# A View of Banhatti and Revan Indices in Chemical Graphs

Authors: Abid Mahboob; G. Muhiuddin; Imran Siddique; Sajid Mahboob Alam
Journal: Journal of Mathematics
 (2022)

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## Abstract

A special type of graph invariant called topological index is the collection of data on algebraic graphs and provides a mathematical way to understand chemical structural features. One of the driving factors behind the wide public attention according to these indices is their remarkable ability to correlate and predict the properties of a wide range of molecular species. Our concern is basically with the study of molecular structures, its shapes, geometries, number of atoms, vertices, bond length, and bond strength. It is very expensive to find the properties of compounds in laboratories due to different expensive apparatus and expensive rare material. It is also time consuming and requires an expert person to perform the experiments. So with the help of graphs and topological index, we want to give an easy approach to find the different characteristics of molecular structures with practical lab experiments. It is a mathematical and theoretical method to estimate the physicochemical properties. Many physicochemical features of a molecular compound can be predicted using topological indices. In this article, we will determine some topological invariants of silicon carbide SiC3–I[t, u] for all values of t and u.


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## Body

## 1. Introduction

Chemical graph theory is a field of mathematical chemistry that has a significant impact on the advancement of the chemical sciences. Physicochemical aspects of chemical compounds are frequently characterized in chemical science using molecular-based structural descriptors, also known as graph indices or topological indices; see [1, 2] for graph indices. A chemical graph is a graph in which chemical bonds are portrayed as edges and atoms of a molecule and are portrayed as vertices.Chemical theory is a branch of graph theory concerned with the discovery of topological indices of chemical graphs that are closely correlated with the chemical properties of chemical compounds. A topological index is a statistical indicator constructed empirically from the network structure. Topological indices have been proven to be effective for assessing correlations between the structure of a molecular compound and its physical characteristics or bioactivities [3–10]. Topological indices are a practical way to convert chemical composition into numerical values that can be utilized in QSPR and QSAR (quantitative-structure property relationship and quantitative-structure activity relationship, respectively) studies to correlate with physical attributes.In recent years, the application of graph invariant in QSPR and QSAR investigations has piqued desire. The study of topological indices is extremely important in nanotechnology and theoretical chemistry. Topological indices have been used in chemistry, physics, mathematics, informatics, biology, and other fields [11, 12]. Many topological descriptors were introduced in the latter decades of the 20th century to meet the needs of chemists [13, 14].The structure utilized in this manuscript is a special isomer of silicon carbide. Silicon carbide has a tetrahedral structure. It is an important gradient for host-guest reaction. It is used in bullet proof jackets, car breaks, LED lights, jewelry, and detectors. Revan and Banhatti indices have a very good correlation with many properties of SiC3–I [t, u].Kulli is an Indian mathematician who proposed the Banhatti indices by inspiring the Zagreb indices given by Milan Randic in 1972. Kulli proposed a series of papers on Banhatti indices and gave their different approaches like modified and hyper-Banhatti indices. These indices give excellent correlation with the characteristics of different chemical and nonchemical graphs.These indices are also helpful in the field of fuzzy mathematics for the study of derived networks [15, 16]. A digital world cannot be imagined without silicon. The discovery of silicon is the great revolution in electronics. Zhao and Zahid explored the structure of silicon with the help of Banhatti and Revan indices [17]. Kulli proposed many papers to study oxide networks, honeycomb networks, drugs, and antibiotic structures with the help of Revan and Banhatti indices [18–20].


### 1.1. Basic Definitions

In this paper, we indicate a simple connected graph byΓ. The shortest distance between ω and ν is Бω,υ. The degree of a vertex ω in graph Γ is denoted by Бω, an edge between the vertices ω and υ is denoted by e  =  ωυ, and the degree of an edge “e” is denoted by Бe, where Бe  =  Бω + Бυ– (2) and the maximum and minimum degree in a graph is represented by БΓandδΓ.We discuss some definitions, and then by utilizing these definitions we can calculate the values of topological indices.Kulli et al. introduced the first k-Banhatti index B1 (Γ) and second k-Banhatti index B2 (Γ) in 2016 [21], mathematically defined as follows:(1)B1Γ=∑ωυ∈EΓ.Бω+Бυ,B2Γ=∑ωυ∈EΓ.Бω×Бυ.In 2020, Kulli checked out the correlation coefficient for chloroquine and hydroxylchloroquine that is used in the pharmaceutical treatment of coronavirus disease 2019. So he mathematically gave the solution to the problem without doing any experiment in the lab. He just deeply studies the structure of medicine and gives information about it by using the Banhatti index [22]. It is the beauty of TIs that we can mathematically estimate the solution to a problem in any field related to graphs or molecular structure. Kulli et al. defined the modified form of first k-Banhatti index ɱB1 (Γ) and second k-Banhatti index ɱB2 (Γ) in 2018 [23]. Mathematically, these are determined as follows:(2)B1.mΓ=∑ωυ∈EΓ1Бω+Бe,B2.mΓ=∑ωυ∈EΓ1Бω×Бe.Tang and Abid utilize these indices to discuss molecular networks [24].In 2016, Kulli [25] proposed first k-hyper-Banhatti index ηB1 (Γ) and second k-hyper-Banhatti index ηB2 (Γ) and are defined as follows:(3)ηB1Γ=∑ωυ∈EΓБω+Бυ2andηB1Γ=∑ωυ∈EΓБω+Бυ2Recently in 2021, Zhao and Zahid give the comparison of Banhatti index, Revan index, and hyperindices for the silicon carbide.In 2017 Kulli [26] proposed first hyper-Revan indices ηŔ1(Γ) and second hyper-Revan indices ηŔ2 (Γ) and are defined as follows:(4)ηŔ1Γ=∑ωυ∈EΓrΓω+rΓυ2andηŔ1Γ=∑ωυ∈EΓrΓω×rΓυ2.The first Revan vertex index and third Revan index of a graphΓ can be defined as follows:(5)Ŕ1Γ=∑ωυ∈ErΓω2andŔ3Γ=∑ωυ∈ErΓω−rΓυ,where rΓω=БΓ+δΓ−Б(e) and ωυ means that the vertex ω and vertex υ are adjacent in Γ. We calculate B1(Γ), B2(Γ), ɱB1(Γ), ɱB2 (Γ), ηB1 (Γ), ηB2 (Γ), ηŔ1 (Γ), ηŔ2 (Γ), Ŕ1 (Γ), and Ŕ3 (Γ) of the nanostructure silicon carbide SiC3–I [t, u].


### 1.2. 2D Silicon CarbideSi2C3−It,u

Silicon carbide, commonly referred to as carborundum, is a semiconductor. Silicones are semiconductors that are used in the assembly of a variety of materials. It is an organism that is found in almost all of today’s electrical devices. Silicon, like carbon, has a two-dimensional allotrope with a honeycomb pattern, abbreviated as silicone. Untainted 2D carbon monolayer-graphene, untainted 2D silicon monolayer-silicon, and 2D silicon-carbon (SiC) monolayer can be considered as a tuneable compound. A great deal of effort have gone into predicting the most stable configurations of the SiC sheet. The cardinality of a vertex set SiC3–I [t, u] is 8tu and the cardinality of an edge set is 12tu-2t-3u. To construct the topological indices for this chemical composition, we divide the vertex set and edge set into subdivisions. Let t, u≥1. The vertex is divided into three groups that relying on thevertices of the degree. Graphical representation of 2D silicon-carbide SiC3–I[t, u] is shown in Figures 1–4.Figure 1
Unit cell of SIC3–I[t, u].Figure 2
SiC3–I[t, u] for t = 3 and u = 1.Figure 3
SiC3–I[t, u] for t = 3 and u = 2.Figure 4
SiC3–I[t, u] for t = 3 and u = 3.


### 1.3. Edge Partition

The silicon carbide structure has five different sets of edges. The first division of edges consist of just two edgesωυ, where Ƌ(ω) = 1 and Ƌ(υ) = 2. The second type of set is composed of one edge ωυ, where Ƌ(ω) = 1and Ƌ(υ) = 3. There are 3t+2u-3 edges in the third pack of edges ωυ, where Ƌ(ω) = Ƌ(υ) = 2. The fourth set of edges have 4t+8u-8 edges ωυ, where Ƌ(ω) = 3 and Ƌ(υ) = 2. The fifth parcel of edges is made up of 12ut−8t−13u+8 edges ωυ, where Ƌ(ω) = Ƌ(υ) = 3. Degrees of these edges are also given in Table 1.Table 1
Edge partitioning of SiC3–I [t, u].
Edges(Ƌ(ω), Ƌ(υ))FrequencyƋ(e)rΓ (ω)rΓ (υ)E1(1, 2)2132E2(1, 3)1231E3(2, 2)2t+2u−3222E4(3, 2)4t+8u−8312E5(3, 3)12tu−8t−13u+8411


### 1.4. Vertex Partition

The degree of a vertex is the number of edges attached to a vertex. The vertex division is degree-based and has three types of vertices, as given in Table2. By utilizing this table, we can find many TIs related to the vertex degree.Table 2
Division of vertices forSiC3−It,u.
Ƌ (ω)Total verticesrΓωω133ω24t+6u−62ω38tu−6u−4t+31


### 1.5. Techniques and Methods

We use various methods to calculate TIs, like combinatorial computation, the neighborhood degree counting technique, the vertex degree method, and the edge partition method. For the generalization and verification of calculations, we utilize Matlab. We use Mathematica and Maple for drawing 3D graphs. The Chem-sketch is helpful for drawing the chemical graphs of silicon. Special software named SPSS is used for correlation analysis. For 3D graphing, Graphing Calculator 3D and Math Grapher can also be used.


### 1.6. Fundamental Computations

In this section, we evaluate our numerical results and graphically show the behavior of all TIs with respect to different values of parameters. We use different techniques to find out our required results.Theorem 1.
LetΓ≅ SiC3-I t,u be the silicon carbide graph, then k-B indices can be determined as follows:(6)B1Γ=168tu−52t−78u+18andB2Γ=288tu–116t–284u+62.Proof.
Table1 shows the edge partitioning of SiC3−It,u based on edge degrees. The B1Γ of Γ is then calculated using Table 1 as follows:(7)=236+9+2t+2u−316+4t+8u−84+12tu−8t−13u+8=21×1+2×1+21×2+3×2+2t+2u−32×2+2×2+4t+8u−83×3+2×3+12t−8t−13u+83×4+3×4=25+8+2t+2u−38+4t+8u−811+12tu−8t−13u+814=10+8+16t+16u–24+44t+88u–88+168tu–112t–182u+112=168tu–52t–78u+18.
B2 (Γ) of Γ is described as follows: (8)B2Γ=∑e=ωυБω×Бe=∑e=ωυБω×Бe+Бυ×Бe=21×1+2×1+11×2+3×2+2t+2u−32×2+2×2+4t+8u−8××××××3×3+2×3+12t−8t−13u+83×4+3×4××××=23+8+2t+2u−38+4t+8u−815+12tu−8t−13u+824=6+8+16t+16u–24+60t+12u–120+288tu–192t–312u+192=288tu–116t–284u+62.Theorem 2.
LetΓ≅SiC3−It,u be the silicon carbide; then, (9)B1.ɱΓ=247tu+19105t+23105u+2068525andB2.ɱΓ=2tu+79t+1918u+10318.Proof.
The ɱB1 (Γ) is calculated using Table 1 as follows:(10)B1.ɱΓ=∑ωυ1Бω+Бe=∑e=ωυ1Бω+Бe+1Бυ+Бe=212+13+113+15+2t+2u−314+14+4t+8u−816+15+12tu−8t−13u+817+17+12tu−8t−13u+817+17=256+815+2t+2u−313+4t+8u−81130+12tu−8t−13u+827+12tu−8t−13u+827=106+815+22t+222u−32+4430t+8830u−8830+247tu−167t−267u+167=53+815+t+u−32+2215t+4415u−4415+247tu−167t−267u+167=247tu+19105t+23105u+2068525.
Now, we determine the equation ofɱB2 (Γ):(11)B2.ɱΓ=∑ωυ1Бω×Бe=∑e=ωυ1Бω×Бe+1Бυ×Бe=21+12+112+16+2t+2u−314+14+4t+8u−819+16+12tu−8t−13u+8112+112=232+23+2t+2u−312+4t+8u−8518+12tu−8t−13u+816+12tu−8t−13u+816=3+23+t+u−32+109t+209u−209+126tu−86t−136u+86=2tu+79t+1918u+10318.Theorem 3.
LetΓ≅SiC3−It,u be the silicon carbide graph; then,(12)ηB1Г=1176tu−476t−722u+260andηB2Г=3456tu−1772t−2744u+1322.Proof.
TheηB1ГΓ is calculated as follows:(13)ηBГ=∑e=ωυБω+Бυ2=∑e=ωυБω+Бe2+Бυ+Бe2=21+12+2+12+11+22+3+22+2t+2u−32+22+2+22++4t+8u−83+32+2+32+12tu–8t–13u+83+42+3+42=213+34+2t+2u−332+4t+8u−861+12tu−8t−13u+898=26+34+64t+64u−96+244t+488u−488+1176tu−784t−1274u+784=1176tu−476t−722u+260.
ηB2 (Γ) is determined as follows:(14)ηB2Γ=∑e=ωυБω×Бυ2=∑e=ωυБω×Бe2+Бυ×Бe2=21×12+2×12+11×22+3×22+2t+2u−32×22+2×22××××××4t+8u−83×32+2×32+12tu–8t–13u+83×42+3×42××××=25+40+2t+2u−332+4t+8u−8117+12tu−8t−13u+8288=10+40+64t+64u−96+468t+936u−936+3456tu−2304t−3744u+2304=3456tu−1772t−2744u+1322.Theorem 4.
LetΓ≅SiC3−It,u be the graph of silicon carbide; then,(15)ηR1′Γ=48tu+36t+52u−22andηR2′Γ=12tu+40t+51u+9.Proof.
LetΓ≅ SiC3–I [t, u] be the graph of silicon carbide. The first hyper-Revan indices ηR1 (Γ) is calculated as follows:(16)ηŔ1Γ=∑ωυϵErΓω+rΓυ2=23+22+13+12+2t+2u−32+22+4t+8u−81+22+12tu–8t–13u+81+12=225+16+2t+2u−316+4t+8u−89+12tu−8t−13u+84=50+16+32t+32u−48+36t+72u−72+48tu−32t−52u+32=48tu+36t+52u−22.
Second hyper-Revan indicesηŔ2 (Γ) is determined as follows: (17)ηŔ2Γ=∑ωυεErΓω×rΓυ2=2322+1312+2t+2u−3222+4t+8u−8122+12tu–8t–13u+81×12=236+9+2t+2u−316+4t+8u−84+12tu−8t−13u+81=72+9+32t+32u−48+16t+32u–32+12tu−8t−13u+8=12tu+40t+51u+9.Theorem 5.
LetΓ≅ SiC3–I [t,u] be the graph of silicon carbide; then,(18)Ŕ1Г=8tu+12t+18u+61Г=8tu+12t+18u+6andŔ3Г=4t+8u–4.Proof.
LetΓ≅ SiC3–I [t, u] be the graph of silicon carbide. The first Revan vertex index Ŕ1 (Γ) is calculated as follows:(19)Ŕ1Γ=∑ωυεErΓω2=∑υr3rΓω2+∑υr2rΓω2+∑υr1rΓω2=3324t+6u−622+8tu–6t–4u+312=39+16t+24u−24+8tu–6t–4u+3=27+16t+24u−24+8tu–6t–4u+3=8tu+12t+18u+6.
Moreover, from the definition ofŔ3 (Γ), we have(20)Ŕ3Γ=∑ωυϵEGrΓω−rΓυ=21+12+2t+2u−30+4t+8u−81+12tu–8t–13u+80.=2+2+4t+8u–8=4+4t+8u–8=4t+8u–4.


## 1.1. Basic Definitions

In this paper, we indicate a simple connected graph byΓ. The shortest distance between ω and ν is Бω,υ. The degree of a vertex ω in graph Γ is denoted by Бω, an edge between the vertices ω and υ is denoted by e  =  ωυ, and the degree of an edge “e” is denoted by Бe, where Бe  =  Бω + Бυ– (2) and the maximum and minimum degree in a graph is represented by БΓandδΓ.We discuss some definitions, and then by utilizing these definitions we can calculate the values of topological indices.Kulli et al. introduced the first k-Banhatti index B1 (Γ) and second k-Banhatti index B2 (Γ) in 2016 [21], mathematically defined as follows:(1)B1Γ=∑ωυ∈EΓ.Бω+Бυ,B2Γ=∑ωυ∈EΓ.Бω×Бυ.In 2020, Kulli checked out the correlation coefficient for chloroquine and hydroxylchloroquine that is used in the pharmaceutical treatment of coronavirus disease 2019. So he mathematically gave the solution to the problem without doing any experiment in the lab. He just deeply studies the structure of medicine and gives information about it by using the Banhatti index [22]. It is the beauty of TIs that we can mathematically estimate the solution to a problem in any field related to graphs or molecular structure. Kulli et al. defined the modified form of first k-Banhatti index ɱB1 (Γ) and second k-Banhatti index ɱB2 (Γ) in 2018 [23]. Mathematically, these are determined as follows:(2)B1.mΓ=∑ωυ∈EΓ1Бω+Бe,B2.mΓ=∑ωυ∈EΓ1Бω×Бe.Tang and Abid utilize these indices to discuss molecular networks [24].In 2016, Kulli [25] proposed first k-hyper-Banhatti index ηB1 (Γ) and second k-hyper-Banhatti index ηB2 (Γ) and are defined as follows:(3)ηB1Γ=∑ωυ∈EΓБω+Бυ2andηB1Γ=∑ωυ∈EΓБω+Бυ2Recently in 2021, Zhao and Zahid give the comparison of Banhatti index, Revan index, and hyperindices for the silicon carbide.In 2017 Kulli [26] proposed first hyper-Revan indices ηŔ1(Γ) and second hyper-Revan indices ηŔ2 (Γ) and are defined as follows:(4)ηŔ1Γ=∑ωυ∈EΓrΓω+rΓυ2andηŔ1Γ=∑ωυ∈EΓrΓω×rΓυ2.The first Revan vertex index and third Revan index of a graphΓ can be defined as follows:(5)Ŕ1Γ=∑ωυ∈ErΓω2andŔ3Γ=∑ωυ∈ErΓω−rΓυ,where rΓω=БΓ+δΓ−Б(e) and ωυ means that the vertex ω and vertex υ are adjacent in Γ. We calculate B1(Γ), B2(Γ), ɱB1(Γ), ɱB2 (Γ), ηB1 (Γ), ηB2 (Γ), ηŔ1 (Γ), ηŔ2 (Γ), Ŕ1 (Γ), and Ŕ3 (Γ) of the nanostructure silicon carbide SiC3–I [t, u].


## 1.2. 2D Silicon CarbideSi2C3−It,u

Silicon carbide, commonly referred to as carborundum, is a semiconductor. Silicones are semiconductors that are used in the assembly of a variety of materials. It is an organism that is found in almost all of today’s electrical devices. Silicon, like carbon, has a two-dimensional allotrope with a honeycomb pattern, abbreviated as silicone. Untainted 2D carbon monolayer-graphene, untainted 2D silicon monolayer-silicon, and 2D silicon-carbon (SiC) monolayer can be considered as a tuneable compound. A great deal of effort have gone into predicting the most stable configurations of the SiC sheet. The cardinality of a vertex set SiC3–I [t, u] is 8tu and the cardinality of an edge set is 12tu-2t-3u. To construct the topological indices for this chemical composition, we divide the vertex set and edge set into subdivisions. Let t, u≥1. The vertex is divided into three groups that relying on thevertices of the degree. Graphical representation of 2D silicon-carbide SiC3–I[t, u] is shown in Figures 1–4.Figure 1
Unit cell of SIC3–I[t, u].Figure 2
SiC3–I[t, u] for t = 3 and u = 1.Figure 3
SiC3–I[t, u] for t = 3 and u = 2.Figure 4
SiC3–I[t, u] for t = 3 and u = 3.


## 1.3. Edge Partition

The silicon carbide structure has five different sets of edges. The first division of edges consist of just two edgesωυ, where Ƌ(ω) = 1 and Ƌ(υ) = 2. The second type of set is composed of one edge ωυ, where Ƌ(ω) = 1and Ƌ(υ) = 3. There are 3t+2u-3 edges in the third pack of edges ωυ, where Ƌ(ω) = Ƌ(υ) = 2. The fourth set of edges have 4t+8u-8 edges ωυ, where Ƌ(ω) = 3 and Ƌ(υ) = 2. The fifth parcel of edges is made up of 12ut−8t−13u+8 edges ωυ, where Ƌ(ω) = Ƌ(υ) = 3. Degrees of these edges are also given in Table 1.Table 1
Edge partitioning of SiC3–I [t, u].
Edges(Ƌ(ω), Ƌ(υ))FrequencyƋ(e)rΓ (ω)rΓ (υ)E1(1, 2)2132E2(1, 3)1231E3(2, 2)2t+2u−3222E4(3, 2)4t+8u−8312E5(3, 3)12tu−8t−13u+8411


## 1.4. Vertex Partition

The degree of a vertex is the number of edges attached to a vertex. The vertex division is degree-based and has three types of vertices, as given in Table2. By utilizing this table, we can find many TIs related to the vertex degree.Table 2
Division of vertices forSiC3−It,u.
Ƌ (ω)Total verticesrΓωω133ω24t+6u−62ω38tu−6u−4t+31


## 1.5. Techniques and Methods

We use various methods to calculate TIs, like combinatorial computation, the neighborhood degree counting technique, the vertex degree method, and the edge partition method. For the generalization and verification of calculations, we utilize Matlab. We use Mathematica and Maple for drawing 3D graphs. The Chem-sketch is helpful for drawing the chemical graphs of silicon. Special software named SPSS is used for correlation analysis. For 3D graphing, Graphing Calculator 3D and Math Grapher can also be used.


## 1.6. Fundamental Computations

In this section, we evaluate our numerical results and graphically show the behavior of all TIs with respect to different values of parameters. We use different techniques to find out our required results.Theorem 1.
LetΓ≅ SiC3-I t,u be the silicon carbide graph, then k-B indices can be determined as follows:(6)B1Γ=168tu−52t−78u+18andB2Γ=288tu–116t–284u+62.Proof.
Table1 shows the edge partitioning of SiC3−It,u based on edge degrees. The B1Γ of Γ is then calculated using Table 1 as follows:(7)=236+9+2t+2u−316+4t+8u−84+12tu−8t−13u+8=21×1+2×1+21×2+3×2+2t+2u−32×2+2×2+4t+8u−83×3+2×3+12t−8t−13u+83×4+3×4=25+8+2t+2u−38+4t+8u−811+12tu−8t−13u+814=10+8+16t+16u–24+44t+88u–88+168tu–112t–182u+112=168tu–52t–78u+18.
B2 (Γ) of Γ is described as follows: (8)B2Γ=∑e=ωυБω×Бe=∑e=ωυБω×Бe+Бυ×Бe=21×1+2×1+11×2+3×2+2t+2u−32×2+2×2+4t+8u−8××××××3×3+2×3+12t−8t−13u+83×4+3×4××××=23+8+2t+2u−38+4t+8u−815+12tu−8t−13u+824=6+8+16t+16u–24+60t+12u–120+288tu–192t–312u+192=288tu–116t–284u+62.Theorem 2.
LetΓ≅SiC3−It,u be the silicon carbide; then, (9)B1.ɱΓ=247tu+19105t+23105u+2068525andB2.ɱΓ=2tu+79t+1918u+10318.Proof.
The ɱB1 (Γ) is calculated using Table 1 as follows:(10)B1.ɱΓ=∑ωυ1Бω+Бe=∑e=ωυ1Бω+Бe+1Бυ+Бe=212+13+113+15+2t+2u−314+14+4t+8u−816+15+12tu−8t−13u+817+17+12tu−8t−13u+817+17=256+815+2t+2u−313+4t+8u−81130+12tu−8t−13u+827+12tu−8t−13u+827=106+815+22t+222u−32+4430t+8830u−8830+247tu−167t−267u+167=53+815+t+u−32+2215t+4415u−4415+247tu−167t−267u+167=247tu+19105t+23105u+2068525.
Now, we determine the equation ofɱB2 (Γ):(11)B2.ɱΓ=∑ωυ1Бω×Бe=∑e=ωυ1Бω×Бe+1Бυ×Бe=21+12+112+16+2t+2u−314+14+4t+8u−819+16+12tu−8t−13u+8112+112=232+23+2t+2u−312+4t+8u−8518+12tu−8t−13u+816+12tu−8t−13u+816=3+23+t+u−32+109t+209u−209+126tu−86t−136u+86=2tu+79t+1918u+10318.Theorem 3.
LetΓ≅SiC3−It,u be the silicon carbide graph; then,(12)ηB1Г=1176tu−476t−722u+260andηB2Г=3456tu−1772t−2744u+1322.Proof.
TheηB1ГΓ is calculated as follows:(13)ηBГ=∑e=ωυБω+Бυ2=∑e=ωυБω+Бe2+Бυ+Бe2=21+12+2+12+11+22+3+22+2t+2u−32+22+2+22++4t+8u−83+32+2+32+12tu–8t–13u+83+42+3+42=213+34+2t+2u−332+4t+8u−861+12tu−8t−13u+898=26+34+64t+64u−96+244t+488u−488+1176tu−784t−1274u+784=1176tu−476t−722u+260.
ηB2 (Γ) is determined as follows:(14)ηB2Γ=∑e=ωυБω×Бυ2=∑e=ωυБω×Бe2+Бυ×Бe2=21×12+2×12+11×22+3×22+2t+2u−32×22+2×22××××××4t+8u−83×32+2×32+12tu–8t–13u+83×42+3×42××××=25+40+2t+2u−332+4t+8u−8117+12tu−8t−13u+8288=10+40+64t+64u−96+468t+936u−936+3456tu−2304t−3744u+2304=3456tu−1772t−2744u+1322.Theorem 4.
LetΓ≅SiC3−It,u be the graph of silicon carbide; then,(15)ηR1′Γ=48tu+36t+52u−22andηR2′Γ=12tu+40t+51u+9.Proof.
LetΓ≅ SiC3–I [t, u] be the graph of silicon carbide. The first hyper-Revan indices ηR1 (Γ) is calculated as follows:(16)ηŔ1Γ=∑ωυϵErΓω+rΓυ2=23+22+13+12+2t+2u−32+22+4t+8u−81+22+12tu–8t–13u+81+12=225+16+2t+2u−316+4t+8u−89+12tu−8t−13u+84=50+16+32t+32u−48+36t+72u−72+48tu−32t−52u+32=48tu+36t+52u−22.
Second hyper-Revan indicesηŔ2 (Γ) is determined as follows: (17)ηŔ2Γ=∑ωυεErΓω×rΓυ2=2322+1312+2t+2u−3222+4t+8u−8122+12tu–8t–13u+81×12=236+9+2t+2u−316+4t+8u−84+12tu−8t−13u+81=72+9+32t+32u−48+16t+32u–32+12tu−8t−13u+8=12tu+40t+51u+9.Theorem 5.
LetΓ≅ SiC3–I [t,u] be the graph of silicon carbide; then,(18)Ŕ1Г=8tu+12t+18u+61Г=8tu+12t+18u+6andŔ3Г=4t+8u–4.Proof.
LetΓ≅ SiC3–I [t, u] be the graph of silicon carbide. The first Revan vertex index Ŕ1 (Γ) is calculated as follows:(19)Ŕ1Γ=∑ωυεErΓω2=∑υr3rΓω2+∑υr2rΓω2+∑υr1rΓω2=3324t+6u−622+8tu–6t–4u+312=39+16t+24u−24+8tu–6t–4u+3=27+16t+24u−24+8tu–6t–4u+3=8tu+12t+18u+6.
Moreover, from the definition ofŔ3 (Γ), we have(20)Ŕ3Γ=∑ωυϵEGrΓω−rΓυ=21+12+2t+2u−30+4t+8u−81+12tu–8t–13u+80.=2+2+4t+8u–8=4+4t+8u–8=4t+8u–4.


## 2. Numerical and Graphical Representation

The numerical results of the above calculated topological descriptors for SiC3–I [t, u] are represented in this section. The numerical representation are shown in Table 3. A diagram shows a system of relationships between variable quantities using dots to represent the points of the system or using lines to represent the relations of continuous variables. There are many types of graphs according to the nature of data.Table 3
Numerical analysis of all TIs forSiC3−It,u.
Banhatti indices(t, u) = [1, 1](t, u) = [2, 2](t, u) = [3, 3](t, u) = [4, 4](t, u) = [5, 5]B1 (Γ)56430114021863568B2 (Γ),50614175434705762ɱB1 (Γ)7.7618.463660.3991.66ɱB2 (Γ)9.5617.429.2349.2864.89ηB1 (Γ)238256872501428423670ηB2 (Γ)2626114188783855465142ηŔ1 (Γ)11434667410981618ηŔ2 (Γ)112239390565764Ŕ1 (Γ)4498168254356Ŕ3 (Γ)820324456The graphs used in this article are 3D because of two variables. The variation in data can be checked out by just discussing numerical values, but for a clear comparison of different TIs, graphs are the best way. Graphs are easy to understand, and so it clearly saves time, but we must have enough knowledge about variables and slopes to use them. Using various values oft and u, we produced numerical tables/results, as shown in Figures 5–9. The values of all TIs increase when we increase the values of their parameters. The rate of variation of the first and second k-B-index is higher than that of the other TIs. These indices have a correlation with many properties in networking and chemistry. The computed topological descriptors increase in proportion to the rise in t and u.Figure 5
B1(Γ) and B2(Γ).Figure 6
ɱB1(Γ) and ɱB2 (Γ).Figure 7
ηB1(Γ) and ηB2(Γ).Figure 8
ηŔ1(Γ) and ηŔ2 (Γ).Figure 9
Ŕ1 (Γ) and Ŕ3 (Γ).Graphical representation of above calculated topological indices. These graphs show the variation in the values of TIs with the change of input parameters.


## 3. Applications

The Banhatti and Revan indices have a very good correlation with different physical and chemical properties of different structures. Asthma drugs are also useful for the treatment of COVID-19. The first Banhatti index has a correlation coefficientr = 0.974 with molar volume, and the second Banhatti index is correlated with the boiling point (r = 0.9192) and enthalpy (r = 0.9125) of these drugs. Hyper-Revan indices are effective to describe the molar volume of asthma drugs. Both these indices have a correlation with the different properties of silicon carbide structures discussed in this manuscript. These indices also tell us about the structural properties of linear alkanes.


## 4. Conclusion

This analysis delves into the multiplicative and degree-based topological indices for SiC3–I [t, u]. Silicon carbide is a compulsory part of almost every electronic device. Due to the great applications and functions, we try to understand the structure of it. To explore the structure, we applied many indices and saw the correlation of these indices with many important properties of SiC. In this article, we evaluate the B1 (Γ), B2(Γ), ɱB1(Γ), ɱB2 (Γ), ηB1 (Γ), ηB2 (Γ), ηŔ1 (Γ), ηŔ2 (Γ), Ŕ1 (Γ), and Ŕ3 (Γ) for Silicon Carbide SiC3–I [t, u]. Our discoveries could assist the understanding of silicon-physical carbon's properties, chemical stability, and bioactivities.



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*Source: 5680712-2022-07-31.xml*