# On Topological Analysis of Entropy Measures for Silicon Carbides Networks

Authors: Xing-Long Wang; Muhammad Kamran Siddiqui; Syed Ajaz K. Kirmani; Shazia Manzoor; Sarfraz Ahmad; Mlamuli Dhlamini
Journal: Complexity
 (2021)

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## Abstract

The silicon material has provoked and stimulated significant research concern to a considerable extent taking into account its marvelous mechanical, optical, and electronic properties. Naturally, silicons are semiconductors and are utilized in the formation of various materials. For example, it is used in assembling the electronic based gadgets. In this article, we have studied the2D structure of silicon carbide Si2C3−Im,n and Si2C3−IIm,n and then continued to discuss some degree grounded topological descriptors in association with their corresponding entropy measures. We extend this computation to the quantitative and pictorial comparisons which could be beneficial in the structure amendment for effective implementation.


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## Body

## 1. Introduction

Chemical graph theory is a fascinating branch of mathematics and is the combination of chemistry and graph theory. In molecular graph, molecules are designed mathematically. In a molecular graph, the vertices and edges portrayed the atoms and the bonds correspondingly. Various approaches are employed on molecular graphs to acquire various structure properties of corresponding compounds. A numerical worth treated theoretically from the atomic graph is named as topological index. It is connected with synthetic constitution exhibiting for association of substance structure with miscellaneous physical and natural exercises and concoction properties. The degree-based topological indices collaborate actively in the exploitation of different fields specifically in pharmaceutical and commercial chemistry. In [1], Randić index is originated, while ABC index, GA index, first Zagreb index, and second Zagreb were discussed in [2, 3]. In [4], hyper-Zagreb index was introduced. In [5], forgotten index was initiated. Furtula [6] established the concept of AZI index. Balaban [7, 8] laid the foundation of Balaban index. The redefined first, second, and third Zagreb indices were investigated in [9]. The forth atom bond connectivity index was presented in [10]. The fifth geometric arithmetic index is declared in [11]. The specific topological indices of some graphs are discussed in [12, 13]. An ordered pair MV,ME, composed of a nonempty set MV of vertices and ME as a set of edges, is called a graph. If any two vertices m and n are attached with each other, they make an edge e. The number of edges adjacent with the vertex m is characterized as τm.Entropy has been a comprehensive and transcendental approach in diverse areas of expertise diverging from logic and biology to physics and engineering. Entropy connects the conception of randomness and uncertainty with physical conditions which are constructed as transformation channels of information. This study represents the special case when these channels are the detectors investigating the response of a system. Formally, entropy came into consideration after the invention of the heat engine, as a consequence of pioneering investigation towards explaining thermodynamical techniques and expanding the efficacy of such machines [14, 15]. This investigation reveals that the entropy of a system or machine cannot decrease over time. In 1948, Claude Shannon [16] introduces a measure of uncertainty as entropy. It can be explicated as the rate of production of new information manufactured by the system [17–19]. As stated by Shannon, uncertainty and information are two sides of the coin: a reduction in uncertainty is the same as the reception of a certain amount of information. Therefore, as the value of the entropy about a system is greater, the uncertainty about its response is also increased. In literature, many graph entropies are estimated; see [20–23].In 2014, Chen et al. [24] introduced the definition of the entropy of edge weighted graph M=MV;ME;ψmn. The entropy formula is represented in(1)EψM=−∑i=1mFiψminiIlogψminiI=logI−1I∑i=1mFiψminilogψmini.


## 2. Crystallographic Structure ofSi2C3−Im,n

The most consistent framework of2D silicon carbon unicellular compound is anticipated which is grounded on the PSO particle swarm optimization approach. It is rich in carbon. In the last decades, silicon carbon was considered to be the hardest material across the world. Due to its Mohs hardness rating, it resembles diamond. Silicon is of low cost and is nontoxic semiconductor. Numerous studies have been done in the refinement, magnification, and device assembling [25]. It is employed for all advanced electronic gadgets. The silicon carbide framework may seem like the honeycomb framework of graphene. Despite that, large scale search proposes that the framework of this compound is absolutely distinct compared to that of graphene. For silicon carbide ISi2C3 layer, the minimum energy of Si2C3 -I demonstrates a planar framework comprised of polygonal rings, where two pentagonal and four heptagonal rings encircling each hexagonal ring. See Figure 1 [26]. Every hexagonal ring carries out three Si and three C atoms in which Si and C atoms are positioned by turn on the vertices. Pentagonal rings are of two kinds in which one is comprised of two Si and three C atoms. Also, three Si and four C atoms establish the heptagonal rings. It is to be noted that there are no Si−Si bonds in Si2C3 -I sheet. Here m denotes the total associated unit cells in each row while n shows the total sum of associated rows with m number of cells.Figure 1
(a) Unit cell ofSi2C3−Im,n; (b) Si2C3−I4,3.
(a)(b)Figure2 illustrates the way in which cells are associated in a row and the way of association of rows with each other. Also, VSi2C3−I=10mn and ESi2C3−I=35mn−n−2m.Figure 2
(a)Si2C3−I4,1, one row with m=4 and n=1; (b) Si2C3−I4,2.
(a)(b)


### 2.1. Formation ofSi2C3−Im,n Formulas

Here, we join one unit cell with another unit cell and then continue this process horizontally tilln unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the Si2C3−Im,n sheet; see Figure 1. The vertex and edge partition are depicted in Tables 1 and 2, respectively.Table 1
Vertex partition ofSi2C3−Im,n.
τmFrequencyKind of vertices12MV124m+6n−4MV2310mn−4m−6n+2MV2Table 2
Edge partition ofSi2C3−Im,n.
τm,τnFrequencySet of edges2,11ME13,11ME22,2m+2nME33,26m+8n−9ME43,315mn−9m−13n+7ME5Table3 shows the edge partition of the chemical graph Si2C3−Im,n based on degree sum of end vertices.Table 3
Edge partition ofSi2C3−Im,n with m,n≥2.
Sm,SnFrequency4,215,315,5m+2n5,517,417,52+2n5,617,6−7+22m+n8,62−1+n8,5−5+2m+n8,718,8−4+2n+m9,7−3+2m+n9,8−7+2m+2n9,915mn−14m−21n+20


### 2.2. Computation of Entropies for Crystallographic Structure ofSi2C3−Im,n

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure ofSi2C3−Im,n.


#### 2.2.1. The Randić Index and Randić Entropy forSi2C3−Im,n

Forα=1, the Randić index and Randić entropy by using Table 2 are(2)R1Si2C3−I=135mn−41m−61n+14,ER1Si2C3−I=log135mn−41m−61n+14−log22135mn−41m−61n+14−log33135mn−41m−61n+14−logm+2n×44135mn−41m−61n+14−log6m+8n−9×66135mn−41m−61n+14−log15mn−9m−13n+7×99135mn−41m−61n+14.Forα=−1, by using Table 2, the Randić index and Randić entropy are(3)R−1Si2C3−I=5mn3+m4+7n18+19,ER−1Si2C3−I=log53mn+144+718n+19−log2−1/25/3mn+1/44+7/18n+1/9−log3−1/35/3mn+1/44+7/18n+1/9−logm+2n×4−1/45/3mn+1/44+7/18n+1/9−log6m+8n−9×6−1/65/3mn+1/44+7/18n+1/9−log15mn−9m−13n+7×9−1/95/3mn+1/44+7/18n+1/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are(4)R1/2Si2C3−I=45mn+66−27m+86−35n+2+3−96+21,ER1/2Si2C3−I=log45mn+66−27m+86−35n+2+3−96+21−log1×21/2⋅245mn+66−27m+86−35n+2+3−96+21−log1×31/2⋅345mn+66−27m+86−35n+2+3−96+21−logm+2n×41/2⋅445mn+66−27m+86−35n+2+3−96+21−log6m+8n−9×61/2⋅645mn+66−27m+86−35n+2+3−96+21−log15mn−9m−13n+7×91/2⋅945mn+66−27m+86−35n+2+3−96+21.Forα=−1/2, by using and Table 2, the Randić index and Randić entropy are(5)R−1/2Si2C3−I=5mn+6−52m+86−103n+12+13−96+73,ER−1/2Si2C3−I=log5mn+6−52m+86−103n+12+13−96+73−log1×2−1/2⋅1/25mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log1×3−1/2⋅1/35mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−logm+2n×4−1/2⋅1/45mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log6m+8n−9×6−1/2⋅1/65mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log15mn−9m−13n+7×9−1/2⋅1/95mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3.


#### 2.2.2. TheABC Index and ABC Entropy of Si2C3−Im,n

By using Table2, the ABC index and ABC entropy are(6)ABCSi2C3−I=10mn+62+12−6m+22+82−263n−82+23+143,EABCSi2C3−I=log10mn+62+12−6m+22+82−263n−82+23+143−log1×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log1×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−logm+2n×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log6m+8n−9×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log15mn−9m−13n+7×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3.


#### 2.2.3. TheGA Index and GA Entropy of Si2C3−Im,n

By using Table2, the GA index and GA entropy are(7)GASi2C3−I=15mn+1265−8m+1665−11n+223+32−1865+7,EGASi2C3−I=log15mn+1265−8m+1665−11n+223+32−1865+7−log1×22/322/315mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×3/23/215mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×m+2n15mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log6m+8n−9×26/526/515mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log15mn−9m−13n+7×115mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7.


#### 2.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−Im,n

By using Table2, the first Zagreb index and entropy are(8)M1Si2C3−I=90mn−20m−30n+4,EM1Si2C3−I=log90mn−20m−30n+4−log1×2790mn−20m−30n+4−log1×25690mn−20m−30n+4−logm+2n×312590mn−20m−30n+4−log15mn−9m−13n+7×4665690mn−20m−30n+4.


#### 2.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−Im,n

By using Table2, the second Zagreb index and second Zagreb entropy are(9)M2Si2C3−I=135mn−9m−61n+14,EM2Si2C3−I=log135mn−9m−61n+14−log1×4135mn−9m−61n+14−log1×27135mn−9m−61n+14−logm+2n×256135mn−9m−61n+14−log6m+8n−9×46656135mn−9m−61n+14−log15mn−9m−13n+7×99135mn−9m−61n+14.


#### 2.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−Im,n

By using Table2, the hyper-Zagreb index and hyper-Zagreb entropy are(10)HMSi2C3−I=540mn−158m−236n+412,EHMSi2C3−I=log540mn−158m−236n+412−log99540mn−158m−236n+412−log1616540mn−158m−236n+412−logm+2n×1616540mn−158m−236n+412−log6m+8n−9×2525540mn−158m−236n+412−log15mn−9m−13n+7×3636540mn−158m−236n+412.


#### 2.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−Im,n

By using Table2, the forgotten index and forgotten entropy are(11)FSi2C3−I=270mn−76m−114n+24,EFSi2C3−I=log270mn−76m−114n+24−log55270mn−76m−114n+24−log1010270mn−76m−114n+24−logm+2n×88270mn−76m−114n+24−log6m+8n−9×1313270mn−76m−114n+24−log15mn−9m−13n+7×1818270mn−76m−114n+24.


#### 2.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−Im,n

By using Table2, the AZI index and AZI entropy are AZISi2C3−I=10935/64mn−3481/64m−4357/64n+1223/64(12)EAZISi2C3−I=log1093564mn−348164m−435764n+122364−log8810935/64mn−3481/64m−4357/64n+1223/64−log27/827/810935/64mn−3481/64m−4357/64n+1223/64−logm+2n×8810935/64mn−3481/64m−4357/64n+1223/64−log15mn−9m−13n+7×93/4393/4310935/64mn−3481/64m−4357/64n+1223/64.


#### 2.2.9. The Balaban Index and Balaban Entropy forSi2C3−Im,n

By using Table2, the Balaban index and Balaban entropy are(13)JSi2C3−I=15mn−2m−3n5mn−2m−3n+212+13+12m+2n+166m+8n−9+1315mn−9m−13n+7,ENTJSi2C3−I=logJG−q/2q−p+2logq/2q−p+2JG−q/3q−p+2logq/3q−p+2JG−q/2q−p+2logm+2n×q/2q−p+2JG−q/6q−p+2logq6m+8n−9/6q−p+2JG−q/3q−p+2log15mn−9m−13n+7×q/3q−p+2JG.


#### 2.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−I=10mn and(14)EReZG1Si2C3−I=log10mn−log3/23/210mn−log4/34/310mn−logm+2n10mn−log6m+8n−9×5/65/610mn−log15mn−9m−13n+7×2/32/310mn.


#### 2.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−I=45/2mn−53/10m−79/10n+67/60 and(15)EReZG2Si2C3−I=log452mn−5310m−7910n+6760−log2/32/345/2mn−53/10m−79/10n+67/60−log3/43/445/2mn−53/10m−79/10n+67/60−logm+2n45/2mn−53/10m−79/10n+67/60−log6/56/56m+8n−945/2mn−53/10m−79/10n+67/60−log3/23/215mn−9m−13n+745/2mn−53/10m−79/10n+67/60.


#### 2.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined third Zagreb index and redefined third Zagreb entropy are ReZG3Si2C3−I=810mn−290m−430n+126 and(16)EReZG3Si2C3−I=log810mn−290m−430n+126−log1×66810mn−290m−430n+126−log1×1212810mn−290m−430n+126−logm+2n×1616810mn−290m−430n+126−log6m+8n−9×3030810mn−290m−430n+126−log15mn−9m−13n+7×5454810mn−290m−430n+126.


#### 2.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−Im,n

By using Table3, the fourth ABC index and fourth ABC entropy are(17)ABC4Si2C3−I=203mn+85+4×1142+2×1140+223+148+56−569m+425+227+21142+1110+722+223+144+256−253n+12+25+85+327+227+310−71142−51140+1356−72−2−721356+719,ENTABC4G=logABC4−1ABC4log121/2−1ABC4log252/5−1ABC4log3273/27−1ABC4logm+2n+1×858/5−1ABC4log2n+2×272/7−1ABC4log3103/10−1ABC4log4m+2n−7×114211/42−1ABC4log2n−2×121/2−1ABC4log2m+2n−5×114011/40−1ABC4log135613/56−1ABC4logm+2n−4×14814/8−1ABC4log2m+2n−3×232/3−1ABC4log2m+4n−7×5265/26−1ABC4log15mn−14m−21n+20×494/9.


#### 2.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−Im,n

By using Table3, the fifth GA index and fifth GA entropy are(18)GA5G=15mn+84213+81013+374+24217−12m+−17+353+44213+837+81013+374+24817n+17−218251+154−67788+353+23011−144213−837−201013+41415,EGA5G=logGA5−1GA5log22322/3−1GA5log15415/4−1GA5logm+2n+1−1GA5log471147/11−1GA5log2n+2×35635/6−1GA5log23011230/11−1GA5log4m+2n−7×24213242/13−1GA5log2n−2×43743/7−1GA5log2m+2n−5×41013410/13−1GA5log41415414/15−1GA5logm+2n−4−1GA5log2m+2n−3×37837/8−1GA5log2m+4n−7×681768/17−1GA5log15mn−14m−21n+20.


## 2.1. Formation ofSi2C3−Im,n Formulas

Here, we join one unit cell with another unit cell and then continue this process horizontally tilln unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the Si2C3−Im,n sheet; see Figure 1. The vertex and edge partition are depicted in Tables 1 and 2, respectively.Table 1
Vertex partition ofSi2C3−Im,n.
τmFrequencyKind of vertices12MV124m+6n−4MV2310mn−4m−6n+2MV2Table 2
Edge partition ofSi2C3−Im,n.
τm,τnFrequencySet of edges2,11ME13,11ME22,2m+2nME33,26m+8n−9ME43,315mn−9m−13n+7ME5Table3 shows the edge partition of the chemical graph Si2C3−Im,n based on degree sum of end vertices.Table 3
Edge partition ofSi2C3−Im,n with m,n≥2.
Sm,SnFrequency4,215,315,5m+2n5,517,417,52+2n5,617,6−7+22m+n8,62−1+n8,5−5+2m+n8,718,8−4+2n+m9,7−3+2m+n9,8−7+2m+2n9,915mn−14m−21n+20


## 2.2. Computation of Entropies for Crystallographic Structure ofSi2C3−Im,n

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure ofSi2C3−Im,n.


### 2.2.1. The Randić Index and Randić Entropy forSi2C3−Im,n

Forα=1, the Randić index and Randić entropy by using Table 2 are(2)R1Si2C3−I=135mn−41m−61n+14,ER1Si2C3−I=log135mn−41m−61n+14−log22135mn−41m−61n+14−log33135mn−41m−61n+14−logm+2n×44135mn−41m−61n+14−log6m+8n−9×66135mn−41m−61n+14−log15mn−9m−13n+7×99135mn−41m−61n+14.Forα=−1, by using Table 2, the Randić index and Randić entropy are(3)R−1Si2C3−I=5mn3+m4+7n18+19,ER−1Si2C3−I=log53mn+144+718n+19−log2−1/25/3mn+1/44+7/18n+1/9−log3−1/35/3mn+1/44+7/18n+1/9−logm+2n×4−1/45/3mn+1/44+7/18n+1/9−log6m+8n−9×6−1/65/3mn+1/44+7/18n+1/9−log15mn−9m−13n+7×9−1/95/3mn+1/44+7/18n+1/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are(4)R1/2Si2C3−I=45mn+66−27m+86−35n+2+3−96+21,ER1/2Si2C3−I=log45mn+66−27m+86−35n+2+3−96+21−log1×21/2⋅245mn+66−27m+86−35n+2+3−96+21−log1×31/2⋅345mn+66−27m+86−35n+2+3−96+21−logm+2n×41/2⋅445mn+66−27m+86−35n+2+3−96+21−log6m+8n−9×61/2⋅645mn+66−27m+86−35n+2+3−96+21−log15mn−9m−13n+7×91/2⋅945mn+66−27m+86−35n+2+3−96+21.Forα=−1/2, by using and Table 2, the Randić index and Randić entropy are(5)R−1/2Si2C3−I=5mn+6−52m+86−103n+12+13−96+73,ER−1/2Si2C3−I=log5mn+6−52m+86−103n+12+13−96+73−log1×2−1/2⋅1/25mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log1×3−1/2⋅1/35mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−logm+2n×4−1/2⋅1/45mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log6m+8n−9×6−1/2⋅1/65mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log15mn−9m−13n+7×9−1/2⋅1/95mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3.


### 2.2.2. TheABC Index and ABC Entropy of Si2C3−Im,n

By using Table2, the ABC index and ABC entropy are(6)ABCSi2C3−I=10mn+62+12−6m+22+82−263n−82+23+143,EABCSi2C3−I=log10mn+62+12−6m+22+82−263n−82+23+143−log1×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log1×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−logm+2n×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log6m+8n−9×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log15mn−9m−13n+7×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3.


### 2.2.3. TheGA Index and GA Entropy of Si2C3−Im,n

By using Table2, the GA index and GA entropy are(7)GASi2C3−I=15mn+1265−8m+1665−11n+223+32−1865+7,EGASi2C3−I=log15mn+1265−8m+1665−11n+223+32−1865+7−log1×22/322/315mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×3/23/215mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×m+2n15mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log6m+8n−9×26/526/515mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log15mn−9m−13n+7×115mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7.


### 2.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−Im,n

By using Table2, the first Zagreb index and entropy are(8)M1Si2C3−I=90mn−20m−30n+4,EM1Si2C3−I=log90mn−20m−30n+4−log1×2790mn−20m−30n+4−log1×25690mn−20m−30n+4−logm+2n×312590mn−20m−30n+4−log15mn−9m−13n+7×4665690mn−20m−30n+4.


### 2.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−Im,n

By using Table2, the second Zagreb index and second Zagreb entropy are(9)M2Si2C3−I=135mn−9m−61n+14,EM2Si2C3−I=log135mn−9m−61n+14−log1×4135mn−9m−61n+14−log1×27135mn−9m−61n+14−logm+2n×256135mn−9m−61n+14−log6m+8n−9×46656135mn−9m−61n+14−log15mn−9m−13n+7×99135mn−9m−61n+14.


### 2.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−Im,n

By using Table2, the hyper-Zagreb index and hyper-Zagreb entropy are(10)HMSi2C3−I=540mn−158m−236n+412,EHMSi2C3−I=log540mn−158m−236n+412−log99540mn−158m−236n+412−log1616540mn−158m−236n+412−logm+2n×1616540mn−158m−236n+412−log6m+8n−9×2525540mn−158m−236n+412−log15mn−9m−13n+7×3636540mn−158m−236n+412.


### 2.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−Im,n

By using Table2, the forgotten index and forgotten entropy are(11)FSi2C3−I=270mn−76m−114n+24,EFSi2C3−I=log270mn−76m−114n+24−log55270mn−76m−114n+24−log1010270mn−76m−114n+24−logm+2n×88270mn−76m−114n+24−log6m+8n−9×1313270mn−76m−114n+24−log15mn−9m−13n+7×1818270mn−76m−114n+24.


### 2.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−Im,n

By using Table2, the AZI index and AZI entropy are AZISi2C3−I=10935/64mn−3481/64m−4357/64n+1223/64(12)EAZISi2C3−I=log1093564mn−348164m−435764n+122364−log8810935/64mn−3481/64m−4357/64n+1223/64−log27/827/810935/64mn−3481/64m−4357/64n+1223/64−logm+2n×8810935/64mn−3481/64m−4357/64n+1223/64−log15mn−9m−13n+7×93/4393/4310935/64mn−3481/64m−4357/64n+1223/64.


### 2.2.9. The Balaban Index and Balaban Entropy forSi2C3−Im,n

By using Table2, the Balaban index and Balaban entropy are(13)JSi2C3−I=15mn−2m−3n5mn−2m−3n+212+13+12m+2n+166m+8n−9+1315mn−9m−13n+7,ENTJSi2C3−I=logJG−q/2q−p+2logq/2q−p+2JG−q/3q−p+2logq/3q−p+2JG−q/2q−p+2logm+2n×q/2q−p+2JG−q/6q−p+2logq6m+8n−9/6q−p+2JG−q/3q−p+2log15mn−9m−13n+7×q/3q−p+2JG.


### 2.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−I=10mn and(14)EReZG1Si2C3−I=log10mn−log3/23/210mn−log4/34/310mn−logm+2n10mn−log6m+8n−9×5/65/610mn−log15mn−9m−13n+7×2/32/310mn.


### 2.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−I=45/2mn−53/10m−79/10n+67/60 and(15)EReZG2Si2C3−I=log452mn−5310m−7910n+6760−log2/32/345/2mn−53/10m−79/10n+67/60−log3/43/445/2mn−53/10m−79/10n+67/60−logm+2n45/2mn−53/10m−79/10n+67/60−log6/56/56m+8n−945/2mn−53/10m−79/10n+67/60−log3/23/215mn−9m−13n+745/2mn−53/10m−79/10n+67/60.


### 2.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined third Zagreb index and redefined third Zagreb entropy are ReZG3Si2C3−I=810mn−290m−430n+126 and(16)EReZG3Si2C3−I=log810mn−290m−430n+126−log1×66810mn−290m−430n+126−log1×1212810mn−290m−430n+126−logm+2n×1616810mn−290m−430n+126−log6m+8n−9×3030810mn−290m−430n+126−log15mn−9m−13n+7×5454810mn−290m−430n+126.


### 2.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−Im,n

By using Table3, the fourth ABC index and fourth ABC entropy are(17)ABC4Si2C3−I=203mn+85+4×1142+2×1140+223+148+56−569m+425+227+21142+1110+722+223+144+256−253n+12+25+85+327+227+310−71142−51140+1356−72−2−721356+719,ENTABC4G=logABC4−1ABC4log121/2−1ABC4log252/5−1ABC4log3273/27−1ABC4logm+2n+1×858/5−1ABC4log2n+2×272/7−1ABC4log3103/10−1ABC4log4m+2n−7×114211/42−1ABC4log2n−2×121/2−1ABC4log2m+2n−5×114011/40−1ABC4log135613/56−1ABC4logm+2n−4×14814/8−1ABC4log2m+2n−3×232/3−1ABC4log2m+4n−7×5265/26−1ABC4log15mn−14m−21n+20×494/9.


### 2.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−Im,n

By using Table3, the fifth GA index and fifth GA entropy are(18)GA5G=15mn+84213+81013+374+24217−12m+−17+353+44213+837+81013+374+24817n+17−218251+154−67788+353+23011−144213−837−201013+41415,EGA5G=logGA5−1GA5log22322/3−1GA5log15415/4−1GA5logm+2n+1−1GA5log471147/11−1GA5log2n+2×35635/6−1GA5log23011230/11−1GA5log4m+2n−7×24213242/13−1GA5log2n−2×43743/7−1GA5log2m+2n−5×41013410/13−1GA5log41415414/15−1GA5logm+2n−4−1GA5log2m+2n−3×37837/8−1GA5log2m+4n−7×681768/17−1GA5log15mn−14m−21n+20.


## 2.2.1. The Randić Index and Randić Entropy forSi2C3−Im,n

Forα=1, the Randić index and Randić entropy by using Table 2 are(2)R1Si2C3−I=135mn−41m−61n+14,ER1Si2C3−I=log135mn−41m−61n+14−log22135mn−41m−61n+14−log33135mn−41m−61n+14−logm+2n×44135mn−41m−61n+14−log6m+8n−9×66135mn−41m−61n+14−log15mn−9m−13n+7×99135mn−41m−61n+14.Forα=−1, by using Table 2, the Randić index and Randić entropy are(3)R−1Si2C3−I=5mn3+m4+7n18+19,ER−1Si2C3−I=log53mn+144+718n+19−log2−1/25/3mn+1/44+7/18n+1/9−log3−1/35/3mn+1/44+7/18n+1/9−logm+2n×4−1/45/3mn+1/44+7/18n+1/9−log6m+8n−9×6−1/65/3mn+1/44+7/18n+1/9−log15mn−9m−13n+7×9−1/95/3mn+1/44+7/18n+1/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are(4)R1/2Si2C3−I=45mn+66−27m+86−35n+2+3−96+21,ER1/2Si2C3−I=log45mn+66−27m+86−35n+2+3−96+21−log1×21/2⋅245mn+66−27m+86−35n+2+3−96+21−log1×31/2⋅345mn+66−27m+86−35n+2+3−96+21−logm+2n×41/2⋅445mn+66−27m+86−35n+2+3−96+21−log6m+8n−9×61/2⋅645mn+66−27m+86−35n+2+3−96+21−log15mn−9m−13n+7×91/2⋅945mn+66−27m+86−35n+2+3−96+21.Forα=−1/2, by using and Table 2, the Randić index and Randić entropy are(5)R−1/2Si2C3−I=5mn+6−52m+86−103n+12+13−96+73,ER−1/2Si2C3−I=log5mn+6−52m+86−103n+12+13−96+73−log1×2−1/2⋅1/25mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log1×3−1/2⋅1/35mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−logm+2n×4−1/2⋅1/45mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log6m+8n−9×6−1/2⋅1/65mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3−log15mn−9m−13n+7×9−1/2⋅1/95mn+6−5/2m+8/6−10/3n+1/2+1/3−9/6+7/3.


## 2.2.2. TheABC Index and ABC Entropy of Si2C3−Im,n

By using Table2, the ABC index and ABC entropy are(6)ABCSi2C3−I=10mn+62+12−6m+22+82−263n−82+23+143,EABCSi2C3−I=log10mn+62+12−6m+22+82−263n−82+23+143−log1×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log1×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−logm+2n×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log6m+8n−9×1/21/210mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3−log15mn−9m−13n+7×2/32/310mn+6/2+1/2−6m+2/2+8/2−26/3n−8/2+2/3+14/3.


## 2.2.3. TheGA Index and GA Entropy of Si2C3−Im,n

By using Table2, the GA index and GA entropy are(7)GASi2C3−I=15mn+1265−8m+1665−11n+223+32−1865+7,EGASi2C3−I=log15mn+1265−8m+1665−11n+223+32−1865+7−log1×22/322/315mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×3/23/215mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log1×m+2n15mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log6m+8n−9×26/526/515mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7−log15mn−9m−13n+7×115mn+126/5−8m+166/5−11n+22/3+3/2−186/5+7.


## 2.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−Im,n

By using Table2, the first Zagreb index and entropy are(8)M1Si2C3−I=90mn−20m−30n+4,EM1Si2C3−I=log90mn−20m−30n+4−log1×2790mn−20m−30n+4−log1×25690mn−20m−30n+4−logm+2n×312590mn−20m−30n+4−log15mn−9m−13n+7×4665690mn−20m−30n+4.


## 2.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−Im,n

By using Table2, the second Zagreb index and second Zagreb entropy are(9)M2Si2C3−I=135mn−9m−61n+14,EM2Si2C3−I=log135mn−9m−61n+14−log1×4135mn−9m−61n+14−log1×27135mn−9m−61n+14−logm+2n×256135mn−9m−61n+14−log6m+8n−9×46656135mn−9m−61n+14−log15mn−9m−13n+7×99135mn−9m−61n+14.


## 2.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−Im,n

By using Table2, the hyper-Zagreb index and hyper-Zagreb entropy are(10)HMSi2C3−I=540mn−158m−236n+412,EHMSi2C3−I=log540mn−158m−236n+412−log99540mn−158m−236n+412−log1616540mn−158m−236n+412−logm+2n×1616540mn−158m−236n+412−log6m+8n−9×2525540mn−158m−236n+412−log15mn−9m−13n+7×3636540mn−158m−236n+412.


## 2.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−Im,n

By using Table2, the forgotten index and forgotten entropy are(11)FSi2C3−I=270mn−76m−114n+24,EFSi2C3−I=log270mn−76m−114n+24−log55270mn−76m−114n+24−log1010270mn−76m−114n+24−logm+2n×88270mn−76m−114n+24−log6m+8n−9×1313270mn−76m−114n+24−log15mn−9m−13n+7×1818270mn−76m−114n+24.


## 2.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−Im,n

By using Table2, the AZI index and AZI entropy are AZISi2C3−I=10935/64mn−3481/64m−4357/64n+1223/64(12)EAZISi2C3−I=log1093564mn−348164m−435764n+122364−log8810935/64mn−3481/64m−4357/64n+1223/64−log27/827/810935/64mn−3481/64m−4357/64n+1223/64−logm+2n×8810935/64mn−3481/64m−4357/64n+1223/64−log15mn−9m−13n+7×93/4393/4310935/64mn−3481/64m−4357/64n+1223/64.


## 2.2.9. The Balaban Index and Balaban Entropy forSi2C3−Im,n

By using Table2, the Balaban index and Balaban entropy are(13)JSi2C3−I=15mn−2m−3n5mn−2m−3n+212+13+12m+2n+166m+8n−9+1315mn−9m−13n+7,ENTJSi2C3−I=logJG−q/2q−p+2logq/2q−p+2JG−q/3q−p+2logq/3q−p+2JG−q/2q−p+2logm+2n×q/2q−p+2JG−q/6q−p+2logq6m+8n−9/6q−p+2JG−q/3q−p+2log15mn−9m−13n+7×q/3q−p+2JG.


## 2.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−I=10mn and(14)EReZG1Si2C3−I=log10mn−log3/23/210mn−log4/34/310mn−logm+2n10mn−log6m+8n−9×5/65/610mn−log15mn−9m−13n+7×2/32/310mn.


## 2.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−I=45/2mn−53/10m−79/10n+67/60 and(15)EReZG2Si2C3−I=log452mn−5310m−7910n+6760−log2/32/345/2mn−53/10m−79/10n+67/60−log3/43/445/2mn−53/10m−79/10n+67/60−logm+2n45/2mn−53/10m−79/10n+67/60−log6/56/56m+8n−945/2mn−53/10m−79/10n+67/60−log3/23/215mn−9m−13n+745/2mn−53/10m−79/10n+67/60.


## 2.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−Im,n

By using Table2, the redefined third Zagreb index and redefined third Zagreb entropy are ReZG3Si2C3−I=810mn−290m−430n+126 and(16)EReZG3Si2C3−I=log810mn−290m−430n+126−log1×66810mn−290m−430n+126−log1×1212810mn−290m−430n+126−logm+2n×1616810mn−290m−430n+126−log6m+8n−9×3030810mn−290m−430n+126−log15mn−9m−13n+7×5454810mn−290m−430n+126.


## 2.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−Im,n

By using Table3, the fourth ABC index and fourth ABC entropy are(17)ABC4Si2C3−I=203mn+85+4×1142+2×1140+223+148+56−569m+425+227+21142+1110+722+223+144+256−253n+12+25+85+327+227+310−71142−51140+1356−72−2−721356+719,ENTABC4G=logABC4−1ABC4log121/2−1ABC4log252/5−1ABC4log3273/27−1ABC4logm+2n+1×858/5−1ABC4log2n+2×272/7−1ABC4log3103/10−1ABC4log4m+2n−7×114211/42−1ABC4log2n−2×121/2−1ABC4log2m+2n−5×114011/40−1ABC4log135613/56−1ABC4logm+2n−4×14814/8−1ABC4log2m+2n−3×232/3−1ABC4log2m+4n−7×5265/26−1ABC4log15mn−14m−21n+20×494/9.


## 2.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−Im,n

By using Table3, the fifth GA index and fifth GA entropy are(18)GA5G=15mn+84213+81013+374+24217−12m+−17+353+44213+837+81013+374+24817n+17−218251+154−67788+353+23011−144213−837−201013+41415,EGA5G=logGA5−1GA5log22322/3−1GA5log15415/4−1GA5logm+2n+1−1GA5log471147/11−1GA5log2n+2×35635/6−1GA5log23011230/11−1GA5log4m+2n−7×24213242/13−1GA5log2n−2×43743/7−1GA5log2m+2n−5×41013410/13−1GA5log41415414/15−1GA5logm+2n−4−1GA5log2m+2n−3×37837/8−1GA5log2m+4n−7×681768/17−1GA5log15mn−14m−21n+20.


## 3. Crystallographic Structure ofSi2C3−IIm,n

The framework ofSi2C3−II sheet contains only hexagonal rings. This framework is greater in energy than that of Si2C3−I sheet. On the other hand, the composition of Si2C3−II is quite different from Si2C3−I sheet as Si−Si bonds can be viewed and the ratio of Si atoms making Si dimers is half. Also, Si2C3−II is more unstable than Si2C3−I. The anticipated minimal energy frameworks Si2C3−I and Si2C3−II sheets have exceptional semiconducting axioms that could be employed for monoelectronic utilization. The mechanic strength of the silicon carbide sheets is also significant. It is widely known that graphene carries magnificent elastic properties having immense elastic constants. Former analyses have demonstrated that silicon carbide also holds well elastic axioms.In Figure3 [27, 28], m denotes the total associated unit cells in each row while n shows the total sum of associated rows with m number of cells. Figure 4 illustrates the way in which cells are associated in a row and the way of association of rows with each other. Also, VSi2C3−II=10mn and ESi2C3−II=35mn−n−mFigure 3
(a) Unit cell ofSi2C3−IIm,n; (b) Si2C3−II3,3.
(a)(b)Figure 4
(a)Si2C3−II5,1; (b) Si2C3−II5,2.
(a)(b)


### 3.1. Methodology of Silicon CarbideSi2C3−IIm,n Formulas

For silicon carbideSi2C3−IIm,n, we join one unit cell with another unit cell and then continue this process horizontally till n unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the Si2C3−IIm,n sheet; see Figure 3. Moreover, Tables 4 and 5 are used for computation of vertices and edges, respectively.Table 4
Vertex partition ofSi2C3−IIm,n established on degrees of each vertex.
τmNo. of repetitionsKinds of vertices13MV126m+6n−6MV2310mn−6m−6n+3MV3Table 5
Edge partition ofSi2C3−IIm,n established on degrees of end vertices.
τr,τsNo. of repetitionsKinds of edges2,12ME13,11ME22,22m+2nME33,28m+8n−14ME43,315mn−13m−13n+11ME5The edge partition ofSi2C3−IIm,n established on the addition of degree of terminal vertices of every edge is illustrated in Table 6.Table 6
Edge partition ofSi2C3−IIm,n, m,n,s≥2.
Ss,StFrequency3,227,314,325,425,5−4+2m+n7,52m+n7,6−2+2m+n8,6−6+2m+n8,5−6+m+n8,728,8−8+2m+n7,729,7−3+2m+n9,8−5+2m+n9,923−19n−19m+15mn


### 3.2. Computation of Entropies for Crystallographic Structure ofSi2C3−IIm,n

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure ofSi2C3−IIm,n.


#### 3.2.1. The Randić Index and Randić Entropy forSi2C3−IIm,n

Forα=1, the Randić index and Randić entropy by Table 5 are(19)R1Si2C3−II=135mn−61m−61n+22,ER1Si2C3−II=log135mn−61m−61n+22−log2×22135mn−61m−61n+22−log1×33135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×66135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.Forα=−1, by using Table 5, the Randić index and Randić entropy are R−1Si2C3−II=5/3mn+7/18m+7/18n+2/9 and(20)ER−1Si2C3−II=log53mn+718m+718n+29−log2×2−1/25/3mn+7/18m+7/18n+2/9−log1×3−1/35/3mn+7/18m+7/18n+2/9−log2m+2n×4−1/45/3mn+7/18m+7/18n+2/9−log8m+8n−14×6−1/65/3mn+7/18m+7/18n+2/9−log15mn−13m−13n+11×9−1/95/3mn+7/18m+7/18n+2/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are R1/2Si2C3−I=45mn+86−35m+n+22+3−146+33 and(21)ER12Si2C3−II=log45mn+86−35m+n+22+3−146+33−log2×22/245mn+86−35m+n+22+3−146+33−log1×33/245mn+86−35m+n+22+3−146+33−log2m+2n×44/245mn+86−35m+n+22+3−146+33−log8m+8n−14×66/245mn+86−35m+n+22+3−146+33−log15mn−13m−13n+11×99/245mn+86−35m+n+22+3−146+33.Forα=−1/2, by using Table 2, the Randić index and Randić entropy are (22)R−1/2Si2C3−II=5mn+8/6−10/3m+n+2+1/3−14/6+11/3,ER−1/2Si2C3−II=log5mn+86−103m+n+2+13−146+113−log2×2−1/225mn+8/6−10/3m+n+2+1/3−14/6+11/3−log1×3−1/235mn+8/6−10/3m+n+2+1/3−14/6+11/3−log2m+2n×4−1/245mn+8/6−10/3m+n+2+1/3−14/6+11/3−log8m+8n−14×6−1/265mn+8/6−10/3m+n+2+1/3−14/6+11/3−log15mn−13m−13n+11×9−1/295mn+8/6−10/3m+n+2+1/3−14/6+11/3.


#### 3.2.2. TheABC Index and ABC Entropy of Si2C3−IIm,n

By using Table5, the ABC index and ABC entropy are (23)ABCSi2C3−II=10mn+10/2−26/3m+n−12/2+2/3+22/3,EABCSi2C3−II=logABCSi2C3−II−log2×1/21/2ABCSi2C3−II−log15mn−13m−13n+11×2/32/3ABCSi2C3−II−log2m+2n×1/21/2ABCSi2C3−II−log8m+8n−14×1/21/2ABCSi2C3−II−log1×2/32/3ABCSi2C3−II.


#### 3.2.3. TheGA Index and GA Entropy of Si2C3−IIm,n

By using Table5, the GA index and GA entropy are(24)GASi2C3−II=15mn+166/5−11m+n+42/3+3/2−286/5+11EGASi2C3−II=logGASi2C3−II−log2×22/322/3GASi2C3−II−log3/23/2GASi2C3−II−log2m+2nGASi2C3−II−log15mn−13m−13n+11GASi2C3−II.


#### 3.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the first Zagreb index and entropy are M1Si2C3−II=90mn−30m−30n+6 and(25)EM1Si2C3−II=log90mn−30m−30n+6−log5490mn−30m−30n+6−log25690mn−30m−30n+6−log2562m+2n90mn−30m−30n+6−log31258m+8n−1490mn−30m−30n+6−log4665615mn−13m−13n+1190mn−30m−30n+6.


#### 3.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−IIm,n

By using Table3, the second Zagreb index and second Zagreb entropy are M2Si2C3−II=135mn−61m−61n+22 and(26)EM2Si2C3−II=log135mn−61m−61n+22−log8135mn−61m−61n+22−log27135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×55135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.


#### 3.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the hyper-Zagreb index and hyper-Zagreb entropy are (27)HMSi2C3−II=540mn−236m+n+71,EHMSi2C3−II=log540mn−236m+n+71−log2×99540mn−236m+n+71−logm+8n−14×2525540mn−236m+n+71−log2m+2n×1616540mn−236m+n+71−log1616540mn−236m+n+71−log363615mn−13m−13n+11540mn−236m+n+71.


#### 3.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−IIm,n

By using Table5, the forgotten index and forgotten entropy are(28)FSi2C3−II=270mn−114m+n+36,EFSi2C3−II=log270mn−114m+n+36−log2×55270mn−114m+n+36−log1010270mn−114m+n+36−log2m+2n×88270mn−114m+n+36−log8m+8n−14×1313270mn−114m+n+36−log15mn−13m−13n+11×1818270mn−114m+n+36.


#### 3.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−IIm,n

By using Table5, it is easy to see that the augmented Zagreb index and augmented Zagreb entropy are(29)AZISi2C3−II=10935/64mn−4357/64m−4357/64n+2091/64,EAZISi2C3−II=log1093564mn−435764m−435764n+209164−log2×8810935/64mn−4357/64m−4357/64n+2091/64−log1×27/827/810935/64mn−4357/64m−4357/64n+2091/64−log2m+2n×8810935/64mn−4357/64m−4357/64n+2091/64−log8m+8n−14×8810935/64mn−4357/64m−4357/64n+2091/64−log15mn−13m−13n+11×93/4393/4310935/64mn−4357/64m−4357/64n+2091/64.


#### 3.2.9. The Balaban Index and Balaban Entropy forSi2C3−IIm,n

By using Table2, it is easy to see that the Balaban index and Balaban entropy are(30)JSi2C3−II=15mn−3m+n5mn−3m+n+22+13+m+n+168m+8n−14+1315mn−13m−13n+11,EJSi2C3−II=logJSi2C3−II−log2×q/2q−p+2q/2q−p+2JSi2C3−II−log1×q/3q−p+2q/3q−p+2JSi2C3−II−log2m+2n×q/2q−p+2q/2q−p+2JSi2C3−II−log8m+8n−14×q/6q−p+2q/6q−p+2JSi2C3−II−log15mn−13m−13n+11×q/3q−p+2q/3q−p+2JSi2C3−II.


#### 3.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−II=10mn and(31)ENTReZG1Si2C3−II=log10mn−log2×3/23/210mn−log4/34/310mn−log2m+2n10mn−log8m+8n−14×5/65/610mn−log15mn−13m−13n+11×2/32/310mn.


#### 3.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−II=45mn/3−79m+n/10+107/60 and(32)EReZG2Si2C3−II=log453mn−7910m−7910n+10760−log2×2/32/345/3mn−79/10m−79/10n+107/60−log1×3/43/445/3mn−79/10m−79/10n+107/60−log2m+2n×145/3mn−79/10m−79/10n+107/60−log8m+8n−14×6/56/545/3mn−79/10m−79/10n+107/60−log15mn−13m−13n+11×3/23/245/3mn−79/10m−79/10n+107/60.


#### 3.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined third Zagreb index and redefined third Zagreb entropy are(33)ReZG3Si2C3−II=810mn−430m+n+198,EReZG3Si2C3−II=log810mn−430m−430n+198−log2×66810mn−430m−430n+198−log1×1212810mn−430m−430n+198−log2m+2n×1616810mn−430m−430n+198−log8m+8n−14×3030810mn−430m−430n+198−log15mn−13m−13n+11×5454810mn−430m−430n+198.


#### 3.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−IIm,n

By using Table6, the fourth ABC index and fourth ABC entropy are(34)ABC4Si2C3−II=23mn3+m+n22215+227+21142+1110+144+223+56−679+2221+53+275−825−21142−31110+21356−14+437−5256+659,ENTABC4G=logABC4−1ABC4log2×121/2−1ABC4log2m+2n−4×22522/5−1ABC4log2×5235/23−1ABC4log2×7257/25−1ABC4log222122/21−1ABC4log2m+2n×272/7−1ABC4log2m+2n−2×114211/42−1ABC4log2m+2n−6×121/2−1ABC4log2m+2n−6×114011/40−1ABC4log2×135613/56−1ABC4log2m+2n−8×14814/8−1ABC4log2×23723/7−1ABC4log2m+2n−3×232/3−1ABC4log2m+2n−5×5265/26−1ABC4log15mn−19m−19n+23×494/9.


#### 3.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−IIm,n

By using Table6, the fifth GA index and fifth GA entropy are(35)GA5G=15mn+m+n−15+353+44213+837+81013+374+24217+465+215+837+859−44213−2437−241013+81415−978−60217+13,EGA5G=logGA5−1GA5log2×26526/5−1GA5log21521/5−1GA5log2×43743/7−1GA5log2×45945/9−1GA5log2m+2n−4−1GA5log2m+2n×35635/6−1GA5log2m+2n−2×24213242/13−1GA5log2m+2n−6×43743/7−1GA5log2m+2n−6×41013410/13−1GA5log2×41415414/15−1GA5log2m+2n−8−1GA5log2−1GA5log2m+2n−3×37837/8−1GA5log2m+2n−5×1227122/7−1GA5log15mn−19m−19n+23.


## 3.1. Methodology of Silicon CarbideSi2C3−IIm,n Formulas

For silicon carbideSi2C3−IIm,n, we join one unit cell with another unit cell and then continue this process horizontally till n unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the Si2C3−IIm,n sheet; see Figure 3. Moreover, Tables 4 and 5 are used for computation of vertices and edges, respectively.Table 4
Vertex partition ofSi2C3−IIm,n established on degrees of each vertex.
τmNo. of repetitionsKinds of vertices13MV126m+6n−6MV2310mn−6m−6n+3MV3Table 5
Edge partition ofSi2C3−IIm,n established on degrees of end vertices.
τr,τsNo. of repetitionsKinds of edges2,12ME13,11ME22,22m+2nME33,28m+8n−14ME43,315mn−13m−13n+11ME5The edge partition ofSi2C3−IIm,n established on the addition of degree of terminal vertices of every edge is illustrated in Table 6.Table 6
Edge partition ofSi2C3−IIm,n, m,n,s≥2.
Ss,StFrequency3,227,314,325,425,5−4+2m+n7,52m+n7,6−2+2m+n8,6−6+2m+n8,5−6+m+n8,728,8−8+2m+n7,729,7−3+2m+n9,8−5+2m+n9,923−19n−19m+15mn


## 3.2. Computation of Entropies for Crystallographic Structure ofSi2C3−IIm,n

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure ofSi2C3−IIm,n.


### 3.2.1. The Randić Index and Randić Entropy forSi2C3−IIm,n

Forα=1, the Randić index and Randić entropy by Table 5 are(19)R1Si2C3−II=135mn−61m−61n+22,ER1Si2C3−II=log135mn−61m−61n+22−log2×22135mn−61m−61n+22−log1×33135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×66135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.Forα=−1, by using Table 5, the Randić index and Randić entropy are R−1Si2C3−II=5/3mn+7/18m+7/18n+2/9 and(20)ER−1Si2C3−II=log53mn+718m+718n+29−log2×2−1/25/3mn+7/18m+7/18n+2/9−log1×3−1/35/3mn+7/18m+7/18n+2/9−log2m+2n×4−1/45/3mn+7/18m+7/18n+2/9−log8m+8n−14×6−1/65/3mn+7/18m+7/18n+2/9−log15mn−13m−13n+11×9−1/95/3mn+7/18m+7/18n+2/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are R1/2Si2C3−I=45mn+86−35m+n+22+3−146+33 and(21)ER12Si2C3−II=log45mn+86−35m+n+22+3−146+33−log2×22/245mn+86−35m+n+22+3−146+33−log1×33/245mn+86−35m+n+22+3−146+33−log2m+2n×44/245mn+86−35m+n+22+3−146+33−log8m+8n−14×66/245mn+86−35m+n+22+3−146+33−log15mn−13m−13n+11×99/245mn+86−35m+n+22+3−146+33.Forα=−1/2, by using Table 2, the Randić index and Randić entropy are (22)R−1/2Si2C3−II=5mn+8/6−10/3m+n+2+1/3−14/6+11/3,ER−1/2Si2C3−II=log5mn+86−103m+n+2+13−146+113−log2×2−1/225mn+8/6−10/3m+n+2+1/3−14/6+11/3−log1×3−1/235mn+8/6−10/3m+n+2+1/3−14/6+11/3−log2m+2n×4−1/245mn+8/6−10/3m+n+2+1/3−14/6+11/3−log8m+8n−14×6−1/265mn+8/6−10/3m+n+2+1/3−14/6+11/3−log15mn−13m−13n+11×9−1/295mn+8/6−10/3m+n+2+1/3−14/6+11/3.


### 3.2.2. TheABC Index and ABC Entropy of Si2C3−IIm,n

By using Table5, the ABC index and ABC entropy are (23)ABCSi2C3−II=10mn+10/2−26/3m+n−12/2+2/3+22/3,EABCSi2C3−II=logABCSi2C3−II−log2×1/21/2ABCSi2C3−II−log15mn−13m−13n+11×2/32/3ABCSi2C3−II−log2m+2n×1/21/2ABCSi2C3−II−log8m+8n−14×1/21/2ABCSi2C3−II−log1×2/32/3ABCSi2C3−II.


### 3.2.3. TheGA Index and GA Entropy of Si2C3−IIm,n

By using Table5, the GA index and GA entropy are(24)GASi2C3−II=15mn+166/5−11m+n+42/3+3/2−286/5+11EGASi2C3−II=logGASi2C3−II−log2×22/322/3GASi2C3−II−log3/23/2GASi2C3−II−log2m+2nGASi2C3−II−log15mn−13m−13n+11GASi2C3−II.


### 3.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the first Zagreb index and entropy are M1Si2C3−II=90mn−30m−30n+6 and(25)EM1Si2C3−II=log90mn−30m−30n+6−log5490mn−30m−30n+6−log25690mn−30m−30n+6−log2562m+2n90mn−30m−30n+6−log31258m+8n−1490mn−30m−30n+6−log4665615mn−13m−13n+1190mn−30m−30n+6.


### 3.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−IIm,n

By using Table3, the second Zagreb index and second Zagreb entropy are M2Si2C3−II=135mn−61m−61n+22 and(26)EM2Si2C3−II=log135mn−61m−61n+22−log8135mn−61m−61n+22−log27135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×55135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.


### 3.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the hyper-Zagreb index and hyper-Zagreb entropy are (27)HMSi2C3−II=540mn−236m+n+71,EHMSi2C3−II=log540mn−236m+n+71−log2×99540mn−236m+n+71−logm+8n−14×2525540mn−236m+n+71−log2m+2n×1616540mn−236m+n+71−log1616540mn−236m+n+71−log363615mn−13m−13n+11540mn−236m+n+71.


### 3.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−IIm,n

By using Table5, the forgotten index and forgotten entropy are(28)FSi2C3−II=270mn−114m+n+36,EFSi2C3−II=log270mn−114m+n+36−log2×55270mn−114m+n+36−log1010270mn−114m+n+36−log2m+2n×88270mn−114m+n+36−log8m+8n−14×1313270mn−114m+n+36−log15mn−13m−13n+11×1818270mn−114m+n+36.


### 3.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−IIm,n

By using Table5, it is easy to see that the augmented Zagreb index and augmented Zagreb entropy are(29)AZISi2C3−II=10935/64mn−4357/64m−4357/64n+2091/64,EAZISi2C3−II=log1093564mn−435764m−435764n+209164−log2×8810935/64mn−4357/64m−4357/64n+2091/64−log1×27/827/810935/64mn−4357/64m−4357/64n+2091/64−log2m+2n×8810935/64mn−4357/64m−4357/64n+2091/64−log8m+8n−14×8810935/64mn−4357/64m−4357/64n+2091/64−log15mn−13m−13n+11×93/4393/4310935/64mn−4357/64m−4357/64n+2091/64.


### 3.2.9. The Balaban Index and Balaban Entropy forSi2C3−IIm,n

By using Table2, it is easy to see that the Balaban index and Balaban entropy are(30)JSi2C3−II=15mn−3m+n5mn−3m+n+22+13+m+n+168m+8n−14+1315mn−13m−13n+11,EJSi2C3−II=logJSi2C3−II−log2×q/2q−p+2q/2q−p+2JSi2C3−II−log1×q/3q−p+2q/3q−p+2JSi2C3−II−log2m+2n×q/2q−p+2q/2q−p+2JSi2C3−II−log8m+8n−14×q/6q−p+2q/6q−p+2JSi2C3−II−log15mn−13m−13n+11×q/3q−p+2q/3q−p+2JSi2C3−II.


### 3.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−II=10mn and(31)ENTReZG1Si2C3−II=log10mn−log2×3/23/210mn−log4/34/310mn−log2m+2n10mn−log8m+8n−14×5/65/610mn−log15mn−13m−13n+11×2/32/310mn.


### 3.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−II=45mn/3−79m+n/10+107/60 and(32)EReZG2Si2C3−II=log453mn−7910m−7910n+10760−log2×2/32/345/3mn−79/10m−79/10n+107/60−log1×3/43/445/3mn−79/10m−79/10n+107/60−log2m+2n×145/3mn−79/10m−79/10n+107/60−log8m+8n−14×6/56/545/3mn−79/10m−79/10n+107/60−log15mn−13m−13n+11×3/23/245/3mn−79/10m−79/10n+107/60.


### 3.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined third Zagreb index and redefined third Zagreb entropy are(33)ReZG3Si2C3−II=810mn−430m+n+198,EReZG3Si2C3−II=log810mn−430m−430n+198−log2×66810mn−430m−430n+198−log1×1212810mn−430m−430n+198−log2m+2n×1616810mn−430m−430n+198−log8m+8n−14×3030810mn−430m−430n+198−log15mn−13m−13n+11×5454810mn−430m−430n+198.


### 3.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−IIm,n

By using Table6, the fourth ABC index and fourth ABC entropy are(34)ABC4Si2C3−II=23mn3+m+n22215+227+21142+1110+144+223+56−679+2221+53+275−825−21142−31110+21356−14+437−5256+659,ENTABC4G=logABC4−1ABC4log2×121/2−1ABC4log2m+2n−4×22522/5−1ABC4log2×5235/23−1ABC4log2×7257/25−1ABC4log222122/21−1ABC4log2m+2n×272/7−1ABC4log2m+2n−2×114211/42−1ABC4log2m+2n−6×121/2−1ABC4log2m+2n−6×114011/40−1ABC4log2×135613/56−1ABC4log2m+2n−8×14814/8−1ABC4log2×23723/7−1ABC4log2m+2n−3×232/3−1ABC4log2m+2n−5×5265/26−1ABC4log15mn−19m−19n+23×494/9.


### 3.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−IIm,n

By using Table6, the fifth GA index and fifth GA entropy are(35)GA5G=15mn+m+n−15+353+44213+837+81013+374+24217+465+215+837+859−44213−2437−241013+81415−978−60217+13,EGA5G=logGA5−1GA5log2×26526/5−1GA5log21521/5−1GA5log2×43743/7−1GA5log2×45945/9−1GA5log2m+2n−4−1GA5log2m+2n×35635/6−1GA5log2m+2n−2×24213242/13−1GA5log2m+2n−6×43743/7−1GA5log2m+2n−6×41013410/13−1GA5log2×41415414/15−1GA5log2m+2n−8−1GA5log2−1GA5log2m+2n−3×37837/8−1GA5log2m+2n−5×1227122/7−1GA5log15mn−19m−19n+23.


## 3.2.1. The Randić Index and Randić Entropy forSi2C3−IIm,n

Forα=1, the Randić index and Randić entropy by Table 5 are(19)R1Si2C3−II=135mn−61m−61n+22,ER1Si2C3−II=log135mn−61m−61n+22−log2×22135mn−61m−61n+22−log1×33135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×66135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.Forα=−1, by using Table 5, the Randić index and Randić entropy are R−1Si2C3−II=5/3mn+7/18m+7/18n+2/9 and(20)ER−1Si2C3−II=log53mn+718m+718n+29−log2×2−1/25/3mn+7/18m+7/18n+2/9−log1×3−1/35/3mn+7/18m+7/18n+2/9−log2m+2n×4−1/45/3mn+7/18m+7/18n+2/9−log8m+8n−14×6−1/65/3mn+7/18m+7/18n+2/9−log15mn−13m−13n+11×9−1/95/3mn+7/18m+7/18n+2/9.Forα=1/2, by using Table 2, the Randić index and Randić entropy are R1/2Si2C3−I=45mn+86−35m+n+22+3−146+33 and(21)ER12Si2C3−II=log45mn+86−35m+n+22+3−146+33−log2×22/245mn+86−35m+n+22+3−146+33−log1×33/245mn+86−35m+n+22+3−146+33−log2m+2n×44/245mn+86−35m+n+22+3−146+33−log8m+8n−14×66/245mn+86−35m+n+22+3−146+33−log15mn−13m−13n+11×99/245mn+86−35m+n+22+3−146+33.Forα=−1/2, by using Table 2, the Randić index and Randić entropy are (22)R−1/2Si2C3−II=5mn+8/6−10/3m+n+2+1/3−14/6+11/3,ER−1/2Si2C3−II=log5mn+86−103m+n+2+13−146+113−log2×2−1/225mn+8/6−10/3m+n+2+1/3−14/6+11/3−log1×3−1/235mn+8/6−10/3m+n+2+1/3−14/6+11/3−log2m+2n×4−1/245mn+8/6−10/3m+n+2+1/3−14/6+11/3−log8m+8n−14×6−1/265mn+8/6−10/3m+n+2+1/3−14/6+11/3−log15mn−13m−13n+11×9−1/295mn+8/6−10/3m+n+2+1/3−14/6+11/3.


## 3.2.2. TheABC Index and ABC Entropy of Si2C3−IIm,n

By using Table5, the ABC index and ABC entropy are (23)ABCSi2C3−II=10mn+10/2−26/3m+n−12/2+2/3+22/3,EABCSi2C3−II=logABCSi2C3−II−log2×1/21/2ABCSi2C3−II−log15mn−13m−13n+11×2/32/3ABCSi2C3−II−log2m+2n×1/21/2ABCSi2C3−II−log8m+8n−14×1/21/2ABCSi2C3−II−log1×2/32/3ABCSi2C3−II.


## 3.2.3. TheGA Index and GA Entropy of Si2C3−IIm,n

By using Table5, the GA index and GA entropy are(24)GASi2C3−II=15mn+166/5−11m+n+42/3+3/2−286/5+11EGASi2C3−II=logGASi2C3−II−log2×22/322/3GASi2C3−II−log3/23/2GASi2C3−II−log2m+2nGASi2C3−II−log15mn−13m−13n+11GASi2C3−II.


## 3.2.4. The First Zagreb Index and First Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the first Zagreb index and entropy are M1Si2C3−II=90mn−30m−30n+6 and(25)EM1Si2C3−II=log90mn−30m−30n+6−log5490mn−30m−30n+6−log25690mn−30m−30n+6−log2562m+2n90mn−30m−30n+6−log31258m+8n−1490mn−30m−30n+6−log4665615mn−13m−13n+1190mn−30m−30n+6.


## 3.2.5. The Second Zagreb Index and Second Zagreb Entropy ofSi2C3−IIm,n

By using Table3, the second Zagreb index and second Zagreb entropy are M2Si2C3−II=135mn−61m−61n+22 and(26)EM2Si2C3−II=log135mn−61m−61n+22−log8135mn−61m−61n+22−log27135mn−61m−61n+22−log2m+2n×44135mn−61m−61n+22−log8m+8n−14×55135mn−61m−61n+22−log15mn−13m−13n+11×99135mn−61m−61n+22.


## 3.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy ofSi2C3−IIm,n

By using Table5, the hyper-Zagreb index and hyper-Zagreb entropy are (27)HMSi2C3−II=540mn−236m+n+71,EHMSi2C3−II=log540mn−236m+n+71−log2×99540mn−236m+n+71−logm+8n−14×2525540mn−236m+n+71−log2m+2n×1616540mn−236m+n+71−log1616540mn−236m+n+71−log363615mn−13m−13n+11540mn−236m+n+71.


## 3.2.7. The Forgotten Index and Forgotten Entropy forSi2C3−IIm,n

By using Table5, the forgotten index and forgotten entropy are(28)FSi2C3−II=270mn−114m+n+36,EFSi2C3−II=log270mn−114m+n+36−log2×55270mn−114m+n+36−log1010270mn−114m+n+36−log2m+2n×88270mn−114m+n+36−log8m+8n−14×1313270mn−114m+n+36−log15mn−13m−13n+11×1818270mn−114m+n+36.


## 3.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy ofSi2C3−IIm,n

By using Table5, it is easy to see that the augmented Zagreb index and augmented Zagreb entropy are(29)AZISi2C3−II=10935/64mn−4357/64m−4357/64n+2091/64,EAZISi2C3−II=log1093564mn−435764m−435764n+209164−log2×8810935/64mn−4357/64m−4357/64n+2091/64−log1×27/827/810935/64mn−4357/64m−4357/64n+2091/64−log2m+2n×8810935/64mn−4357/64m−4357/64n+2091/64−log8m+8n−14×8810935/64mn−4357/64m−4357/64n+2091/64−log15mn−13m−13n+11×93/4393/4310935/64mn−4357/64m−4357/64n+2091/64.


## 3.2.9. The Balaban Index and Balaban Entropy forSi2C3−IIm,n

By using Table2, it is easy to see that the Balaban index and Balaban entropy are(30)JSi2C3−II=15mn−3m+n5mn−3m+n+22+13+m+n+168m+8n−14+1315mn−13m−13n+11,EJSi2C3−II=logJSi2C3−II−log2×q/2q−p+2q/2q−p+2JSi2C3−II−log1×q/3q−p+2q/3q−p+2JSi2C3−II−log2m+2n×q/2q−p+2q/2q−p+2JSi2C3−II−log8m+8n−14×q/6q−p+2q/6q−p+2JSi2C3−II−log15mn−13m−13n+11×q/3q−p+2q/3q−p+2JSi2C3−II.


## 3.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined first Zagreb index and redefined first Zagreb entropy are ReZG1Si2C3−II=10mn and(31)ENTReZG1Si2C3−II=log10mn−log2×3/23/210mn−log4/34/310mn−log2m+2n10mn−log8m+8n−14×5/65/610mn−log15mn−13m−13n+11×2/32/310mn.


## 3.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined second Zagreb index and redefined second Zagreb entropy are ReZG2Si2C3−II=45mn/3−79m+n/10+107/60 and(32)EReZG2Si2C3−II=log453mn−7910m−7910n+10760−log2×2/32/345/3mn−79/10m−79/10n+107/60−log1×3/43/445/3mn−79/10m−79/10n+107/60−log2m+2n×145/3mn−79/10m−79/10n+107/60−log8m+8n−14×6/56/545/3mn−79/10m−79/10n+107/60−log15mn−13m−13n+11×3/23/245/3mn−79/10m−79/10n+107/60.


## 3.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy forSi2C3−IIm,n

By using Table5, the redefined third Zagreb index and redefined third Zagreb entropy are(33)ReZG3Si2C3−II=810mn−430m+n+198,EReZG3Si2C3−II=log810mn−430m−430n+198−log2×66810mn−430m−430n+198−log1×1212810mn−430m−430n+198−log2m+2n×1616810mn−430m−430n+198−log8m+8n−14×3030810mn−430m−430n+198−log15mn−13m−13n+11×5454810mn−430m−430n+198.


## 3.2.13. The FourthABC Index and Fourth ABC Entropy of Si2C3−IIm,n

By using Table6, the fourth ABC index and fourth ABC entropy are(34)ABC4Si2C3−II=23mn3+m+n22215+227+21142+1110+144+223+56−679+2221+53+275−825−21142−31110+21356−14+437−5256+659,ENTABC4G=logABC4−1ABC4log2×121/2−1ABC4log2m+2n−4×22522/5−1ABC4log2×5235/23−1ABC4log2×7257/25−1ABC4log222122/21−1ABC4log2m+2n×272/7−1ABC4log2m+2n−2×114211/42−1ABC4log2m+2n−6×121/2−1ABC4log2m+2n−6×114011/40−1ABC4log2×135613/56−1ABC4log2m+2n−8×14814/8−1ABC4log2×23723/7−1ABC4log2m+2n−3×232/3−1ABC4log2m+2n−5×5265/26−1ABC4log15mn−19m−19n+23×494/9.


## 3.2.14. The FifthGA Index and Fifth GA Entropy of Si2C3−IIm,n

By using Table6, the fifth GA index and fifth GA entropy are(35)GA5G=15mn+m+n−15+353+44213+837+81013+374+24217+465+215+837+859−44213−2437−241013+81415−978−60217+13,EGA5G=logGA5−1GA5log2×26526/5−1GA5log21521/5−1GA5log2×43743/7−1GA5log2×45945/9−1GA5log2m+2n−4−1GA5log2m+2n×35635/6−1GA5log2m+2n−2×24213242/13−1GA5log2m+2n−6×43743/7−1GA5log2m+2n−6×41013410/13−1GA5log2×41415414/15−1GA5log2m+2n−8−1GA5log2−1GA5log2m+2n−3×37837/8−1GA5log2m+2n−5×1227122/7−1GA5log15mn−19m−19n+23.


## 4. Comparisons and Discussion forSi2C3−Im,n

In this segment, we discussed many topological indices with their respective entropies. The quantitative and pictorial comparisons are illustrated in Tables7–11 and Figures 5–12, respectively. Due to the complexity of molecular structures, different molecular descriptors in association with their entropy measures are used to anticipate the maximum structural information.Table 7
Comparison of Randić entropies forSi2C3−Im,n.
m,nER1ER−1ER1/2ER−1/22,22.480.462.201.163,32.941.032.491.574,43.231.332.771.845,53.451.552.992.066,63.621.723.162.237,73.761.873.302.378,83.891.993.452.499,93.992.113.532.5910,104.092.203.622.69Table 8
Comparison ofEABC, EGA, EM1, and EM2 entropies for Si2C3−Im,n.
m,nEABCEGAEM1EM22,21.461.802.352.483,31.862.172.792.944,42.142.423.083.235,52.352.613.303.456,62.522.773.533.627,72.662.903.683.808,82.783.013.733.899,92.893.113.833.9910,102.983.203.934.11Table 9
Comparison ofEHM, EF, EAZI, and EJ entropies for Si2C3−Im,n.
m,nEHMEFEAZIEJ2,23.172.782.611.663,33.583.243.062.064,43.853.543.342.335,54.073.753.562.536,64.233.933.732.697,74.374.073.872.838,84.504.193.992.959,94.614.304.103.0610,104.704.404.203.15Table 10
Comparison of the redefined Zagreb entropies forSi2C3−Im,n.
m,nEReZG1EReZG2EReZG32,21.511.7613.193,31.902.193.694,42.172.473.995,52.382.694.216,62.542.864.397,72.682.994.548,82.803.124.669,92.903.234.7710,102.993.324.87Table 11
Comparison of theABC4 and GA5 entropies for Si2C3−Im,n.
m,nEABC4EGA52,21.341.693,31.732.084,41.992.355,52.202.556,62.362.717,72.502.858,82.622.979,92.723.0710,102.823.17Figure 5
(a) Randić entropy forα=1; (b) Randić entropy for α=−1.
(a)(b)Figure 6
(a) Randić entropy forα=1/2; (b) Randić entropy for α=−1/2.
(a)(b)Figure 7
(a) The atom bond connectivity entropy; (b) the geometric arithmetic entropy.
(a)(b)Figure 8
(a) The first Zagreb entropy; (b) the second Zagreb entropy.
(a)(b)Figure 9
(a) The hyper-Zagreb entropy; (b) the forgotten entropy.
(a)(b)Figure 10
(a) The augmented Zagreb entropy; (b) the Balaban entropy.
(a)(b)Figure 11
(a) The redefined first Zagreb entropy; (b) the redefined second Zagreb entropy; (c) the redefined third Zagreb entropy.
(a)(b)(c)Figure 12
(a) TheABC4 entropy; (b) the GA5 entropy.
(a)(b)These descriptors possess a lot of chemical properties like Randić index which has good correlation between various physicochemical characteristics of alkanes such as retention times of chromatographic parameters in the Antoine equation for surface areas, vapor pressure, and boiling points, etc. Recently, the utilization of this index is related with medicine and pharmacological matters. AlsoABC index can be a useful prognostic tool in the investigation of the heat formation in alkanes. It is also associated with stability of linear and branched alkanes both quantitatively and qualitatively. Furthermore, the geometric arithmetic index provides better correlation coefficients as compared to other descriptors.The Zagreb type indices are used to measure the totalπ-electron energy of the molecules [2]. In recent times, Zagreb indices along with their invariant have been employed to examine complexity and heterosystems [29]. These indices are also used for acquiring multilinear regression models as well as in QSPR and QSAR studies [30]. The forgotten index is useful in the analysis of pharmacological and chemical properties of medication atomic structures. The AZI index determines an excellent correlation of branched alkanes as well as linear alkanes. The strain energy of cycloalkanes is also estimated by this index. Furthermore, Balaban index is correlated with acute toxicity of mediums and variety of physicochemical properties. It is also helpful for transformation of heteroatoms. These indices are also used in designing the fractional bounds. Entropy function is monotonic as, in all cases, an increase of entropy measure is observed with the increase in the size of the molecular structure.


## 5. Comparisons and Discussion forSi2C3−IIm,n

Physicochemical properties depend upon their molecular structures. Thus modeling and prediction of these physicochemical properties and biological activities is an important field of study. Physical organic chemistry will be intended for clarifying thoroughly how these properties are established by the structure. So in this consideration, one of the essential points is the choice of suitable topological descriptors embracing the information stored in the molecular structure. It is to be taken into consideration that due to the complexity of these molecular structures, it is no possible to anticipate that a single descriptor would be able to contain all the structural information. This is the foremost purpose why the study for molecular structure descriptors remains through a usual procedure grounded on the required characteristics that a molecular structure descriptor requires to possess.In this section, we estimated degree-based entropy measures. In [31, 32], many algorithms were recommended to examine the structural complexity. But the entropy approach is reviewed to be the most substantial approach to distinguish the structural information of the complex networks. Therefore, we have listed mathematically some degree-based entropies for small considerations of parameters for Si2C3−IIm,n. Also, we produce Tables 12–16 with the help of Matlab for small estimations of m,n. From tables, we can note that all the evaluation of entropy is in growing request as the values of parameters are expanded. These estimations are very helpful for the chemist to analyse chemical properties of the structures. The graphical representation of computed findings is demonstrated in Figures 13–20 for certain measurements of m,n.Table 12
Comparison of Randić entropies forSi2C3−IIm,n.
m,nER1ER−1ER1/2ER−1/22,22.440.602.011.143,32.991.022.461.564,43.221.322.761.845,53.441.552.982.056,63.611.733.192.227,73.761.883.292.368,83.882.013.412.489,93.992.113.522.5910,104.092.213.622.68Table 13
Comparison ofEABC, EGA, EM1, and EM2 entropies for Si2C3−IIm,n.
m,nEABCEGAEM1EM22,21.431.802.342.443,31.852.172.782.924,42.132.423.073.225,52.342.613.283.446,62.522.773.463.617,72.662.903.603.768,82.783.013.723.889,92.883.113.833.9910,102.983.203.924.09Table 14
Comparison ofEHM, EF, EAZI, and EJ entropies for Si2C3−IIm,n.
m,nEHMEFEAZIEJ2,22.992.732.561.793,33.513.223.032.164,43.813.523.332.415,54.043.703.552.616,64.213.923.722.767,74.364.063.872.898,84.484.193.993.019,94.594.294.103.1010,104.694.394.193.20Table 15
Comparison of the redefined Zagreb entropies forSi2C3−IIm,n.
m,nEReZG1EReZG2EReZG32,21.561.423.183,31.931.933.674,42.192.243.985,52.392.464.206,62.542.644.387,72.682.794.538,82.802.924.659,92.903.034.7610,102.993.134.86Table 16
Comparison of theABC4 and GA5 entropies for Si2C3−IIm,n.
m,nEABC4EGA52,21.341.643,31.732.054,41.992.325,52.202.536,62.362.707,72.502.848,82.622.969,92.723.0610,102.813.16Figure 13
(a) Randić entropy forα=1; (b) Randić entropy for α=−1.
(a)(b)Figure 14
(a) Randić entropy forα=1/2; (b) Randić entropy for α=−1/2.
(a)(b)Figure 15
(a) The atom bond connectivity entropy; (b) the geometric arithmetic entropy.
(a)(b)Figure 16
(a) The first Zagreb entropy; (b) the second Zagreb entropy.
(a)(b)Figure 17
(a) The hyper-Zagreb entropy; (b) the forgotten entropy.
(a)(b)Figure 18
(a) The augmented Zagreb entropy; (b) the Balaban entropy.
(a)(b)Figure 19
(a) The redefined first Zagreb entropy; (b) the redefined second Zagreb entropy; (c) the redefined third Zagreb entropy.
(a)(b)(c)Figure 20
(a) TheABC4 entropy; (b) the GA5 entropy.
(a)(b)


## 6. Conclusion

In this study, we considered the molecular structure of silicon carbideSi2C3−Im,n and Si2C3−IIm,n. We estimated degree-based indices like general Randić index, ABC index, GA index, Zagreb type indices, forgotten index, hyper-Zagreb index, AZI index, Balaban index, redefined Zagreb indices, fourth atom bond connectivity index, and fifth geometric arithmetic index. Afterwards, we extend our computation towards the estimation of corresponding entropies of aforementioned degree-based indices. We compared these estimations both numerically and graphically. These entropies associate particular physicochemical characteristics like distortion, stability, melting points, and strain energy of chemical compounds. The mathematical findings for these graphs are helpful for the chemist to understand the biochemical utilization of these structures.



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