2.24. Let p be odd and p < q. Then the ABC index of

Theorem 2.25. Let p be odd and p < q. Then the forgotten topological polynomial of 

3(p + 1)y18 + (3pq + p + 2q − 5)y

Proof. Using the atom-bond partition from Table 2, in the formula of forgotten topological 

polynomial (3), we get 

��SC�
�

, �� =  �

����~��

This gives 

��SC�
�

, �� = 3(� + 1)���

By taking the first derivative of the polynomial in Theorem 2.25 at 

forgotten topological index of chain of 

Corollary 2.26. Let p be odd and p < q. Then the forgotten topological i

− 189p − 54q − 27. 

Theorem 2.27. Let p be odd and p < q. Then the geometric arithmetic polynomial of   

3(� + 1)�
√�

� +

Proof. Using the atom-bond partition from Table 2, in the formula of geometric 

polynomial (4), we get 

 

����SC�
�, �� =  

�

)�
�

� + (3�� + � + 2� − 5)�
�

�� + (3�� − 2(2� +

By taking the first derivative of the polynomial in Theorem 2.23 at y = 1, we get the 

) as follows: 

Let p be odd and p < q. Then the ABC index of(SC�
�)   is

�������

��

Let p be odd and p < q. Then the forgotten topological polynomial of 

y45 + (3pq − 2(2p + q − 1))y72. 

bond partition from Table 2, in the formula of forgotten topological 

�[�����]

��

+ � �[�����]

����~����

+ � �[

����~����

+ (3�� + � + 2� − 5)��� + 2�3�� − 2(2� +

By taking the first derivative of the polynomial in Theorem 2.25 at y 

forgotten topological index of chain of SiO4 SCqp as follows: 

Let p be odd and p < q. Then the forgotten topological index of 

Let p be odd and p < q. Then the geometric arithmetic polynomial of   

+ (3�� + � + 2� − 5)�
�

� + 2(2� + � − 1)�
�

√� 

bond partition from Table 2, in the formula of geometric 

� � �
�√���

���

����~����

+ � �
�√���

���

��~�

+ � �
�√�

��

����~����

+ � − 1)�
�

�� 

= 1, we get the ABC 

�������� 

��
 

Let p be odd and p < q. Then the forgotten topological polynomial of SCqp is 

bond partition from Table 2, in the formula of forgotten topological 

[�����] 

� − 1)��� 

 

y = 1, we get the 

ndex of   is 351pq 

Let p be odd and p < q. Then the geometric arithmetic polynomial of   is  

 

bond partition from Table 2, in the formula of geometric arithmetic 

���

��  

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This gives  

���SC�
�, �� = 3(� + 1)�

√�

�

By taking the first derivative of the polynomial in Theorem 2.27 at 

geometric arithmetic index of chain of 

Corollary 2.28. Let p be odd and p < q. Then the geometric arithmetic index of 

2(� + 3�� + 2� − 5) + √3�3�� − 2(2

3.

Theorem 2.29. Let p be odd and p < q. Then the randic polynomial polynomial of

3(� + 1)�
�

� + (3�� + � + 2� −

Proof. Using the atom-bond partition from Table 2, in the formula of randic polynomial 

polynomial (5), we get 

��SC�
�, �� =  �

����~�

This gives 

��SC�
�

, �� = 3(� + 1)�
�

� + (3��

By taking the first derivative of the polynomial in Theorem 2.29 at 

randic polynomial index of chain of 

Corollary 2.30. Let p be odd and p < q.Then the randic polynomial index o

(3�� + � + 2� − 5)�
�

�√� + 2�3��

Theorem 2.31. Let p be odd and p < q. Then the reciprocal randic polynomial Polynomial

of SC�
�is 3(� + 1)�� + (3�� + �

Proof. Using the atom-bond partition from Table 2, in the

polynomial polynomial (6), we get

��SC�
�, �� =  �

����~

)
�

� + (3�� + � + 2� − 5)�
�

� + 2�3�� − 2(2� +

By taking the first derivative of the polynomial in Theorem 2.27 at y = 1, we get the 

geometric arithmetic index of chain of SiO4 SC�
�

as follows: 

Let p be odd and p < q. Then the geometric arithmetic index of 

(2� + � − 1)� + 3 √6 � + 3√6
 

Let p be odd and p < q. Then the randic polynomial polynomial of

5)�
�

�√� + 2�3�� − 2(2� + � − 1)��
�

� 

bond partition from Table 2, in the formula of randic polynomial 

� �
�

�(�)(�)

����

+ � �
�

�(�)(�)

����~����

+ � ��

����~����

( �� + � + 2� − 5)�
�

�√� + 2�3�� − 2(2� + � − 1)

By taking the first derivative of the polynomial in Theorem 2.29 at y 

randic polynomial index of chain of SiO4 (SC�
�)as follows: 

Let p be odd and p < q.Then the randic polynomial index of SC

� �� − 2(2� + � − 1)��
�

�. 

Let p be odd and p < q. Then the reciprocal randic polynomial Polynomial

� + 2� − 5)��√� + �3�� − 2(2� + � − 1)��� 

bond partition from Table 2, in the formula of reciprocal randic 

polynomial polynomial (6), we get 

� ��(�)(�)

~����

+ � ��(�)(�)

��~�

+ � ��(�)

����~����

� − 1)��
�

√� 

= 1, we get the 

Let p be odd and p < q. Then the geometric arithmetic index of SC�
�

is 

Let p be odd and p < q. Then the randic polynomial polynomial of   is 

bond partition from Table 2, in the formula of randic polynomial 

�

�(�)(�) 

)��
�

� 

 

y = 1, we get the 

C�
�is 

Let p be odd and p < q. Then the reciprocal randic polynomial Polynomial 

�  

formula of reciprocal randic 

� )(�) 

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��SC�
�

, �� = 3(� + 1)�� + (3�� + � + 2� − 5)��√� + �3�� − 2(2� + � − 1)��� 

 

By taking the first derivative of the polynomial in Theorem 2.31 at y = 1, we get the 

reciprocal randic polynomial index of chain of SiO4 SC�
�as follows: 

Corollary 2.32. Let p be odd and p < q. Then the reciprocal randic polynomial index of 

SC�
� �� 9�2�� + 18�� + 3�2� − 15� + 6√2� + 21 − 12� − 15√2. 

Theorem 2.33. Let p be odd and p < q. Then the sigma polynomial of SC�
�is(3�� − � − 2� +

1)�� + (4�� + � + 2� − 5)�
��

�  

 

Proof. Using the atom-bond partition from Table 2, in the formula of symmetric division 

degree polynomial (7), we get 

����SC�
�

, �� =  � �
[(�)��(�)�]

(�)(�)

����~����

+ � �
[(�)��(�)�]

(�)(�)

����~����

+ � �
[(�)��(�)�]

(�)(�)

��~�

 

This gives 

����SC�
�

, �� = 3(� + 1)�� + (3�� + � + 2� − 5)�
��

� + (3�� − 2(2� + � − 1))�� 

= (3�� − � − 2� + 1)�� + (3�� + � + 2� − 5)�
��

�  

 

By taking the first derivative of the polynomial in Theorem 2.33 at y = 1, we get the 

symmetric division degree index of chain of SiO4 (SC�
�

) as follows: 

Corollary 2.34. Let p be odd and p < q. Then the symmetric division degree index of SC�
�

is 
 

�

�
(69�� + 7� + 14� − 567). 

Theorem 2.35. Let p be odd and p < q. Then the inverse symmetric division polynomial of 

SC�
�is(3�� − � − 2� + 1)�

�

� + (3�� + � + 2� − 5)�
�

��. 

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Proof. Using the atom-bond partition from Table 2, in the formula of inverse symmetric 

division polynomial (8), we get

�����SC�
�, �� =  �

����~

This gives 

�����SC�
�, �� = 3(� + 1)�

�

� +

= (3�� − � − 2

By taking the first derivative of the polynomial in Theorem 2.39 at 

inverse symmetric division index of chain of 

Corollary 2.36. Let p be odd and p < q. Then the inverse symmetric division index of 

. 

Theorem 2.37. Let p be odd and p < q. Then the sigma polynomial of 

y9+ 3pq − p − 2q + 1. 

Proof. Using the atom-bond partition from Table 2, in the formul

we get 

��SC�
�

, �� =  �

����

This gives 

����SC�
�

, �� = 3(� + 1) +

= (3��

By taking the first derivative of the polynomial in Theorem 2.39 at 

sigma index of chain of

Corollary 2.38. Let p be odd and p < q. Then the sigma index of

bond partition from Table 2, in the formula of inverse symmetric 

polynomial (8), we get 

� �
(�)(�)

[(�)��(�)�]

~����

+ � �
(�)(�)

[(�)��(�)�]

��~�

+ � �[

����~����

+ (3�� + � + 2� − 5)�
�

�� + (3�� − 2(2� − 2(2

2� + 1)�
�

� + (3�� + � + 2� − 5)�
�

��. 

By taking the first derivative of the polynomial in Theorem 2.39 at y 

inverse symmetric division index of chain of SiO4 SC�
�

as follows: 

Let p be odd and p < q. Then the inverse symmetric division index of 

Let p be odd and p < q. Then the sigma polynomial of SC�
�is (3

bond partition from Table 2, in the formula of sigma polynomial (8), 

� �(���)�

�~����

+ � �(���)�

����~����

+ � �(���

��~�

) + (3�� + � + 2� − 5)�� + (3�� − 2(2� − 2(2

( + � + 2� − 5)�� + 3�� − � − 2� + 1. 

By taking the first derivative of the polynomial in Theorem 2.39 at y 

 as follows: 

Let p be odd and p < q. Then the sigma index of SC�
�is (3pq + p 

bond partition from Table 2, in the formula of inverse symmetric 

(�)(�)

[(�)��(�)�] 

(2� + � − 1))�
�

�

 

y = 1, we get the 

Let p be odd and p < q. Then the inverse symmetric division index of is

(3pq + p + 2q − 5) 

a of sigma polynomial (8), 

�)�
 

(2� + � − 1)) 

 

y = 1, we get the 

p + 2q − 5). 

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Theorem 2.39. Let p be odd and p < q. Then the somber polynomial of SC�
�

 is 3(� + 1)��√� +

(3�� + � + 2� − 5)��√� + (3�� − 2(2� + � − 1)��√�. 

Proof. Using the atom-bond partition from Table 2, in the formula of somber polynomial 

(8), we get 

���SC�
�

, �� =  � ��(�)��(�)�

����~����

+ � ��(�)��(�)�

����~����

+ � ��(�)��(�)�

��~�

 

This gives 

���SC�
�

, �� = 3(� + 1)��√� + (3�� + � + 2� − 5)��√� + (3�� − 2(2� + � − 1)��√� 

 

By taking the first derivative of the polynomial in Theorem 2.39 at y = 1, we get the 

somber index of chain of SiO4 (SCqp) as follows: 

Corollary 2.40. Let p be odd and p < q. Then the somber index of SC�
� �� (3�� + � + 2� −

5)3√5 + (6�� − 5� − 4� + 5))3√2 

Conclusion 

In this article, two important silicon tetrahedron compound structures are considered, and 

the accurate formulas of some important valency-based topological indices are calculated 

using the technique of atom-bonds partitioning of these molecular structures. Our 

investigated results, such as the H-index, ABC-index, F-index, GA-index, R-index, RR- index, 

SDD-index, ISDD-index, S-index and SO-index are useful for determining physio-chemical 

properties of chemical compounds, as in 2005, Zhou explain in [34], such as formation 

enthalpies, boiling points, chromatographic retention times, vapour pressure, and surface 

areas. The obtained results are also innovative and noteworthy contributions to network 

science, providing a foundation for understanding the deep topology of these important 

networks. These findings, may also be useful in determining the role of silicon-carbon in 

electronics and industry. We also present a numerical comparison of topological 

characterizations for p = q for the SiO4 chain in (SCqp) in Table 3 and a graphical comparison 

in Figure 2. 

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Table 3: Topological characterizations of

p=q H ABC F 

2 6.11 9.56 918 

3 12.61 20.39 2430 

4 21.44 35.22 4644 

5 32.61 54.06 7560 

6 46.11 76.89 11178

7 61.94 103.72 15498

8 80.11 134.56 20520

9 100.61 169.39 26244

10 123.44 208.22 32670

Figure 2: Graphical comparison

 

 

Open problems For the characterization of the chain of 

discuss or research the these open problems.

Table 3: Topological characterizations of  and p is odd

GA R RR SDD ISDD

 17.02 6.30 88.30 72.5 8.73

 36.66 13.04 195.71 164 19.53

 63.78 22.20 343.27 290 34.93

 98.34 33.77 530.99 450.5 54.93

11178 140.38 47.76 758.86 645.5 79.53

15498 189.89 64.16 1026.89 875 108.73

20520 246.86 82.97 1335.07 1139 142.53

26244 311.29 104.20 1683.39 1437.5 180.93

32670 383.18 127.84 2071.88 1770.5 223.93

 

 

 

Figure 2: Graphical comparison 

For the characterization of the chain of SiO4 the followers are invited to 

discuss or research the these open problems. 

is odd 

ISDD SO 

8.73 144.83 

19.53 354.67 

34.93 655.67 

54.93 1047.84 

79.53 1531.16 

108.73 2105.65 

142.53 2771.30 

180.93 3528.11 

223.93 4376.08 

 

the followers are invited to 

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	Introduction
	1 Chain of SiO4
	1.1 Result and discussion
	1.2 Results for p<q and p is odd

	Conclusion
	References 

