# Structural Analysis and Topological Characterization of Sudoku Nanosheet

Authors: Shahid Zaman; Mehwish Jalani; Asad Ullah; Ghulamullah Saeedi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The physical and biological properties of chemical compounds are modelled using chemical graph theory. The geometric structure of chemical compounds can be modelled using a variety of topological indices derived from graph theory. The chemical structures, physicochemical characteristics, and biological activities are predicted by the topological indices using the real numbers derived from the molecular compound. The topological index’s first use was to identify the physical characteristics of alkenes. A topological index is a molecular structure descriptor calculated from a chemical compound’s molecular graph describing its topology. When applied to a chemical compound’s molecular structure, it tells the theoretical properties. The chemical structure is studied as a graph, where elements are denoted as vertices, and chemical bonds are called edges. In this study, we have computed some novel topological indices named as modified neighborhood version of the forgotten topological indexFN∗, the neighborhood version of the first multiplicative Zagreb M1∗, the neighborhood version of the second Zagreb index M2∗, the neighborhood version of hyper-Zagreb index HMN, the Sambor topological index SOG, and the Sambor reduced topological index SOredG for the Sudoku nanosheet and derived formulas for them. Based on the derived formulas, the numerical results of the understudy nanosheet’s physical and chemical properties are investigated. Our computed results are undoubtedly helpful in understanding the topology of the understudy nanosheet. These computed indices have the best correlation with acentric factor and entropy; therefore, they are effective in QSPRs and QSARs analysis with complete accuracy.


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## Body

## 1. Introduction

The subject of intense research is the famous 2-dimensional nano-material graphene, the world’s [1–3] lightest 2D material with a hexagonal grid. Since their inventions, carbon nanotubes have gained a lot of attention [4, 5]. Carbon nanotubes (CNTs) are made of carbon using nanometer diameters. The one-wall carbon nanotubes are a bore in the variety of nanometers by Iijima and Ichihashi [4], and Bethune et al. [5] independently. The thickness of the 2D nanostructure varies from 1 to 100 nm [6]. These 2D nanosheets have interesting physical, electronic, biological, and chemical properties that are essential for various applications. Therefore, it is critical to predicting these nanostructures to attain discrimination and bring about the network topology, meliorate, and their physical characteristics. Chemical graph theory is used to represent better and characterise molecules to understand chemical compounds’ physical properties [7]. Graphs are mathematical structures that are used to model relationships. A Chemical graph is a basic calculable graph with atoms at the edge in an implicit system. In [8], Randic' gives the concept of finding a suitable path to reach the desire vertex.A TI describes a lot of prominent properties of some chemical compounds. These are essential in biological and chemical science and engineering. Physicochemical characteristics or theoretical molecular descriptors are used as forecasters in QSAR modelling [9–11], whilst the expression QSPR models as replication variables [12–15]. Wiener [16] introduced the TI concept while functioning on paraffin’s boiling point and set it as a path number. From both a practical and theoretical standpoint, it is the first and maximum considered TI. It is described as the total distance of the distances between all pairs of vertices in graph G, with more information available in ref [17]. Degree-based TI is often used in chemical network graph theory, mainly in chemistry. Gutman et al. established the premature topological indices and the Zagreb indices [18–21], which have been used to investigate in molecular difficulty, boiling point, and chirality.The discipline of graph theory is rapidly expanding and performing a vital function, cheminformatics, the investigation of various chemical formations and their physiochemical qualities that combine chemistry and mathematics, and information on technology. For example, Chemical network graph theory is the subsection of mathematics that deals with chemistry that uses the help of topological indices. Graph theory allows us to focus on the physiochemical aspects of the molecular graph (TIs). TIs are numerical numbers connected with various chemical structures in the molecular graph that forecast toxicological, organic, and physiochemical elements of chemical compounds present in the diagram. Furthermore, IT is for the isomorphic graphs to be the same. Period before, many researchers have functioned on TIs, see [22, 23]. The most common application of these TIs has been used most frequently to explore the evaluative construct movement and correlations in nature (QSAR and QSPR). In [24], Guo and Dong, gave the energy and analytical applications of chemical graphs. Li et al. [25] used some matrix techniques to obtain the eigenvalues and their multiplicity to find the kirchhoff index. In [26], Khan et al., established someproperties and applications of toxicities. Zaman et al. [27] determined the maximum H-index of Halin graph. The results on Cacti with maximal general sum-connectivity index are given in [28].The degree of a vertex is an essential invariant in examining a graph. Calculate the number of edges with that vertex as an endpoint to get the degree.The vertex degreev of any graph G expressed by dGv, that is detailed as the total amount of edges linked with a confers vertex v. Besides, we define(1)αGn=∑m∈NGndGm,where(2)NGn=m∈VG:mn∈EG.LetFN∗ denote the modified neighborhood version of the forgotten topological index and is defined as follows [18]:(3)FN∗G=∑mn∈EGαGm2+αGn2.LetM1∗ denote the first multiplicative Zagreb index, which is defined as follows [29]:(4)M1∗G=∑mn∈EGαGm+αGn.LetM2∗ denote the second Zagreb index, which is defined in [29] as follows:(5)M2∗G=∑mn∈EGαGm×αGn.The neighborhood form of the hyper-Zagreb index is expressed byHMN, moreover, represented as follows [30]:(6)HMNG=∑mn∈EGαGm+αGn2.The Sambor indexSOG and reduced Sambor index SOredG are introduced by Gutman in [31], which are defined as follows:(7)SOG=∑mn∈EGαGm2+αGn2,SOredG=∑mn∈EGαm−12+αn−12.LetAZI denote the Argumanted Zagreb topological index, which is defined by [32] as follows:(8)AZIG=∑mn∈EGαGm×αGnαGm+αGn−2.For a few essentials valuesn≥2, the Sudoku nanosheet (SKn×n) is a graphical description of the Sudoku puzzle in the form of a n×n grid chording of n2 cells, where every single cell is a precise graph of nodes with four vertices of degree 5, 4 vertices of degree 7, and 1 vertex of degree 8. Besides, Sudoku graphs VSKn×n=9n2, belong to the number of vertices and ESKn×n=n34n−6 are the number of edges, accordingly. For n=3, the Sudoku nanosheet SK3×3 is demonstrated in Figure 1. In Table 1, we ciphered the consequence for the linked number-based partition of edges for n ≥ 3 of the Sudoku graph SKn×n.Figure 1
The Sudoku nanosheetSKn×n for r=3.Table 1
Edge partition of Sudoku nanosheetSKn×n.
αiEdge partitionFrequencyα1α5,78α2α5,812α3α6,64r−4α4α4,78rα5α6,832r−40α6α7,74r4α7α7,820r2−36r+20α8α8,810r2−14r+4


### 1.1. Main Results on Sudoku NanosheetSKr×r

Theorem 1.
Assume thatG denotes the Sudoku nanosheet, then the Modified neighborhood version of forgotten topological index of G is(9)FN∗G=392r4+3540r2+1852r+144.Proof.
By definition ofFN∗ (G) and Table 1, we have(10)FN∗G=∑mn∈EGαGm2+αGn2=α5,725+49+α5,825+64+α6,636+36+α4,716+49+α6,836+64+α7,749+49+α7,849+64+α8,864+64=874+1289+4r−472+8r65+32r−40100+4r498+20r2−36r+20113+10r2−14r+4128=392r4+3540r2+1852r+144.Theorem 2.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of first multiplicative Zagreb index is M1∗(G) =56r4+460r2−180r+8Proof.
By using the definition ofM1∗ and Table 1, we can write as follows:(11)M1∗G=∑mn∈EGαGm+αGn=∑mn∈E5,7αGu+αGv+∑mn∈E5,8αGu+αGv+∑mn∈E6,6αGu+αGv+∑mn∈E4,7αGu+αGv+∑mn∈E6,8αGu+αGv+∑mn∈E7,7αGu+αGv+∑mn∈E7,8αGu+αGv+∑mn∈E8,8αGu+αGv=α5,75+7+α5,85+8+α6,66+6+α4,74+7+α6,86+8+α7,77+7+α7,87+8+α8,88+8=128+1312+124r−4+118r+1432r−40+144r4+1520r2−36r+20+1610r2−14r+4M1∗G=56r4+460r2−180r+8.Theorem 3.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of second Zagreb index is M2∗(G) =196r4+1760r2−1008r+72Proof.
According to the definition ofM2∗ and Table 1, we can write as follows:(12)M2∗G=∑mn∈EGαGm×αGn=∑mn∈E5,7αGm×αGn+∑mn∈E5,8αGm×αGn+∑mn∈E6,6αGm×αGn+∑mn∈E4,7αGm×αGn+∑mn∈E6,6αGm×αGn+∑mn∈E7,7αGm×αGn+∑mn∈E7,8αGm×αGn+∑mn∈E7,7αGm×αGn=α5,75×7+α5,85×8+α6,66×6+α4,74×7+α6,86×8+α7,77×7+α7,87×8+α8,88×8=280+480+144r−144+224r+1536r−1920+196r4+1120r2−2016+1120+640r2−896r+256M2∗G=196r4+1760r2−1008r+72.Theorem 4.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of hyper-Zagreb index is HMNG = 784r4+7060r2−3868r+288.Proof.
According to the definition ofHMN and Table 1, we can write as follows:(13)HMNG=∑mn∈EGαGm+αGn2,HMNG=∑mn∈E5,7αGm+αGn2+∑mn∈E5,8αGm+αGn2+∑mn∈E6,6αGm+αGn2+∑mn∈E4,7αGm+αGn2+∑mn∈E6,8αGm+αGn2+∑mn∈E7,7αGm+αGn2+∑mn∈E7,8αGm+αGn2+∑mn∈E8′8αGm+αGn2,HMNG=α5,75+72+α5,85+82+α6,66+62+α4,74+72+α6,86+82+α7,77+72+α7,87+82+α8,88+82=1152+2028+576r−576+968r+6272r−7840+784r4+4500r2−8100r+4500+2560r2−3584r+1024=784r4+7060r2−3868r+288.Theorem 5.
Assume thatG denotes the Sudoku nanosheet, then the sambor index is(14)SOG=39.59r4+325.73r2−122.631r+5.919.Proof.
According to the definition ofS0G and Table 1, we have(15)S0G=∑mn∈EGαGm2+αGn2=25+498+25+6412+36+364r−4+16+498r+36+6432r−40+49+494r4+49+6420r2−36r+20+64+6410r2−14r+4=748+8912+624r−4+658r+1032r−40+724r4+11320r2−36r+20+8210r2−14r+4S0G=39.59r4+325.73r2−122.631r+5.91.Theorem 6.
Assume thatG denotes the Sudoku nanosheet, then the reduced sambor index is S0redG=33.941r4+218.331r2−113.275r+6.049.Proof.
Through analysis of the definition ofSOredG as well as Table 1, we can write as follows:(16)S0redG=∑mn∈EGαm−12+αn−12+16+368+16+4912+25+254r−4+9+368r+25+4932r−40+36+364r4+36+4920r2−36r+20+49+4910r2−14r+4=528+6512+504r−4+458r+7432r−40+724r4+8520r2−36r+20+9810r2−14r+4=33.941r4+218.331r2−113.275r+6.049.Theorem 7.
Assume thatG denotes the Sudoku nanosheet, then the Argumented Zagreb Index is AZIG=272.33r4+2553.6r2−1738.966r+153.955Proof.
Through analysis of the definition ofAZIG as well as Table 1, we have(17)5×75+7−238+5×85+8−2312+6×66+6−234r−4+4×74+7−238r+6×86+8−2332r−40+7×77+7−234r4+7×87+8−2320r2−36r+20+8×88+8−2310r2−14r+4=343+576.979+186.624r−186.624+240.89r+2048r−2560+272.33r4+1598.6r2−2877.48+1598.6+955r2−1337r+382=272.33r4+2553.6r2−1738.966r+153.955.


## 1.1. Main Results on Sudoku NanosheetSKr×r

Theorem 1.
Assume thatG denotes the Sudoku nanosheet, then the Modified neighborhood version of forgotten topological index of G is(9)FN∗G=392r4+3540r2+1852r+144.Proof.
By definition ofFN∗ (G) and Table 1, we have(10)FN∗G=∑mn∈EGαGm2+αGn2=α5,725+49+α5,825+64+α6,636+36+α4,716+49+α6,836+64+α7,749+49+α7,849+64+α8,864+64=874+1289+4r−472+8r65+32r−40100+4r498+20r2−36r+20113+10r2−14r+4128=392r4+3540r2+1852r+144.Theorem 2.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of first multiplicative Zagreb index is M1∗(G) =56r4+460r2−180r+8Proof.
By using the definition ofM1∗ and Table 1, we can write as follows:(11)M1∗G=∑mn∈EGαGm+αGn=∑mn∈E5,7αGu+αGv+∑mn∈E5,8αGu+αGv+∑mn∈E6,6αGu+αGv+∑mn∈E4,7αGu+αGv+∑mn∈E6,8αGu+αGv+∑mn∈E7,7αGu+αGv+∑mn∈E7,8αGu+αGv+∑mn∈E8,8αGu+αGv=α5,75+7+α5,85+8+α6,66+6+α4,74+7+α6,86+8+α7,77+7+α7,87+8+α8,88+8=128+1312+124r−4+118r+1432r−40+144r4+1520r2−36r+20+1610r2−14r+4M1∗G=56r4+460r2−180r+8.Theorem 3.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of second Zagreb index is M2∗(G) =196r4+1760r2−1008r+72Proof.
According to the definition ofM2∗ and Table 1, we can write as follows:(12)M2∗G=∑mn∈EGαGm×αGn=∑mn∈E5,7αGm×αGn+∑mn∈E5,8αGm×αGn+∑mn∈E6,6αGm×αGn+∑mn∈E4,7αGm×αGn+∑mn∈E6,6αGm×αGn+∑mn∈E7,7αGm×αGn+∑mn∈E7,8αGm×αGn+∑mn∈E7,7αGm×αGn=α5,75×7+α5,85×8+α6,66×6+α4,74×7+α6,86×8+α7,77×7+α7,87×8+α8,88×8=280+480+144r−144+224r+1536r−1920+196r4+1120r2−2016+1120+640r2−896r+256M2∗G=196r4+1760r2−1008r+72.Theorem 4.
Assume thatG denotes the Sudoku nanosheet, then the Neighborhood version of hyper-Zagreb index is HMNG = 784r4+7060r2−3868r+288.Proof.
According to the definition ofHMN and Table 1, we can write as follows:(13)HMNG=∑mn∈EGαGm+αGn2,HMNG=∑mn∈E5,7αGm+αGn2+∑mn∈E5,8αGm+αGn2+∑mn∈E6,6αGm+αGn2+∑mn∈E4,7αGm+αGn2+∑mn∈E6,8αGm+αGn2+∑mn∈E7,7αGm+αGn2+∑mn∈E7,8αGm+αGn2+∑mn∈E8′8αGm+αGn2,HMNG=α5,75+72+α5,85+82+α6,66+62+α4,74+72+α6,86+82+α7,77+72+α7,87+82+α8,88+82=1152+2028+576r−576+968r+6272r−7840+784r4+4500r2−8100r+4500+2560r2−3584r+1024=784r4+7060r2−3868r+288.Theorem 5.
Assume thatG denotes the Sudoku nanosheet, then the sambor index is(14)SOG=39.59r4+325.73r2−122.631r+5.919.Proof.
According to the definition ofS0G and Table 1, we have(15)S0G=∑mn∈EGαGm2+αGn2=25+498+25+6412+36+364r−4+16+498r+36+6432r−40+49+494r4+49+6420r2−36r+20+64+6410r2−14r+4=748+8912+624r−4+658r+1032r−40+724r4+11320r2−36r+20+8210r2−14r+4S0G=39.59r4+325.73r2−122.631r+5.91.Theorem 6.
Assume thatG denotes the Sudoku nanosheet, then the reduced sambor index is S0redG=33.941r4+218.331r2−113.275r+6.049.Proof.
Through analysis of the definition ofSOredG as well as Table 1, we can write as follows:(16)S0redG=∑mn∈EGαm−12+αn−12+16+368+16+4912+25+254r−4+9+368r+25+4932r−40+36+364r4+36+4920r2−36r+20+49+4910r2−14r+4=528+6512+504r−4+458r+7432r−40+724r4+8520r2−36r+20+9810r2−14r+4=33.941r4+218.331r2−113.275r+6.049.Theorem 7.
Assume thatG denotes the Sudoku nanosheet, then the Argumented Zagreb Index is AZIG=272.33r4+2553.6r2−1738.966r+153.955Proof.
Through analysis of the definition ofAZIG as well as Table 1, we have(17)5×75+7−238+5×85+8−2312+6×66+6−234r−4+4×74+7−238r+6×86+8−2332r−40+7×77+7−234r4+7×87+8−2320r2−36r+20+8×88+8−2310r2−14r+4=343+576.979+186.624r−186.624+240.89r+2048r−2560+272.33r4+1598.6r2−2877.48+1598.6+955r2−1337r+382=272.33r4+2553.6r2−1738.966r+153.955.


## 2. Comparison and Concluding Remarks

This study ciphered a few novel topological indices for SK(r×r) and their formulas. The knowing topological indices are the modified neighborhood version of Forgotten topological index FN∗, the neighborhood version of the first multiplicative Zagreb M1∗, the neighborhood version of the second Zagreb index M2∗, the neighborhood version of hyper-Zagreb index HMN. The Sambor topological index SOG and the Sambor reduced topological index SOred(G). Our ciphered consequence is really beneficial in the apprehension of the topology and ameliorates the material characteristics of reserve SK(r×r). As previously stated, the beyond-ciphered indices FN∗, M1∗, M2∗,HMN,SO and SOred have the best tie-up with entropy and acentric factor; therefore, the particular dynamic in QSAR and QSPR anatomy with robust precision. Their ability to distinguish between isomer momentous as opposed to other degree-based indices is extraneous. Meanwhile, Table 2 and Figures 2–8 indicate that all the considered indices also increase when we increase. This implies that there is a direct proportion between r the considered indices. In Figure 9, we collate all the indices together, reflecting that HMN they increase more speedily than other indices. Similarly, SOredG it increases slower than other indices.Table 2
Numerical table corresponding with the structure of the Sudoku graph.
rFN∗GM1∗(G)M2∗(G)HMN(G)SOGSOredGAZIG1213434410204264248.608145.0461240.91921687223848232333361697.0171195.87911247.7033582008144287641157285776.3864380.42439978.1874149728209847437628848014862.11511735.141103772.17153243844560816153264744832279.76426111.074225505.375662450488064311400124730462305.05351173.851434589.43711018321557445498522201536110163.86291403.684766971.923818175202573849074643632448182032.231152095.3691265136.3079284212840306414195165681160285036.36239358.2861978101.99110425562460420821259928507608427252.609360116.392961424.295Figure 2
The correlation ofr and FN∗.Figure 3
The correlation ofr and M1∗.Figure 4
The correlation ofr and M2∗.Figure 5
The correlation ofr and HMN.Figure 6
The correlation ofr and SOG.Figure 7
The correlation ofr and SOredG.Figure 8
The correlation ofr and AZI.Figure 9
Correlation ofr with obtained indices.



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