# On Generalized Topological Indices of Silicon-Carbon

Authors: Xing-Long Wang; Jia-Bao Liu; Akbar Jahanbani; Muhammad Kamran Siddiqui; Nader Jafari Rad; Roslan Hasni
Journal: Journal of Mathematics
 (2020)

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## Abstract

LetGbe a graph with n vertices and Γu be the degree of its u-th vertex (Γx is the degree of u). In this article, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−Ip,q,Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q.


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## Body

## 1. Introduction

Mathematical chemistry is a branch of theoretical chemistry that discusses about the molecular structure by mathematical methods without necessarily referring to quantum mechanics. Molecular descriptors play a significant role in mathematical chemistry especially in QSPR/QSAR investigations [1]. A chemical structure can be represented by using graph theory, where vertices denote atoms and edges denote chemical bonds. Chemical graph theory is a branch of mathematical chemistry that it is a subject that connects mathematics, chemistry, and graph theory and solves problems arising in chemistry mathematically (for more details you can see [2–8]).In chemical graph theory, a graph of molecule is a simple connected graph, in which atoms and chemical bonds are represented by vertices and edges, respectively. A graph is connected if there is a connection between any pair of vertices. Among them, special place is reserved for so-called topological descriptors or topological indices. Actually, topological indices are numeric quantities that tell us about the whole structure of graph. The topological indices are useful in the prediction of physicochemical properties and the bioactivity of the chemical compounds [9–11].The topological indices of 2-dimensional silicon-carbons are computed in [12], in [13], Kwunet al. On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3−Ip,q and Si2C3−IIp,q, in [14], Imran et al.. On Topological Properties of Symmetric Chemical Structures in [15], Idrees et al. Molecular Descriptors of Benzenoid System, in [16], Kulli. F-indices of Chemical Networks, in [6], in [17], Gao et al. the Redefined first, second and third Zagreb Indices of Titania Nanotubes TiO2m,n and in [18], Kang et al. computed the topological indices of 2-dimensional silicon-carbon. For more details, see [19–23].Bearing this in mind, it seems to be purposeful to compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index of Si2C3−Ip,q,Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q.Throughout this paper, all graphs will be assumed simple that is without loops, multiple, or directed edges. LetG=n,m be a simple graph with vertex set VG=v1,v2,v3,…,vn and edge set EG,EG=m. Also, let di be the degree of vertex vi in graph G, for i=1,2,…,n. The concept of valence in chemistry and the concept of degree in a graph are somehow closely related. For details on bases of graph theory, we refer to the book [24]. If two vertices u and v of the graph G are adjacent, then the edge connecting them will be denoted by uv. The number of first neighbors of the vertex u∈VG is its degree and will be denoted by Γu.In [25], the authors defined a new index, named generalization of Zagreb index:(1)Mα,β=Mα,βG=∑uv∈EGΓu×ΓvαΓu+Γvβ,where α and β are arbitrary real numbers. Few years later, the same index was proposed in [26] under the name second Gourava index, obtained as a special case of the generalized Zagreb index Mr,s introduced in [27]:(2)Mr,s=Mr,sG=∑uv∈EGΓur×Γvs+Γus×Γvr.In [28], Ghobadi et al. defined the hyper F-index or the first hyper F-index of a graph G as(3)HF1G=∑uv∈EGΓu2+Γv22.In [16], the second hyper F-index of a graph is defined as(4)HF2G=∑uv∈EGΓu2×Γv22.In [16], the author introduced the sum connectivity F-index and the product connectivity F-index of a graph G, defined as(5)SFG=∑uv∈EG1Γu2+Γv2,PFG=∑uv∈EG1Γu2×Γv2.The concept of silicon carbide was introduced by an American scientist in 1891. But nowadays, we can produce silicon carbide artificially by silica and carbon. Till 1929, silicon carbide was known as the hardest material on Earth. Here, we will find out the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index.This paper is organized as follows. In Section2, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−Ip,q. In Section 3, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−IIp,q. In Section 4, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−IIIp,q. In Section 5, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of SiC3−IIIp,q.For more details about these indices, see [12, 13, 29–39].


## 2. Results for Silicon-CarbonSi2C3−Ip,q

In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−Ip,q. In Figure 1, one unit of Si2C3−I is shown. Molecular graph of Si2C3−I is shown in Figure 2, in which p denotes the number of cells attached in a single row and q denotes the number of total rows where each row contains p cells. In Figures 3 and 4, we demonstrate how cells are connected in one row (chain) and how one row is connected to another row. In Figures 1–4, carbon atoms are shown as brown, and silicon atoms Si are shown as blue.Figure 1
Unit cell ofSi2C3−Ip,q.Figure 2
Sheet ofSi2C3−I4,3.Figure 3
Sheet ofSi2C3−I4,1.Figure 4
Sheet ofSi2C3−I4,2.Remark 1 (see [12]).
The graphSi2C3−Ip,q contains 10pq vertices and 15pq−2p−3q edges.
We start by proving the carbon nanocones for the redefined Zagreb indices.Theorem 1.
LetSi2C3−Ip,q be the silicon carbide. Then,(6)Mα,βSi2C3−Ip,q=9α×15pq−9p−13q+76β+6α×6p+8q−95β+p+2q×4α−β+2α3β+3α4β,Mr,sSi2C3−Ip,q=6p+8q−9×2r×3s+2s×3r+30pq−18p−26q+14×3s+r+2p+4q×2s+r+2r+2s+3r+3s.Proof.
Consider the graph silicon carbideSi2C3−Ip,q. By Remark 1, the graph Si2C3−Ip,q contains 10pq vertices and 15pq−2p−3q edges. From the graph of Si2C3−Ip,q silicon carbide, we can see that there are three partitions, V1=v∈VSi2C3−Ip,qΓv=1, V2=v∈VSi2C3−Ip,qΓv=2, and V3=v∈VSi2C3−Ip,qΓv=3. The edge set of the Si2C3−Ip,q can be partitioned as follows:(7)E1=e=uv∈ESi2C3−Ip,qΓu=1andΓv=2,E2=e=uv∈ESi2C3−Ip,qΓu=1andΓv=3,E3=e=uv∈ESi2C3−Ip,qΓu=2andΓv=2,E4=e=uv∈ESi2C3−Ip,qΓu=2andΓv=3,E5=e=uv∈ESi2C3−Ip,qΓu=3andΓv=3.From the molecular graph of Si2C3−Ip,q, we can observe that E1=1,E2=1,E3=p+2q,E4=6p−1+8q−1, and E3=15pq−9p−13q+7. Thus, by definition generalization Zagreb index of Si2C3−Ip,q, we have(8)Mα,βSi2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓu×ΓvαΓu+Γvβ=∑uv∈E1Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E2Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E3Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E4Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E5Si2C3−Ip,qΓu×ΓvαΓu+Γvβ=1×2α1+2β+1×3α1+3β+p+2q×2×2α2+2β+6p−1+8q−1×2×3α2+3β+15pq−9p−13q+7×3×3α3+3β=9α×15pq−9p−13q+76β+6α×6p+8q−95β+p+2q×4α−β+2α3β+3α4β.which is the required (18) result.
By definition of the generalized Zagreb index ofSi2C3−Ip,q, we have(9)Mr,sSi2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓur×Γvs+Γus×Γvr=∑uv∈E1Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E2Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E3Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E4Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E5Si2C3−Ip,qΓur×Γvs+Γus×Γvr=1r×2s+1s×2r+1r×3s+1s×3r+p+2q×2r×2s+2s×2r+6p−1+8q−1×2r×3s+2s×3r+15pq−9p−13q+7×3r×3s+3s×3r=6p+8q−9×2r×3s+2s×3r+30pq−18p−26q+14×3s+r+2p+4q×2s+r+2r+2s+3r+3s.which is the required (19) result.Theorem 2.
LetSi2C3−Ip,q be the silicon carbide. Then,(10)HF1Si2C3−Ip,q=4860pq−1838p−2732q+872,(11)HF2Si2C3−Ip,q=98415pq−51017p−74413q+34360.Proof.
Consider the graph silicon carbideSi2C3−Ip,q. By Remark 1, the graph Si2C3−Ip,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the first hyper F-index of Si2C3−Ip,q, we have(12)HF1Si2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓu2+Γv22=∑uv∈E1Si2C3−Ip,qΓu2+Γv22+∑uv∈E2Si2C3−Ip,qΓu2+Γv22+∑uv∈E3Si2C3−Ip,qΓu2+Γv22+∑uv∈E4Si2C3−Ip,qΓu2+Γv22+∑uv∈E5Si2C3−Ip,qΓu2+Γv22=12+222+12+322+p+2q×22+222+6p−1+8q−1×22+322+15pq−9p−13q+7×32+322=4860pq−1838p−2732q+872.which is the required (10) result.
By definition of the second hyper F-index ofSi2C3−Ip,q, we have(13)HF2Si2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓu2+Γv22=∑uv∈E1Si2C3−Ip,qΓu2+Γv22+∑uv∈E2Si2C3−Ip,qΓu2+Γv22+∑uv∈E3Si2C3−Ip,qΓu2+Γv22+∑uv∈E4Si2C3−Ip,qΓu2+Γv22+∑uv∈E5Si2C3−Ip,qΓu2+Γv22=12×222+12×322+p+2q×22×222+6p−1+8q−1×22×322+15pq−9p−13q+7×32×322=98415pq−51017p−74413q+34360.which is the required (11) result.Theorem 3.
LetSi2C3−Ip,q be the silicon carbide. Then,(14)SFSi2C3−Ip,q=2×5pq2−3p2−13q6+76+2×pq4+q2+13×6p+8q−913+q2+55+1010,(15)PFSi2C3−Ip,q=9kn24−27kn20+k10.Proof.
Consider the graph silicon carbideSi2C3−Ip,q. By Remark 1, the graph Si2C3−Ip,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the sum connectivity F-index of Si2C3−Ip,q, we have(16)SFSi2C3−Ip,q=∑uv∈ESi2C3−Ip,q1Γu2+Γv2=∑uv∈E1Si2C3−Ip,q1Γu2+Γv2+∑uv∈E2Si2C3−Ip,q1Γu2+Γv2+∑uv∈E3Si2C3−Ip,q1Γu2+Γv2+∑uv∈E4Si2C3−Ip,q1Γu2+Γv2+∑uv∈E5Si2C3−Ip,q1Γu2+Γv2=112+22+112+32+p+2q×122+22+6p−1+8q−1×122+32+15pq−9p−13q+7×132+32=2×5pq2−3p2−13q6+76+2×pq4+q2+13×6p+8q−913+q2+55+1010.which is the required (14) result.
By definition of product connectivity F-index ofSi2C3−Ip,q, we have(17)PFSi2C3−Ip,q=∑uv∈ESi2C3−Ip,q1Γu2+Γv2=∑uv∈E1Si2C3−Ip,q1Γu2+Γv2+∑uv∈E2Si2C3−Ip,q1Γu2+Γv2+∑uv∈E3Si2C3−Ip,q1Γu2+Γv2+∑uv∈E4Si2C3−Ip,q1Γu2+Γv2+∑uv∈E5Si2C3−Ip,q1Γu2+Γv2=112×22+112×32+p+2q×122×22+6p−1+8q−1×122×32+15pq−9p−13q+7×132×32=5pq3+p4+7q18+19.which is the required (15) result.


## 3. Results for Silicon-CarbonSi2C3−IIp,q

In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−IIp,q.In Figure 5, one unit of Si2C3−II is given. By connecting p cells in a row and then connecting q rows where each row contains p cells, we get molecular graph of Si2C3−II. The molecular graph of Si2C3−II is shown in Figure 6 for p=3 and q=4. Figures 7 and 8 demonstrate how cells are connected in a row (chain) and how a row is connected to another row. We will use Si2C3−IIp,q to represent this molecular graph.Figure 5
One unit ofSi2C3−IIp,q.Figure 6
Sheet ofSi2C3−II3,3.Figure 7
Sheet ofSi2C3−II5,1.Figure 8
Sheet ofSi2C3−II5,2.Remark 2 (see [12]).
The graphSi2C3−IIp,q contains 10pq vertices and 15pq−3p−3q edges.
We start by proving the silicon carbide for the generalization of Zagreb index.Theorem 4.
LetSi2C3−IIp,q be the silicon carbide. Then,(18)Mα,βSi2C3−IIp,q=9α×15pq−9p−13q+76β+6α×6p+8q−95β+p+2q×4α−β+2α3β+3α4β,(19)Mr,sSi2C3−IIp,q=6p+8q−9×2r×3s+2s×3r+30pq−18p−26q+14×3s+r+2p+4q×2s+r+2r+2s+3r+3s.Proof.
Consider the graph silicon carbideSi2C3−IIp,q. By Remark 2, the graph Si2C3−IIp,q contains 10pq vertices and 15pq−2p−3q edges. From the graph of Si2C3−IIp,q silicon carbide, we can see that there are three partitions, V1=v∈VSi2C3−IIp,qΓv=1, V2=v∈VSi2C3−IIp,qΓv=2, and V3=v∈VSi2C3−IIp,qΓv=3. The edge set of the Si2C3−IIp,q can be partitioned as follows:(20)E1=e=uv∈ESi2C3−IIp,qΓu=1andΓv=2,E2=e=uv∈ESi2C3−IIp,qΓu=1andΓv=3,E3=e=uv∈ESi2C3−IIp,qΓu=2andΓv=2,E4=e=uv∈ESi2C3−IIp,qΓu=2andΓv=3,E5=e=uv∈ESi2C3−IIp,qΓu=3andΓv=3.From the molecular graph of Si2C3−IIp,q, we can observe that E1=2,E2=1,E3=2p+2q,E4=8p+8q−14, and E3=15pq−13p−13q+11.
Thus, by definition generalization ofSi2C3−IIp,q, we have(21)Mα,βSi2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓu×ΓvαΓu+Γvβ=∑uv∈E1Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E2Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E3Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E4Si2C3−Ip,qΓu×ΓvαΓu+Γvβ+∑uv∈E5Si2C3−Ip,qΓu×ΓvαΓu+Γvβ=2×1×2α1+2β+1×3α1+3β+2p+2q×2×2α2+2β+8p+8q−14×2×3α2+3β+15pq−13p−13q+11×3×3α3+3β=9α×15pq−13p−13q+116β+6α×8p+8q−145β+2p+2q×4α−β+2×2α3β+3α4β.which is the required (18) result.
By definition of the generalized Zagreb index ofSi2C3−IIp,q, we have(22)Mr,sSi2C3−Ip,q=∑uv∈ESi2C3−Ip,qΓur×Γvs+Γus×Γvr=∑uv∈E1Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E2Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E3Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E4Si2C3−Ip,qΓur×Γvs+Γus×Γvr+∑uv∈E5Si2C3−Ip,qΓur×Γvs+Γus×Γvr=2×1r×2s+1s×2r+1r×3s+1s×3r+2p+2q×2r×2s+2s×2r+8p+8q−14×2r×3s+2s×3r+15pq−13p−13q+11×3r×3s+3s×3r=8p−6×2r×3s+2s×3r+30pq−26p−26q+22×3s+r+4p+4q×2s+r+2×2r+2×2s+3r+3s.which is the required (19) result.Theorem 5.
LetSi2C3−IIp,q be the silicon carbide. Then,(23)HF1Si2C3−IIp,q=4860pq−2732p−2732q+1348,(24)HF2Si2C3−IIp,q=98415pq−74413p−74413q+54140.Proof.
Consider the graph silicon carbideSi2C3−IIp,q. By Remark 2, the graph Si2C3−IIp,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the first hyper F-index of Si2C3−IIp,q, we have(25)HF1Si2C3−IIp,q=∑uv∈ESi2C3−IIp,qΓu2+Γv22=∑uv∈E1Si2C3−IIp,qΓu2+Γv22+∑uv∈E2Si2C3−IIp,qΓu2+Γv22+∑uv∈E3Si2C3−IIp,qΓu2+Γv22+∑uv∈E4Si2C3−IIp,qΓu2+Γv22+∑uv∈E5Si2C3−IIp,qΓu2+Γv22=2×12+222+12+322+2p+2q×22+222+8p+8q−14×22+322+15pq−13p−13q+11×32+322=4860pq−2732p−2732q+1348.which is the required (23) result.
By definition of the second hyper F-index ofSi2C3−IIp,q, we have(26)HF2Si2C3−IIp,q=∑uv∈ESi2C3−IIp,qΓu2×Γv22=∑uv∈E1Si2C3−IIp,qΓu2×Γv22+∑uv∈E2Si2C3−IIp,qΓu2×Γv22+∑uv∈E3Si2C3−IIp,qΓu2×Γv22+∑uv∈E4Si2C3−IIp,qΓu2×Γv22+∑uv∈E5Si2C3−IIp,qΓu2×Γv22=2×12×222+12×322+2p+2q×22×222+8p+8q−14×22×322+15pq−13p−13q+11×32×322=98415pq−74413p−74413q+54140.which is the required (24) result.Theorem 6.
LetSi2C3−IIp,q be the silicon carbide. Then,(27)SFSi2C3−IIp,q=2×5pq2−13p6−13q6+116+2×p+q2+13×8p+8q−1413+2×55+1010,(28)PFSi2C3−IIp,q=5pq3+7p18+7q18+29.Proof.
Consider the graph silicon carbideSi2C3−IIp,q. By Remark 2, the graph Si2C3−IIp,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the sum connectivity F-index of Si2C3−IIp,q, we have(29)SFSi2C3−IIp,q=∑uv∈ESi2C3−IIp,q1Γu2+Γv2=∑uv∈E1Si2C3−IIp,q1Γu2+Γv2+∑uv∈E2Si2C3−IIp,q1Γu2+Γv2+∑uv∈E3Si2C3−IIp,q1Γu2+Γv2+∑uv∈E4Si2C3−IIp,q1Γu2+Γv2+∑uv∈E5Si2C3−IIp,q1Γu2+Γv2=2×112+22+112+32+2p+2q×122+22+8p+8q−14×122+32+15pq−13p−13q+11×132+32=2×5pq2−13p6−13q6+116+2×p+q2+13×8p+8q−1413+2×55+1010.which is the required (27) result.
By definition of product connectivity F-index ofSi2C3−IIp,q, we have(30)PFSi2C3−IIp,q=∑uv∈ESi2C3−IIp,q1Γu2×Γv2=∑uv∈E1Si2C3−IIp,q1Γu2×Γv2+∑uv∈E2Si2C3−IIp,q1Γu2×Γv2+∑uv∈E3Si2C3−IIp,q1Γu2×Γv2+∑uv∈E4Si2C3−IIp,q1Γu2×Γv2+∑uv∈E5Si2C3−IIp,q1Γu2×Γv2=2×112×22+112×32+2p+2q×122×22+8p+8q−14×122×32+15pq−13p−13q+11×132×32=5pq3+7p18+7q18+29.which is the required (28) result.


## 4. Results for Silicon-CarbonSi2C3−IIIp,q

In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−IIIp,q. The 2D silicon-carbon Si−C single layers can be seen as configurable (or tunable) materials between the pure 2D carbon single-layer graphene and the pure 2D silicon single-layer silicene. Lots of attempts have been conducted trying anticipating the most stable structures of the SiC sheet (for more details, see [40, 41]).The2D molecular graph of silicon carbide Si2C3−IIIp,q is given in Figure 9, where carbon atom C is shown in brown color and silicon atom Si is shown in blue color (for more details, see [42]). In Figure 10, we gave a demonstration how the cells connect in a row (chain) and how one row connects to another row; red lines (edges) show the connection between the unit cell in a chain and green lines (edges) connect the upper and lower rows (chains). We will denote this molecular graph by Si2C3−IIIp,q.Figure 9
2
D structure of Si2C3−IIIp,q: (a) unit cell of Si2C3−Ip,q and (b) Si2C3−I5,4.
(a)
(b)Figure 10
2
D structure of Si2C3−IIIp,q: (a)Si2C3−I5,1 and (b) Si2C3−I5,2.
(a)
(b)Remark 3 (see [13]).
The graphSi2C3−Ip,q contains 10pq vertices and 15pq−2p−3q edges.
We start by proving the carbon nanocones for the redefined Zagreb indices.Theorem 7.
LetSi2C3−IIIp,q be the silicon carbide. Then,(31)Mα,βSi2C3−IIIp,q=9α×15pq−10p−13q+86β+6α×8p+8q−125β+4+4q×4α−β+2×3α4β,(32)Mr,sSi2C3−IIIp,q=8p+8q−12×2r×3s+2s×3r+30pq−20p−26q+16×3s+r+2+2q×4s+r+2×3r+2×3s.Proof.
Consider the graph silicon carbideSi2C3−IIIp,q. By Remark 3, the graph Si2C3−IIIp,q contains 10pq vertices and 15pq−2p−3q edges. From the graph of Si2C3−IIIp,q silicon carbide, we can see that there are three partitions, V1=v∈VSi2C3−IIIp,qΓv=1, V2=v∈VSi2C3−IIIp,qΓv=2, and V3=v∈VSi2C3−IIIp,qΓv=3. The edge set of the Si2C3−IIIp,q can be partitioned as follows:(33)E1=e=uv∈ESi2C3−IIIp,qΓu=1andΓv=3,E2=e=uv∈ESi2C3−IIIp,qΓu=2andΓv=2,E3=e=uv∈ESi2C3−IIIp,qΓu=2andΓv=3,E4=e=uv∈ESi2C3−IIIp,qΓu=3andΓv=3.From the molecular graph of Si2C3−IIIp,q, we can observe that E1=2,E2=p+2q,E3=8p+8q−12, and E4=15pq−10p−13q+8.
Thus, by definition generalization ofSi2C3−IIIp,q, we have(34)Mα,βSi2C3−IIIp,q=∑uv∈ESi2C3−IIIp,qΓu×ΓvαΓu+Γvβ=∑uv∈E1Si2C3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E2Si2C3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E3Si2C3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E4Si2C3−IIIp,qΓu×ΓvαΓu+Γvβ=2×1×3α1+3β+p+2q×2×2α2+2β+8p+8q−12×2×3α2+3β+15pq−10p−13q+8×3×3α3+3β=9α×15pq−10p−13q+86β+6α×8p+8q−125β+2+2q×4α−β+2×3α4β.which is the required (31) result.
By definition of the generalized Zagreb index ofSi2C3−IIIp,q, we have(35)Mr,sSi2C3−IIIp,q=∑uv∈ESi2C3−IIIp,qΓur×Γvs+Γus×Γvr=∑uv∈E1Si2C3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E2Si2C3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E3Si2C3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E4Si2C3−IIIp,qΓur×Γvs+Γus×Γvr=2×1r×3s+1s×3r+p+2q×2r×2s+2s×2r+8p+8q−12×2r×3s+2s×3r+15pq−10p−13q+8×3r×3s+3s×3r=8p+8q−12×2r×3s+2s×3r+30pq−20p−26q+16×3s+r+4+4q×2s+r+2×3r+2×3s.which is the required (32) result.Theorem 8.
LetSi2C3−IIIp,q be the silicon carbide. Then,(36)HF1Si2C3−IIIp,q=4860pq−2732p−2732q+1348,(37)HF2Si2C3−IIIp,q=98415pq−74413p−74413q+54140.Proof.
Consider the graph silicon carbideSi2C3−IIIp,q. By Remark 3, the graph Si2C3−IIIp,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the first hyper F-index of Si2C3−IIIp,q, we have(38)HF1Si2C3−IIIp,q=∑uv∈ESi2C3−IIIp,qΓu2+Γv22=∑uv∈E1Si2C3−IIIp,qΓu2+Γv22+∑uv∈E2Si2C3−IIIp,qΓu2+Γv22+∑uv∈E3Si2C3−IIIp,qΓu2+Γv22+∑uv∈E4Si2C3−IIIp,qΓu2+Γv22=2×12+322+2p+2×22+222+8p+8q−12×22+322+15pq−10p−13q+8×32+322=4860pq−1888p−2732q+892.which is the required (36) result.
By definition of the second hyper F-index ofSi2C3−IIIp,q, we have(39)HF2Si2C3−IIp,q=∑uv∈ESi2C3−IIIp,qΓv2×Γv22=∑uv∈E1Si2C3−IIIp,qΓv2×Γv22+∑uv∈E2Si2C3−IIIp,qΓv2×Γv22+∑uv∈E3Si2C3−IIIp,qΓv2×Γv22+∑uv∈E4Si2C3−IIIp,qΓv2×Γv22+∑uv∈E5Si2C3−IIIp,qΓv2×Γv22=212×322+2+2q×22×222+8p+8q−12×22×322+15pq−10p−13q+8×32×322=98415pq−54730p−74413q+37136.which is the required (37) result.Theorem 9.
LetSi2C3−IIIp,q be the silicon carbide. Then,(40)SFSi2C3−IIIp,q=2×5pq2−5p3−13q6+43+13×8p+8q−1213+2×q+12+105,(41)PFSi2C3−IIIp,q=5pq3+2p9+7q18−518.Proof.
Consider the graph silicon carbideSi2C3−IIIp,q. By Remark 3, the graph Si2C3−IIIp,q contains 10pq vertices and 15pq−2p−3q edges. By definition of the sum connectivity F-index of Si2C3−IIIp,q, we have(42)SFSi2C3−IIIp,q=∑uv∈ESi2C3−IIIp,q1Γu2+Γv2=∑uv∈E1Si2C3−IIIp,q1Γu2+Γv2+∑uv∈E2Si2C3−IIIp,q1Γu2+Γv2+∑uv∈E3Si2C3−IIIp,q1Γu2+Γv2+∑uv∈E4Si2C3−IIIp,q1Γu2+Γv2=2×112+32+2+2q×122+22+8p+8q−12×122+32+15pq−10p−13q+8×132+32=2×5pq2−5p3−13q6+43+13×8p+8q−1213+2×q+12+105.which is the required (40) result.
By definition of product connectivity F-index ofSi2C3−IIIp,q, we have(43)PFSi2C3−IIIp,q=∑uv∈ESi2C3−IIIp,q1Γu2×Γv2=∑uv∈E1Si2C3−IIIp,q1Γu2×Γv2+∑uv∈E2Si2C3−IIIp,q1Γu2×Γv2+∑uv∈E3Si2C3−IIIp,q1Γu2×Γv2+∑uv∈E4Si2C3−IIIp,q1Γu2×Γv2=2×112×32+2+2q×122×22+8p+8q−12×122×32+15pq−10p−13q+8×132×32=5pq3+2p9+7q18−518.which is the required (41) result.


## 5. Results for Silicon-CarbonSi2C3−IIIp,q

In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−IIIp,q.The2D molecular graph of silicon carbide SiC3−III is given in Figure 11, where carbon atom C is shown in brown color and silicon atom Si is shown in blue color (for more details, see [42]). In Figure 12, we gave a demonstration how the cells connect in a row (chain) and how one row connects to another row; red lines show the connection between the unit cells and green lines (edges) connect the upper and lower rows. We will denote this molecular graph by Si2C3−IIIp,q.Figure 11
2
D structure of SiC3−IIIp,q: (a) SiC3−I5,1 and (b) SiC3−I5,2.
(a)
(b)Figure 12
2
D structure of SiC3−IIIp,q: (a) SiC3−I5,1 and (b) SiC3−I5,2.
(a)
(b)Remark 4 (see [13]).
The graphSi2C3−IIIp,q contains 8pq vertices and 12pq−3p−2q. edges.
We start by proving the carbon nanocones for the redefined Zagreb indices.Theorem 10.
LetSi2C3−IIIp,q be the silicon carbide. Then,(44)Mα,βSiC3−IIIp,q=9α×12pq−12p−8q+86β+6α×6p+4q−85β+3p+2q−3×4α−β+3α4β+2×2α3β,(45)Mr,sSiC3−IIIp,q=6p+4q−8×2r×3s+2s×3r+24pq−24p−16q+16×3s+r+6p+4q−6×2s+r+3r+3s+2×2r+2×2s.Proof.
Consider the graph silicon carbideSi2C3−IIIp,q. By Remark 4, the graph Si2C3−IIIp,q contains 8pq vertices and 12pq−3p−2q edges. From the graph of Si2C3−IIIp,q silicon carbide, we can see that there are three partitions, V1=v∈VSiC3−IIIp,qΓv=1, V2=v∈VSiC3−IIIp,qΓv=2, and V3=v∈VSiC3−IIIp,qΓv=3. The edge set of the Si2C3−IIIp,q can be partitioned as follows:(46)E1=e=uv∈ESiC3−IIIp,qΓu=1andΓv=2,E2=e=uv∈ESiC3−IIIp,qΓu=1andΓv=3,E3=e=uv∈ESiC3−IIIp,qΓu=2andΓv=2,E4=e=uv∈ESiC3−IIIp,qΓu=2andΓv=3,E1=e=uv∈ESiC3−IIIp,qΓu=3andΓv=3.
From the molecular graph ofSi2C3−IIIp,q, we can observe that E1=2,E2=1,E3=3p+2q−3,E4=6p+4q−8, and E5=12pq−12p−8q+8.
Thus, by definition generalization ofSi2C3−IIIp,q, we have(47)Mα,βSiC3−IIIp,q=∑uv∈ESiC3−IIIp,qΓu×ΓvαΓu+Γvβ=∑uv∈E1SiC3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E2SiC3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E3SiC3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E4SiC3−IIIp,qΓu×ΓvαΓu+Γvβ+∑uv∈E5SiC3−IIIp,qΓu×ΓvαΓu+Γvβ=2×1×2α1+2β+1×3α1+3β+3p+2q−3×2×2α2+2β+6p+4q−8×2×3α2+3β+12pq−12p−8q+8×3×3α3+3β=9α×12pq−12p−8q+86β+6α×6p+4q−85β+3p+2q−3×4α−β+3α4β+2×2α3β.which is the required (44) result.
By definition of the generalized Zagreb index ofSiC3−IIIp,q, we have(48)Mr,sSiC3−IIIp,q=∑uv∈ESiC3−IIIp,qΓur×Γvs+Γus×Γvr=∑uv∈E1SiC3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E2SiC3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E3SiC3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E4SiC3−IIIp,qΓur×Γvs+Γus×Γvr+∑uv∈E5SiC3−IIIp,qΓur×Γvs+Γus×Γvr=2×1r×2s+1s×2r+1r×3s+1s×3r+3p+2q−3×2r×2s+2s×2r+6p+4q−8×2r×3s+2s×3r+12pq−12p−8q+8×3r×3s+3s×3r=6p+4q−8×2r×3s+2s×3r+24pq−24p−16q+16×3s+r+6p+4q−6×2s+r+3r+3s+2×2r+2×2s.which is the required (45) result.Theorem 11.
LetSiC3−IIIp,q be the silicon carbide. Then,(49)HF1SiC3−IIIp,q=3888pq−2682p−1788q+1198,(50)HF2SiC3−IIIp,q=78732pq−70188p−46792q+41465.Proof.
Consider the graph silicon carbideSiC3−IIIp,q. By Remark 4, the graph SiC3−IIIp,q contains 8pq vertices and 12pq−3p−2q. edges. By definition of the first hyper F-index of SiC3−IIIp,q, we have(51)HF1SiC3−IIIp,q=∑uv∈ESiC3−IIIp,qΓu2+Γv22=∑uv∈E1SiC3−IIIp,qΓu2+Γv22+∑uv∈E2SiC3−IIIp,qΓu2+Γv22+∑uv∈E3SiC3−IIIp,qΓu2+Γv22+∑uv∈E4SiC3−IIIp,qΓu2+Γv22+∑uv∈E5SiC3−IIIp,qΓu2+Γv22=2×12+222+12+322+3p+2q−3×22+222+6p+4q−8×22+322+12pq−12p−8q+8×32+322=3888pq−2682p−1788q+1198.which is the required (49) result.
By definition of the second hyper F-index ofSi2C3−IIIp,q, we have(52)HF2SiC3−IIIp,q=∑uv∈ESiC3−IIIp,qΓu2×Γv22=∑uv∈E1SiC3−IIIp,qΓu2×Γv22+∑uv∈E2SiC3−IIIp,qΓu2×Γv22+∑uv∈E3SiC3−IIIp,qΓu2×Γv22+∑uv∈E4SiC3−IIIp,qΓu2×Γv22+∑uv∈E5SiC3−IIIp,qΓu2×Γv22=2×12×222+12×322+3p+2q−3×22×222+6p+4q−8×22×322+12pq−12p−8q+8×32×322=78732pq−70188p−46792q+41465.which is the required (50) result.Theorem 12.
LetSiC3−IIIp,q be the silicon carbide. Then,(53)SFSiC3−IIIp,q=2×2pq−4p3−2p+43+13×6p+4q−813+2×q2+3q2−34+1010+255,(54)PFSiC3−IIIp,q=4pq3+5p12+5p18+536.Proof.
Consider the graph silicon carbideSiC3−IIIp,q. By Remark 4, the graph SiC3−IIIp,q contains 8pq vertices and 12pq−3p−2q edges. By definition of the sum connectivity F-index of SiC3−IIIp,q, we have(55)SFSiC3−IIIp,q=∑uv∈ESiC3−IIIp,q1Γu2+Γv2=∑uv∈E1SiC3−IIIp,q1Γu2+Γv2+∑uv∈E2SiC3−IIIp,q1Γu2+Γv2+∑uv∈E3SiC3−IIIp,q1Γu2+Γv2+∑uv∈E4SiC3−IIIp,q1Γu2+Γv2+∑uv∈E5SiC3−IIIp,q1Γu2+Γv2=2×112+22+112+32+3p+2q−3×122+22+6p+4q−8×122+32+12pq−12p−8q+8×132+32=2×2pq−4p3−2p+43+13×6p+4q−813+2×q2+3q2−34+1010+255.which is the required (53) result.
By definition of product connectivity F-index ofSi2C3−IIIp,q, we have(56)PFSiC3−IIIp,q=∑uv∈ESiC3−IIIp,q1Γu2×Γv2=∑uv∈E1SiC3−IIIp,q1Γu2×Γv2+∑uv∈E2SiC3−IIIp,q1Γu2×Γv2+∑uv∈E3SiC3−IIIp,q1Γu2×Γv2+∑uv∈E4SiC3−IIIp,q1Γu2×Γv2+∑uv∈E5SiC3−IIIp,q1Γu2×Γv2=2×112×22+112×32+3p+2q−3×122×22+6p+4q−8×122×32+12pq−12p−8q+8×132×32=4pq3+5p12+5p18+536.which is the required (54) result.


## 6. Conclusion

In this paper, we computed the generalization of Zagreb index, the generalized Zagreb index, the first and second hyperF-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−Ip,q,Si2C3−IIp,q, Si2C3−IIIp,q and Si2C3−IIIp,q.



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