IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Jyothi.Mj 1 * https://ijojournals.com/ Volume 08 || Issue 04 || April, 2025 || *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 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 1 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. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 2 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 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 3 ����(�, �) = � � (��)(��) [ (��)��[(��)�] ��∈�(�) & ����(�) = � (��)(��) [ (��)� + [(��)�] (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. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 4 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)� � � + IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 5 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. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 6 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)� � � IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 7 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)� �� � IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 8 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 ����������� �� . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 9 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�� + IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) IJO JOURNALS Volume 08 | Issue 04 | April 2025 | https://ijojournals.com/index.php/m/index 10 1.2 Results for p