# Analysis of Complex Networks via Some Novel Topological Indices

Authors: Shahid Amin; M. A. Rehman; Amir Naseem; Ilyas Khan; Nawa Alshammari; Nawaf N. Hamadneh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Chemical graph theory is a field of mathematical chemistry that links mathematics, chemistry, and graph theory to solve chemistry-related issues quantitatively. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. A graphical representation of chemical molecules, known as the molecular graph of the chemical substance, is one of these tools. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we calculate a new topological index namely, the Sombor index, the Super Sombor index, and its reduced version for chemical networks. We also plot our computed results to examine how they were affected by the parameters involved. This document lists the distinct degrees and degree sums of enhanced mesh network, triangular mesh network, star of silicate network, and rhenium trioxide lattice. The edge partitions of these families of networks are tabled which depend on the sum of degrees of end vertices and the sum of the degree-based edges. These edge partitions are used to find closed formulae for numerous degree-based topological indices of the networks.


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## Body

## 1. Introduction

In mathematics, the graph is an abstract structure formed by set of end points called nodes or vertices V (G) which are joined by a set of lines called edges E(G). The order and size of a graph is the total number of vertices and total number of edges of the graphG, respectively. In the theory of graphs, we usually analyze the chemical structures that appeared in different branches of chemistry.Mathematical chemistry is the branch of mathematics in which mathematical tools are used to solve the problems arising in chemistry. One of these tools is a graphical representation of chemical compounds and this representation is known as the molecular graph of the concerned chemical compound [1, 2]. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. A topological index is a molecular graph invariant that links the physicochemical features of a molecular graph with a number in the fields of chemical graph theory, molecular topology and mathematical chemistry [3]. Chemical graphs are graphs in which the vertices and edges represent the atoms and bonds of a chemical molecule. The vertices and edges are identified according to the atoms and types of bonds they represent [4, 5]. Topological invariants of a molecular graph can be used to learn about a chemical structure and its hidden properties without having to conduct experiments [6, 7]. Distance, degree, and eccentricity are the three major topological indices. Among them, degree-based indices have much importance and can be used as a tool for chemists [8]. Distance-based topological indices are concerned with graph distances, degree-based topological indices are concerned with the concept of degree, and counting-based topological indices are concerned with the counting of edges [9]. Topological indices have a wide range of applications in material science, math, informatics, biology, and other fields [10, 11]. The structure of the chemical graph is connected to topological indices [12, 13]. Physical and chemical features of the underlying molecule that can be determined from the TIs include heat of evaporation, heat of creation, boiling point, chromatographic retention, surface tension, and vapor pressure [14, 15].The first topological index, known as the Wiener index, was developed by Wiener while examining the boiling point of alkanes [16]. The Randic index is a simple topological index which was introduced by Milan Randic [17]. Zagreb indices are the oldest topological indices written by Gutman and Trinajstic [18, 19]. In QSPRs, topological indices are utilized to guess the properties of the concerned compound [20]. There are no such type of topological index which give us an idea of all the properties of the concerned compound. As a result, there is constantly a need for new topological indices to be defined. The Sambor indices and reduced Sambor indices have lately been characterized by Gutman [21] as follows:(1)SOG=∑ij∈EGdi2+dj2,SOredG=∑ij∈EGdi−12+dj−12.Only the Super Sambor and Super-reduced Sambor indices can be calculated if we can find the edge partition of these interconnection chemical networks based on the sum of the degrees of the end vertices of each edge in these graphs [9]as follows:(2)SSOG=∑ij∈EGSi2+Sj2,SSOredG=∑ij∈EGSi−12+Sj−12.We computeSOG,SOred,SSOG and SSOredG indices for certain chemical networks, as well as their graphical representations, in this study.


## 2. Enhanced Mesh

The degree-based topological descriptors of enhanced mesh networks are examined in this section. Also, the graphical comparison ofSO,SOred,SSO, and SSOred indices is depicted in Figure 1.Figure 1
(a) 2D and (b) 3D graphical comparison ofSO,SOred,SSO, and SSOred indices.
(a)(b)


### 2.1. Construction

The end points of a common type of lattice graph relate to points in the grid with integer parameters, x-coordinates in the region 1, 2, …,s, and two vertices are associated by an edge whenever the corresponding points are at a distance of 1. In other words, for the point set mentioned, it is a unit distance graph. In literature, the name t-mesh has also been applied to a variety of different sorts of graphs with a specific structure, such as the Cartesian product of a number of path graphs. When we apply the Cartesian product of edges of order x1,x2,…,xt, we get such a t-mesh Mx1,x2,…,xt, which is written as follows:(3)Mx1,x2,…,xt=Px1×Px2×⋯×Pxt.If we take t-mesh,Mx1,x2,…,xt then it has order as follows:(4)VMx1,x2,…,xt=x1,x2,…,xt.and its size is(5)EMx1,x2,…,xt=x1,x2,…,xtt−1x1−1x2−⋯−1xt.Now, we discuss enhance 2-mesh networkMx1×x2, its vertex set is as follows:(6)V=i,j:1≤i≤s,1≤j≤t.and the set is(7)E=i,j,i,j+1:1≤i≤s,1≤j≤t−1∪i,j,i+1,j:1≤i≤s−1,1≤j≤t.is the edge set. Now, enhanced meshEMPx1×Px2 is the result by replacing each 4-cycle of MPx1×Px2 by a wheel W4 which has 4 vertices. As a result, a wheel W4 is a graph obtained by connecting the central vertex of cycle C4 to each vertex of cycle C4. A new vertex is the hub (central vertex) of W4. Let the collection of all hub (central) vertices be hij,1≤i≤s−1,1≤j≤t−1. The graph of enhanced mesh is shown in Figure 2.Figure 2
Enhanced mesh M (5, 5) network.Theorem 1.
SO and SOred indices for the enhanced mesh network are as follows:(i)
SOG=82.7475+73.6081s−69.4994t+58.4045st(ii)
SOredG=78.4639−62.9848s+t+50.2618stProof.
From the edge partitioning of enhanced mesh network given in Table1, we have the following computations for SO and SOred indices:(8)SOG=∑ij∈EGdi2+dj2=32+424+32+528+42+524s+t−4+42+824st−2s−2t+4+52+522s+t−6+52+822s+t−4+82+822st−5s−5t+12=82.7475+73.6081s−69.4994t+58.4045st.SOredG=∑ij∈EGdi−12+dj−12=3−12+4−124+3−12+5−128+4−12+5−124s+t−4+4−12+8−12st−2s−2t+4+5−12+5−122s+t−6+5−12+8−122s+t−4+8−12+8−1222st−5s−5t+12=78.4639−62.9848s+t+50.2618st.Table 1
Edge partitioning centered on the degree of each edge’s ending vertices of EM (s, t).
ϕϕdi,djFrequencyϕ1ϕ3,44ϕ2ϕ3,58ϕ3ϕ4,54s+t−4ϕ4ϕ4,84st−2s−2t+4ϕ4ϕ5,52s+t−6ϕ4ϕ5,82s+t−4ϕ5ϕ8,82st−5s−5t+12Theorem 2.
SSO and SSOred indices for the enhanced mesh network are as follows:(i)
SSOG=726.3922−521.7772s+t+366.5203st(ii)
SSOredG=−1855.5167−522.7773s+t+358.147stProof.
From the edge partitioning of enhanced mesh network given in Table2, we have the following computations for SSO and SSOred indices:(9)SSOG=∑ij∈EGSi2+Sj2=142+2124+142+2428+212+2428+212+4224+242+26216+212+4224+242+4228+262+26223s+3t−26+262+4228+262+4526s+t−8+322+4224+322+4524s+t−8+322+4824st−4s−4t+16+422+4528+452+4522s+t−10+452+4822s+t−8+482+4822st−9s−9t−40=726.3922−521.7772s+t+366.5203st.SSOredG=∑ij∈EGSi−12+Sj−12=14−12+21−124+14−12+24−128+21−12+24−128+21−12+42−124+24−12+26−1216+24−12+42−128+26−12+26−1223s+3t−26+26−12+42−128+26−12+45−126s+t−8+32−12+42−124+32−12+45−124s+t−8+32−12+48−1243s+3t−26+26−12+26−122st−4s−4t−16+42−12+45−128+45−12+45−122s+t−10+45−12+48−122s+t−8+48−12+48−122s−9s−9t+40=−1855.5167−522.7773s+t+358.147st.Table 2
Edge partitioning on the basis of the sum of vertices which is a distance unit from the end vertices of each edge of EM (s, t).
ϕϕdi,djFrequencyϕ1ϕ14,214ϕ2ϕ14,248ϕ3ϕ21,248ϕ4ϕ21,424ϕ5ϕ24,2616ϕ6ϕ24,428ϕ7ϕ26,2623s+3t−26ϕ8ϕ26,428ϕ9ϕ26,456s+t−8ϕ10ϕ32,424ϕ11ϕ32,454s+t−8ϕ12ϕ32,484st−4s−4t+16ϕ13ϕ42,458ϕ14ϕ45,452s+t−10ϕ15ϕ45,482s+t−8ϕ16ϕ48,482st−9s−9t+40


## 2.1. Construction

The end points of a common type of lattice graph relate to points in the grid with integer parameters, x-coordinates in the region 1, 2, …,s, and two vertices are associated by an edge whenever the corresponding points are at a distance of 1. In other words, for the point set mentioned, it is a unit distance graph. In literature, the name t-mesh has also been applied to a variety of different sorts of graphs with a specific structure, such as the Cartesian product of a number of path graphs. When we apply the Cartesian product of edges of order x1,x2,…,xt, we get such a t-mesh Mx1,x2,…,xt, which is written as follows:(3)Mx1,x2,…,xt=Px1×Px2×⋯×Pxt.If we take t-mesh,Mx1,x2,…,xt then it has order as follows:(4)VMx1,x2,…,xt=x1,x2,…,xt.and its size is(5)EMx1,x2,…,xt=x1,x2,…,xtt−1x1−1x2−⋯−1xt.Now, we discuss enhance 2-mesh networkMx1×x2, its vertex set is as follows:(6)V=i,j:1≤i≤s,1≤j≤t.and the set is(7)E=i,j,i,j+1:1≤i≤s,1≤j≤t−1∪i,j,i+1,j:1≤i≤s−1,1≤j≤t.is the edge set. Now, enhanced meshEMPx1×Px2 is the result by replacing each 4-cycle of MPx1×Px2 by a wheel W4 which has 4 vertices. As a result, a wheel W4 is a graph obtained by connecting the central vertex of cycle C4 to each vertex of cycle C4. A new vertex is the hub (central vertex) of W4. Let the collection of all hub (central) vertices be hij,1≤i≤s−1,1≤j≤t−1. The graph of enhanced mesh is shown in Figure 2.Figure 2
Enhanced mesh M (5, 5) network.Theorem 1.
SO and SOred indices for the enhanced mesh network are as follows:(i)
SOG=82.7475+73.6081s−69.4994t+58.4045st(ii)
SOredG=78.4639−62.9848s+t+50.2618stProof.
From the edge partitioning of enhanced mesh network given in Table1, we have the following computations for SO and SOred indices:(8)SOG=∑ij∈EGdi2+dj2=32+424+32+528+42+524s+t−4+42+824st−2s−2t+4+52+522s+t−6+52+822s+t−4+82+822st−5s−5t+12=82.7475+73.6081s−69.4994t+58.4045st.SOredG=∑ij∈EGdi−12+dj−12=3−12+4−124+3−12+5−128+4−12+5−124s+t−4+4−12+8−12st−2s−2t+4+5−12+5−122s+t−6+5−12+8−122s+t−4+8−12+8−1222st−5s−5t+12=78.4639−62.9848s+t+50.2618st.Table 1
Edge partitioning centered on the degree of each edge’s ending vertices of EM (s, t).
ϕϕdi,djFrequencyϕ1ϕ3,44ϕ2ϕ3,58ϕ3ϕ4,54s+t−4ϕ4ϕ4,84st−2s−2t+4ϕ4ϕ5,52s+t−6ϕ4ϕ5,82s+t−4ϕ5ϕ8,82st−5s−5t+12Theorem 2.
SSO and SSOred indices for the enhanced mesh network are as follows:(i)
SSOG=726.3922−521.7772s+t+366.5203st(ii)
SSOredG=−1855.5167−522.7773s+t+358.147stProof.
From the edge partitioning of enhanced mesh network given in Table2, we have the following computations for SSO and SSOred indices:(9)SSOG=∑ij∈EGSi2+Sj2=142+2124+142+2428+212+2428+212+4224+242+26216+212+4224+242+4228+262+26223s+3t−26+262+4228+262+4526s+t−8+322+4224+322+4524s+t−8+322+4824st−4s−4t+16+422+4528+452+4522s+t−10+452+4822s+t−8+482+4822st−9s−9t−40=726.3922−521.7772s+t+366.5203st.SSOredG=∑ij∈EGSi−12+Sj−12=14−12+21−124+14−12+24−128+21−12+24−128+21−12+42−124+24−12+26−1216+24−12+42−128+26−12+26−1223s+3t−26+26−12+42−128+26−12+45−126s+t−8+32−12+42−124+32−12+45−124s+t−8+32−12+48−1243s+3t−26+26−12+26−122st−4s−4t−16+42−12+45−128+45−12+45−122s+t−10+45−12+48−122s+t−8+48−12+48−122s−9s−9t+40=−1855.5167−522.7773s+t+358.147st.Table 2
Edge partitioning on the basis of the sum of vertices which is a distance unit from the end vertices of each edge of EM (s, t).
ϕϕdi,djFrequencyϕ1ϕ14,214ϕ2ϕ14,248ϕ3ϕ21,248ϕ4ϕ21,424ϕ5ϕ24,2616ϕ6ϕ24,428ϕ7ϕ26,2623s+3t−26ϕ8ϕ26,428ϕ9ϕ26,456s+t−8ϕ10ϕ32,424ϕ11ϕ32,454s+t−8ϕ12ϕ32,484st−4s−4t+16ϕ13ϕ42,458ϕ14ϕ45,452s+t−10ϕ15ϕ45,482s+t−8ϕ16ϕ48,482st−9s−9t+40


## 3. Triangular Mesh

In the present section, we examined degree-based topological descriptors of the triangular mesh network. Also, the graphical comparison ofSO,SOred,SSO, and SSOred indices is depicted in Figure 3.Figure 3
(a) 2D and (b) the 3D graphical comparison ofSO,SOred,SSO, and SSOred indices.
(a)(b)


### 3.1. Construction

LetTt represent the radix—t triangular mesh network, which has the following node set:(10)VTt=u,v:0≤u,v≤tand0≤u+v≤t,and there always exist a mesh arc between the nodes u1,v1 and u2,v2 if u1−u2+v1−v2≤t−1 and u1+v1≤u2+v2. In a Tt, then the number of vertices (nodes) is equals to t(t + 1)/2. In the aforementioned network, the degree of node can be 2, 4, or 6. Three vertices of degree two are referred to as “corner vertices.” In the present section, G is symbolised by the graph of a triangular mesh network. The graph of a triangular mesh is shown in Figure 4.Figure 4
Triangular meshT5 network.Theorem 3.
SO and SOred indices for the triangular mesh network are as follows:(i)
SOG=155.4646+41.1213t+2.7279t2(ii)
SOredG=153.7212−110.7795t+21.213t2Proof.
From the edge partitioning of triangular the mesh network given in Table3, we have the following computations for SO and SOred indices:(11)SOG=∑ij∈EGdi2+dj2=22+426+42+423t−2+42+626t−3+62+623t2−7t+12=155.4646+41.1213t+2.7279t2.SOredG=∑ij∈EGdi−12+dj−12=2−12+4−126+4−12+4−123t−2+4−12+6−126t−3+6−12+6−12t2−7t+12=153.7212−110.7795t+21.213t2.Table 3
Edge partitioning depends on the degree of the end vertices ofTt.
ϕϕdi,djFrequencyϕ1ϕ2,46ϕ2ϕ4,43t−2ϕ3ϕ4,66t−3ϕ4ϕ6,63t2−7t+12/2Theorem 4.
SSO and SSOred indices for the triangular mesh network are as follows:(i)
SSOG=122.2908−256.8792t+76.3675t2(ii)
SSOredG=222−254.3856t+74.2462t2Proof.
From the edge partitioning of triangular mesh network given in Table4, we have the following computations for SSO and SSOred indices:(12)SSOG=∑ij∈EGSi2+Sj2=82+1626+162+1623+162+2026+162+2826+202+2023t−5+202+2826+202+3226t−5+282+3226+322+3223t−5+322+3626t−6+362+3623t2−13t+422=122.2908−256.8792t+76.3675t2.SSOredG=∑ij∈EGSi−12+Sj−12=8−12+16−126+16−12+20−126+16−12+28−126+20−12+20−123t−5+20−12+28−126+20−12+32−126t−5+28−12+32−126+32−12+32−123t−5+32−12+36−126t−6+36−12+36−123t2−13t+422=222−254.3856t+74.2462t2.Table 4
Edge partitioning for SSO ofTt.
ϕϕdi,djFrequencyϕ1ϕ8,166ϕ2ϕ16,163ϕ3ϕ16,206ϕ4ϕ16,286ϕ5ϕ20,203t−5ϕ6ϕ20,286ϕ7ϕ20,326t−5ϕ8ϕ28,326ϕ9ϕ32,323t−5ϕ10ϕ32,366t−6ϕ11ϕ36,363t2−13t+42/2


## 3.1. Construction

LetTt represent the radix—t triangular mesh network, which has the following node set:(10)VTt=u,v:0≤u,v≤tand0≤u+v≤t,and there always exist a mesh arc between the nodes u1,v1 and u2,v2 if u1−u2+v1−v2≤t−1 and u1+v1≤u2+v2. In a Tt, then the number of vertices (nodes) is equals to t(t + 1)/2. In the aforementioned network, the degree of node can be 2, 4, or 6. Three vertices of degree two are referred to as “corner vertices.” In the present section, G is symbolised by the graph of a triangular mesh network. The graph of a triangular mesh is shown in Figure 4.Figure 4
Triangular meshT5 network.Theorem 3.
SO and SOred indices for the triangular mesh network are as follows:(i)
SOG=155.4646+41.1213t+2.7279t2(ii)
SOredG=153.7212−110.7795t+21.213t2Proof.
From the edge partitioning of triangular the mesh network given in Table3, we have the following computations for SO and SOred indices:(11)SOG=∑ij∈EGdi2+dj2=22+426+42+423t−2+42+626t−3+62+623t2−7t+12=155.4646+41.1213t+2.7279t2.SOredG=∑ij∈EGdi−12+dj−12=2−12+4−126+4−12+4−123t−2+4−12+6−126t−3+6−12+6−12t2−7t+12=153.7212−110.7795t+21.213t2.Table 3
Edge partitioning depends on the degree of the end vertices ofTt.
ϕϕdi,djFrequencyϕ1ϕ2,46ϕ2ϕ4,43t−2ϕ3ϕ4,66t−3ϕ4ϕ6,63t2−7t+12/2Theorem 4.
SSO and SSOred indices for the triangular mesh network are as follows:(i)
SSOG=122.2908−256.8792t+76.3675t2(ii)
SSOredG=222−254.3856t+74.2462t2Proof.
From the edge partitioning of triangular mesh network given in Table4, we have the following computations for SSO and SSOred indices:(12)SSOG=∑ij∈EGSi2+Sj2=82+1626+162+1623+162+2026+162+2826+202+2023t−5+202+2826+202+3226t−5+282+3226+322+3223t−5+322+3626t−6+362+3623t2−13t+422=122.2908−256.8792t+76.3675t2.SSOredG=∑ij∈EGSi−12+Sj−12=8−12+16−126+16−12+20−126+16−12+28−126+20−12+20−123t−5+20−12+28−126+20−12+32−126t−5+28−12+32−126+32−12+32−123t−5+32−12+36−126t−6+36−12+36−123t2−13t+422=222−254.3856t+74.2462t2.Table 4
Edge partitioning for SSO ofTt.
ϕϕdi,djFrequencyϕ1ϕ8,166ϕ2ϕ16,163ϕ3ϕ16,206ϕ4ϕ16,286ϕ5ϕ20,203t−5ϕ6ϕ20,286ϕ7ϕ20,326t−5ϕ8ϕ28,326ϕ9ϕ32,323t−5ϕ10ϕ32,366t−6ϕ11ϕ36,363t2−13t+42/2


## 4. Star of the Silicate Network

This section contains the analysis of degree-based topological descriptors for the star of the silicate network. Also, the graphical comparison ofSO,SOred,SSO, and SSOred indices is depicted in Figure 5.Figure 5
(a) 2D and (b) the 3D graphical comparison ofSO,SOred,SSO, and SSOred indices.
(a)(b)


### 4.1. Construction

The star of the David network in Figure6 indicates the expansion of a new star in a silicate network.Step 1: Create an H-dimensional star of the David graph (Figure6).Step 2: To divide H into 2t  1 edges, apply 2t − 2 vertices to each edge.Step 3:Excluding the pairings at a distance of 1, 3, 5, 7, …, (6t − 1) from one corner vertex, join any two vertices u and v by an edge in such a way that the first is the complete reflection of the second one, the distance between them is odd from one corner vertices (2t − 1).Step 4: Except for the pairings at a distance, join every two verticesu and v with the help of an edge if one is the mirror image of the other and they are at an odd distance 1, 3, 5, 7, …, (6t − 1) from one corner vertices (2t − 1).Step 5: A tetrahedron is used to replace each K3 subgraph. SSL denotes the t-dimensional star silicate star, also known as the star of silicate network star (t). Figure7 depicts the graph for the SSL(2) silicate network star.Figure 6
Construction of star of the David graph of dimension 2.Figure 7
Graph of star of silicate network SSL (2).Here,G denotes the star silicate network SSL(t). Each tetrahedron has 6 corner vertices and a central vertex of degree 3 in G. The vertices that were added in the second step have degree four. The remaining vertices are all of degree 6. Thus, for u∈VSSLt, the degrees du are 3, 4, and 6. The edge partitioning of G is shown in the table below.Theorem 5.
SO and SOred indices for the star of a silicate network are as follows:(i)
SOG=291.4702−930.7296t+546.9624t2(ii)
SOredG=279.5208−583.2501t+448.4238t2Proof.
From the edge partitioning of the star silicate network given in Table5, we have the following computations for SO and SOred indices:(13)SOG=∑ij∈EGdi2+dj2=32+326+32+42122t−1+32+6236t260t+36+42+4224t−30+42+6224t−1+62+6236t2−72t+48=291.4702−930.7296t+546.9624t2.SOredG=∑ij∈EGdi−12+dj−12=3−12+3−126+3−12+4−122t−1+3−12+6−1236t2−60n+36+4−12+4−1224t−30+4−12+6−1224t−1+6−12+6−1236n2−72t+48=279.5208−583.2501t+448.4238t2.Table 5
Edge partitioning centered on the degree of the end vertices of SSL (t).
ϕϕdi,djFrequencyϕ1ϕ3,36ϕ2ϕ3,4122t−1ϕ3ϕ3,636t2−60t+36ϕ4ϕ4,424t−30ϕ5ϕ4,624t−1ϕ6ϕ6,636t2−72t+48Theorem 6.
SSO and SSOred indices for the star of a silicate network are as follows:(i)
SSOG=115.4495−3998.9124t+2679.807t2(ii)
SSOredG=1299.4434−2964n+2579.3676t2Proof.
From the edge partitioning of the star silicate network given in Table6, we have the following computations for SSO and SSOred indices:(14)SSOG=∑ij∈EGSi2+Sj2=112+1126+112+14224+142+1426+142+17224t−6+142+26212n−24+162+19212+162+26212+162+28212+172+17224t−60+172+19212+172+26224t−48+182+26212t2−48t+48+182+28224t−36+182+30212n2−48n+24+192+26212+192+28212+262+2626t2−24n+24+262+28212+262+30212t2−48n+48+282+28212t−18+282+30224t−36+302+30218t2−36t+18=115.4495−3998.9124t+2679.807t2.SSOredG=∑ij∈EGSi−12+Sj−12=11−12+11−126+11−12+44−1224+14−12+14−126+14−12+17−1224t−36+14−12+26−1212t−24+16−12+19−1212+16−12+26−1212+16−12+28−1212+17−12+17−1224t−60+17−12+19−1212+17−12+26−1224n−48+18−12+26−1212t2−48t+48+18−12+28−1224n−36+18−12+30−12242−48t+24+19−12+26−1212+19−12+28−1212+26−12+26−126t2−24t+24+26−12+28−1212+26−12+30−1212t2−48t+48+28−12+28−1212t−18+28−12+30−1224t−36+30−12+30−1218t2−36t+18.=1299.4434−2964t+2579.3676t2.Table 6
Edge partitioning for SSO of SSL (t).
ϕϕdi,djFrequencyϕ1ϕ11,116ϕ2ϕ11,1424ϕ3ϕ14,146ϕ4ϕ14,1724t−36ϕ5ϕ14,2612t−24ϕ6ϕ16,1912ϕ7ϕ16,2612ϕ8ϕ16,2812ϕ9ϕ17,1724n−60ϕ10ϕ17,1912ϕ11ϕ17,2624t−48ϕ12ϕ18,2612t2−48n+48ϕ13ϕ18,2824t−36ϕ14ϕ18,3024n2−48n+24ϕ15ϕ19,2612ϕ16ϕ19,2812ϕ17ϕ26,266t2−24t+24ϕ18ϕ26,2812ϕ19ϕ26,3012t2−48t+48ϕ20ϕ28,2812t−18ϕ21ϕ28,3024t−36ϕ22ϕ30,3018t2−36t+18


## 4.1. Construction

The star of the David network in Figure6 indicates the expansion of a new star in a silicate network.Step 1: Create an H-dimensional star of the David graph (Figure6).Step 2: To divide H into 2t  1 edges, apply 2t − 2 vertices to each edge.Step 3:Excluding the pairings at a distance of 1, 3, 5, 7, …, (6t − 1) from one corner vertex, join any two vertices u and v by an edge in such a way that the first is the complete reflection of the second one, the distance between them is odd from one corner vertices (2t − 1).Step 4: Except for the pairings at a distance, join every two verticesu and v with the help of an edge if one is the mirror image of the other and they are at an odd distance 1, 3, 5, 7, …, (6t − 1) from one corner vertices (2t − 1).Step 5: A tetrahedron is used to replace each K3 subgraph. SSL denotes the t-dimensional star silicate star, also known as the star of silicate network star (t). Figure7 depicts the graph for the SSL(2) silicate network star.Figure 6
Construction of star of the David graph of dimension 2.Figure 7
Graph of star of silicate network SSL (2).Here,G denotes the star silicate network SSL(t). Each tetrahedron has 6 corner vertices and a central vertex of degree 3 in G. The vertices that were added in the second step have degree four. The remaining vertices are all of degree 6. Thus, for u∈VSSLt, the degrees du are 3, 4, and 6. The edge partitioning of G is shown in the table below.Theorem 5.
SO and SOred indices for the star of a silicate network are as follows:(i)
SOG=291.4702−930.7296t+546.9624t2(ii)
SOredG=279.5208−583.2501t+448.4238t2Proof.
From the edge partitioning of the star silicate network given in Table5, we have the following computations for SO and SOred indices:(13)SOG=∑ij∈EGdi2+dj2=32+326+32+42122t−1+32+6236t260t+36+42+4224t−30+42+6224t−1+62+6236t2−72t+48=291.4702−930.7296t+546.9624t2.SOredG=∑ij∈EGdi−12+dj−12=3−12+3−126+3−12+4−122t−1+3−12+6−1236t2−60n+36+4−12+4−1224t−30+4−12+6−1224t−1+6−12+6−1236n2−72t+48=279.5208−583.2501t+448.4238t2.Table 5
Edge partitioning centered on the degree of the end vertices of SSL (t).
ϕϕdi,djFrequencyϕ1ϕ3,36ϕ2ϕ3,4122t−1ϕ3ϕ3,636t2−60t+36ϕ4ϕ4,424t−30ϕ5ϕ4,624t−1ϕ6ϕ6,636t2−72t+48Theorem 6.
SSO and SSOred indices for the star of a silicate network are as follows:(i)
SSOG=115.4495−3998.9124t+2679.807t2(ii)
SSOredG=1299.4434−2964n+2579.3676t2Proof.
From the edge partitioning of the star silicate network given in Table6, we have the following computations for SSO and SSOred indices:(14)SSOG=∑ij∈EGSi2+Sj2=112+1126+112+14224+142+1426+142+17224t−6+142+26212n−24+162+19212+162+26212+162+28212+172+17224t−60+172+19212+172+26224t−48+182+26212t2−48t+48+182+28224t−36+182+30212n2−48n+24+192+26212+192+28212+262+2626t2−24n+24+262+28212+262+30212t2−48n+48+282+28212t−18+282+30224t−36+302+30218t2−36t+18=115.4495−3998.9124t+2679.807t2.SSOredG=∑ij∈EGSi−12+Sj−12=11−12+11−126+11−12+44−1224+14−12+14−126+14−12+17−1224t−36+14−12+26−1212t−24+16−12+19−1212+16−12+26−1212+16−12+28−1212+17−12+17−1224t−60+17−12+19−1212+17−12+26−1224n−48+18−12+26−1212t2−48t+48+18−12+28−1224n−36+18−12+30−12242−48t+24+19−12+26−1212+19−12+28−1212+26−12+26−126t2−24t+24+26−12+28−1212+26−12+30−1212t2−48t+48+28−12+28−1212t−18+28−12+30−1224t−36+30−12+30−1218t2−36t+18.=1299.4434−2964t+2579.3676t2.Table 6
Edge partitioning for SSO of SSL (t).
ϕϕdi,djFrequencyϕ1ϕ11,116ϕ2ϕ11,1424ϕ3ϕ14,146ϕ4ϕ14,1724t−36ϕ5ϕ14,2612t−24ϕ6ϕ16,1912ϕ7ϕ16,2612ϕ8ϕ16,2812ϕ9ϕ17,1724n−60ϕ10ϕ17,1912ϕ11ϕ17,2624t−48ϕ12ϕ18,2612t2−48n+48ϕ13ϕ18,2824t−36ϕ14ϕ18,3024n2−48n+24ϕ15ϕ19,2612ϕ16ϕ19,2812ϕ17ϕ26,266t2−24t+24ϕ18ϕ26,2812ϕ19ϕ26,3012t2−48t+48ϕ20ϕ28,2812t−18ϕ21ϕ28,3024t−36ϕ22ϕ30,3018t2−36t+18


## 5. Rhenium Trioxide Lattice

The topological indices of the RO(s,t, r) rhenium trioxide lattice on the basis of degree are studied in this part. It is a nonmetallic compound made up of rhenium and oxygen atoms. Rhenium trioxide, often known as ReO3, is a red solid that forms a crystal. The corners of a hollow circle denote rhenium atoms, whereas the corners of solid circles denote oxygen atoms. 12 oxygen atoms and 8 rhenium atoms make up the unit cell of ReO3 (Figure 8). Also, the graphical comparison of SO,SOred,SSO, and SSOred indices is depicted in Figure 9.Figure 8
Unit cell of the rhenium trioxide latticeReO3.Figure 9
(a) 2D and (b) 3D graphical comparison ofSO,SOred,SSO, and SSOred indices.
(a)(b)


### 5.1. Construction

Thes×t×rReO3 lattice is represented by ReO3. There are s unit cell rows along the X-axis, t unit cell columns along the Y-axis, and r unit cell pages along the Z-axis. The RO(s, t, r) graph is the rhenium trioxide graph RO(s, t, r), which is formed as follows:Step 1: Here, we create a 3D grid M(s,q, r), that is defined by a Cartesian product of the paths Ps□Pt□Pr. The atoms of rhenium are displayed by such vertices.Step 2: In this phase, partition each of the M(s,t, r) edges. The new vertices correspond to oxygen atoms. The vertices of RO will now be marked (s, t, r). The identical 3D-grid mark will be applied to the corners and rhenium atoms. Between two rhenium atoms x′,y′,z′ and x′′,y′′,z′′, the oxygen atom will receive the label x,y,z, where x=x′+x′′/2,y=y′+y′′/2 and z=z′+z′′/2. Two vertices l′,m′,n′ and l′′,m′′,n′′ are adjacent if l′−l′′+m′−m′′+n′−n′′=1/2. The number of vertices and edges in RO(s, t, r) are 4str—st—tr—sr and 6str-2st—2tr—2sr, respectively. The graph of rhenium trioxide RO is denoted by the letter G throughout this section (s, t, r). The degree for every corner can correspond to the set 2,3,4,5,6 as a result of the formation of ROs,t,r.Theorem 7.
SO and SOred indices for the rhenium trioxide lattice are as follows:(i)
SOG=−0.1506+7.9369s+t+r−22.043st+tr+sr−37.9473str(ii)
SOredG=0.1064+8.0488s+t+8.6788r−19.9572st+tr+sr+30.5941strProof.
From the edge partitioning of the rhenium trioxide lattice given in Table7, we have the following computations for SO and SOred indices:(15)SOG=∑ij∈EGdi2+dj2=22+3224+22+4216s+t+r−6+22+5210st+tr+sr−4s−4t−4r+12+22+626str−2st−2tr−2sr+4s+4t+4r−8=−0.1506+7.9369s+t+r−22.043st+tr+sr−37.9473str.SOredG=∑ij∈EGdi−12+dj−12=2−12+3−1224+2−12+4−1216s+t+r−6+2−12+5−1210st+tr+sr−4s−4t−4r+12+2−12+6−126str−2st−2tr−2sr+4s+4t+4r−8=0.1064+8.0488s+t+8.6788r−19.9572st+tr+sr+30.5941str.Table 7
Edge partitioning centered on degree of final (end) vertices of RO (s,t, r).
ϕϕdi,djFrequencyϕ1ϕ2,324ϕ2ϕ2,416s+t+r−6ϕ3ϕ2,510st+tr+sr−4s−4t−4r+12ϕ4ϕ2,66str−2st−2tr−2sr+4s+4t+4r−8Theorem 8.
SSO and SSOred indices for the rhenium trioxide lattice are as follows:(i)
SSOG=255.2827+152.9129s+45.286t+r−94.7185st+tr+sr+305.4699str(ii)
SSOredG=−356.6413+141.6272s+t+r−89.0583st+tr+sr+93.3381strProof.
From the edge partitioning of the rhenium trioxide lattice given in Table8, we have the following computations for SSO and SSOred indices:(16)SSOG=∑ij∈EGSi2+Sj2=62+7224+72+8224+82+828s+t+r−9+82+928s+t+r−6+92+1028s+t+r−6+102+1028st+tr+sr−5s−5t−5r+18+102+1122st+tr+sr−4s−4t−4r+12+122+12223str−7st−7tr−7sr+16s+16t+16r−36=255.2827+152.9129s+45.286t+r−94.7185st+tr+sr+305.4699str.SSOredG=∑ij∈EGSi−12+Sj−12=6−12+7−1224+7−12+8−1224+8−12+8−128s+t+r−9+8−12+9−128s+q+r−6+9−12+10−128s+t+r−6+10−12+10−128st+tr+sr−5s−5q−5r+18+10−12+11−122st+tr+sr−4s−4t−4r+12+12−12+12−1223str−7st−7tr−7sr+16s+16q+16r−36=−356.6413+141.6272s+t+r−89.0583st+tr+sr+93.3381str.Table 8
Edge partitioning for SSO of RO (s,t, r).
ϕϕdi,djFrequencyϕ1ϕ6,724ϕ2ϕ7,824ϕ3ϕ8,88s+t+r−9ϕ4ϕ8,98s+t+r−6ϕ5ϕ9,108s+t+r−6ϕ6ϕ10,108st+tr+sr−5p−5t−5r+18ϕ7ϕ10,112st+tr+sr−4s−4t−4r+12ϕ8ϕ12,1223str−7st−7tr−7sr+16s+16t+16r−36


## 5.1. Construction

Thes×t×rReO3 lattice is represented by ReO3. There are s unit cell rows along the X-axis, t unit cell columns along the Y-axis, and r unit cell pages along the Z-axis. The RO(s, t, r) graph is the rhenium trioxide graph RO(s, t, r), which is formed as follows:Step 1: Here, we create a 3D grid M(s,q, r), that is defined by a Cartesian product of the paths Ps□Pt□Pr. The atoms of rhenium are displayed by such vertices.Step 2: In this phase, partition each of the M(s,t, r) edges. The new vertices correspond to oxygen atoms. The vertices of RO will now be marked (s, t, r). The identical 3D-grid mark will be applied to the corners and rhenium atoms. Between two rhenium atoms x′,y′,z′ and x′′,y′′,z′′, the oxygen atom will receive the label x,y,z, where x=x′+x′′/2,y=y′+y′′/2 and z=z′+z′′/2. Two vertices l′,m′,n′ and l′′,m′′,n′′ are adjacent if l′−l′′+m′−m′′+n′−n′′=1/2. The number of vertices and edges in RO(s, t, r) are 4str—st—tr—sr and 6str-2st—2tr—2sr, respectively. The graph of rhenium trioxide RO is denoted by the letter G throughout this section (s, t, r). The degree for every corner can correspond to the set 2,3,4,5,6 as a result of the formation of ROs,t,r.Theorem 7.
SO and SOred indices for the rhenium trioxide lattice are as follows:(i)
SOG=−0.1506+7.9369s+t+r−22.043st+tr+sr−37.9473str(ii)
SOredG=0.1064+8.0488s+t+8.6788r−19.9572st+tr+sr+30.5941strProof.
From the edge partitioning of the rhenium trioxide lattice given in Table7, we have the following computations for SO and SOred indices:(15)SOG=∑ij∈EGdi2+dj2=22+3224+22+4216s+t+r−6+22+5210st+tr+sr−4s−4t−4r+12+22+626str−2st−2tr−2sr+4s+4t+4r−8=−0.1506+7.9369s+t+r−22.043st+tr+sr−37.9473str.SOredG=∑ij∈EGdi−12+dj−12=2−12+3−1224+2−12+4−1216s+t+r−6+2−12+5−1210st+tr+sr−4s−4t−4r+12+2−12+6−126str−2st−2tr−2sr+4s+4t+4r−8=0.1064+8.0488s+t+8.6788r−19.9572st+tr+sr+30.5941str.Table 7
Edge partitioning centered on degree of final (end) vertices of RO (s,t, r).
ϕϕdi,djFrequencyϕ1ϕ2,324ϕ2ϕ2,416s+t+r−6ϕ3ϕ2,510st+tr+sr−4s−4t−4r+12ϕ4ϕ2,66str−2st−2tr−2sr+4s+4t+4r−8Theorem 8.
SSO and SSOred indices for the rhenium trioxide lattice are as follows:(i)
SSOG=255.2827+152.9129s+45.286t+r−94.7185st+tr+sr+305.4699str(ii)
SSOredG=−356.6413+141.6272s+t+r−89.0583st+tr+sr+93.3381strProof.
From the edge partitioning of the rhenium trioxide lattice given in Table8, we have the following computations for SSO and SSOred indices:(16)SSOG=∑ij∈EGSi2+Sj2=62+7224+72+8224+82+828s+t+r−9+82+928s+t+r−6+92+1028s+t+r−6+102+1028st+tr+sr−5s−5t−5r+18+102+1122st+tr+sr−4s−4t−4r+12+122+12223str−7st−7tr−7sr+16s+16t+16r−36=255.2827+152.9129s+45.286t+r−94.7185st+tr+sr+305.4699str.SSOredG=∑ij∈EGSi−12+Sj−12=6−12+7−1224+7−12+8−1224+8−12+8−128s+t+r−9+8−12+9−128s+q+r−6+9−12+10−128s+t+r−6+10−12+10−128st+tr+sr−5s−5q−5r+18+10−12+11−122st+tr+sr−4s−4t−4r+12+12−12+12−1223str−7st−7tr−7sr+16s+16q+16r−36=−356.6413+141.6272s+t+r−89.0583st+tr+sr+93.3381str.Table 8
Edge partitioning for SSO of RO (s,t, r).
ϕϕdi,djFrequencyϕ1ϕ6,724ϕ2ϕ7,824ϕ3ϕ8,88s+t+r−9ϕ4ϕ8,98s+t+r−6ϕ5ϕ9,108s+t+r−6ϕ6ϕ10,108st+tr+sr−5p−5t−5r+18ϕ7ϕ10,112st+tr+sr−4s−4t−4r+12ϕ8ϕ12,1223str−7st−7tr−7sr+16s+16t+16r−36


## 6. Conclusion

Various tools of mathematics are used to predict different properties of chemical compounds such as the topological index. We can associate a single number with a molecular graph of a chemical complex using the topological index. For the characterization of the unambiguous graph theoretical system of biological concern, we have appropriated various degree-based indices. We employed the Sombor index (SO), which has chemical relevance in determining numerous physicochemical properties of octane isomers since it has a coefficient of correlation near to one in the field of mathematical chemistry. Drugs and other chemical substances are frequently shown as polygonal shapes, trees, graphs, and other geometrical shapes. In this paper, we discuss the newly introduced Sombor index, Super Sombor index, and its reduced version for specific networks. These networks include triangular mesh, enhanced mesh, rhenium trioxide lattice, and star of silicate network. We also give a graphical comparison of computed results for the abovementioned chemical network. For these groups of chemical networks, the analytical closed formulae are derived.



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