# Computing Some Degree-Based Topological Indices of Honeycomb Networks

Authors: Lili Gu; Shamaila Yousaf; Akhlaq Ahmad Bhatti; Peng Xu; Adnan Aslam
Journal: Complexity
 (2022)

---

## Abstract

A topological index is a numeric quantity related with the chemical composition claiming to correlate the chemical structure with different chemical properties. Topological indices serve to predict physicochemical properties of chemical substance. Among different topological indices, degree-based topological indices would be helpful in investigating the anti-inflammatory activities of certain chemical networks. In the current study, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for oxide networkOXn, silicate network SLn, chain silicate network CSn, and hexagonal network HXn. Also, we determine the neighborhood second Zagreb index and the first extended first-order connectivity index for honeycomb network HCn.


---

## Body

## 1. Introduction

Chemical graph theory/mathematical chemistry has made easier for the scientists to predict physicochemical properties of different chemical structures using topological indices. A numeric value assigned to a molecular graph (a graph portraying chemical compounds) is recognized as the topological index, and it remains unchanged under the isomorphism of the graph. For a given period, there is an increasing interest in topological indices that reflect some structural features of molecular compounds. Topological indices hold attention because they correlate well with certain chemical properties of molecular graphs. Also, they play an important role in the quantitative structure-activity relationship (QSAR) and quantitative structure property relationship (QSPR) studies [1]. To overview the structure-activity relationship, topological indices are required to effectively characterize structural features and bioactivity of chemical compounds [2–4]. In the current study, under discussion, molecular structures are nanostructures due to their design and applications.All the graphs in present study are simple, undirected, finite, and connected. For a vertexw∈VG, we use the notation Nw for the set containing the vertices adjacent to w. The degree of a vertex w is the cardinality of the set Nw and is denoted by dw. Let Sw denote the sum of degrees of the vertices adjacent to w. In other words, Sw=∑uw∈EGdu and Nt=v∈VG|tv∈EG. For undefined terminologies related to graph theory, the author can read [5–7]. Consider the following general graph invariant:(1)IG=∑vw∈EGfSv,Sw.Some special cases of the above invariantsI have already been appeared in mathematical chemistry. For example, if we take fSv,Sw=SvSw or 1/SvSw, then I gives neighborhood second Zagreb index NM2 [8] and the first extended first-order connectivity index 1χ1 [9, 10], respectively. These indices are defined as(2)1χ1G=∑vw∈EG1SvSw,(3)NM2G=∑vw∈EGSvSw.Toropov et al. [11] proved that extended molecular connectivity indices have good correlation with the boiling point of chemical structures. Detail about theoretical and computation aspects of some families of chemical structures can be viewed in [12–16].In this report, we are interested in five well-known chemical structures, name as oxide networkOXn, honeycomb network HCn, silicate network SLn, chain silicate network CSn, and hexagonal network HXn. As far as authors are aware, neighborhood second Zagreb index and the first extended first-order connectivity index of oxide network OXn, silicate network SLn, chain silicate network CSn, hexagonal network HXn, and honeycomb network HCn have not been investigated until now, and therefore, the study under consideration is an addition in the aforesaid direction.


## 2. Motivation

Since topological indices are useful to predict the physicochemical properties of chemical compounds, therefore, it is important to analyze the correlation of the topological indices investigated. Yousaf et al. [17] proved that the correlation of entropy and eccentric factor of octane isomers with neighborhood second Zagreb index and the first extended first-order connectivity index has high accuracy. In addition, topological indices are moreover used for discrimination against octane isomers. The discrimination ability of an index has remarkable application within the coding and the computer processing of molecular structures (isomers).Denote byxq,rH (or xq,r, when terms are not confusing) the number of edges in H with end-vertex degrees q and r. Two graphs H1 and H2 are called edge-equivalent graphs if they are satisfying xq,rH1=xq,rH2 for all q and r with 1≤q≤r≤Δ. A topological index TI is called edge-equivalent topological index if it satisfies TIH1=TIH2 for every pair of edge-equivalent graphs H1 and H2. We notice that the bond incident degree indices [18, 19] (BID indices for short [20]) are edge-equivalent topological indices. General form of the BID indices is(4)BIDH=∑vw∈EGgdv,dw=∑q≤rxq,rgq,r.where gq,r is a bivariate symmetric function. It is worth noting that most of the degree-based topological indices used in mathematical chemistry are BID indices. These indices may be produced from equation (4) based on the selection of gq,r.To study the discriminatory performance of the topological indexNM2 and 1χ1, the class of octane isomers are considered which are representing the 8-vertex trees. These 18 molecular graphs of octane isomers are shown in Figure 1. Figure 1 makes it easy to see that trees O3 and O4 are edge-equivalent, but their corresponding first extended first-order connectivity indices and neighborhood second Zagreb indices are not the same: 1χ1O3=0.097, 1χ1O4=0.096, NM2O3=106, and NM2O4=107. Similarly, the trees O11 and O12 are edge-equivalent graphs possessing different first extended first-order connectivity indices and neighborhood second Zagreb indices: 1χ1O11=0.086, 1χ1O12=0.087, NM2O13=136, and NM2O4=137.Figure 1
Graphs of octane isomers. (a)O1. (b) O2. (c) O3. (d) O4. (e) O5. (f) O6. (g) O7. (h) O8. (i) O9. (j) O10. (k) O11. (l) O12. (m) O13. (n) O14. (o) O15. (p) O16. (q) O17. (r) O18.Thus, the topological indexNM2 and 1χ1 does not belong to the class of edge-equivalent topological indices. Subsequently, we can presume that the topological indices NM2 and 1χ1 are identified by a better discriminatory power than the conventional BID indices. Particularly, Wang et al. [21] proved that all the molecular graphs of 18 octane isomers have different NM2 and 1χ1 values. It is worth mentioning that, in certain specific cases, the discriminatory capacity of NM2 and 1χ1 is limited, but still better than that of the BID indices. This can be explained by examining the corresponding edge-equivalent graphs illustrated in Figure 2: we have NM2D1=451 and 1χ1D1=1.09, but NM2D2=NM2D3=448 and 1χ1D2=1χ1D3=1.08 (of course, these three graphs are identical for any arbitrary BID index; in other words, with the help of an arbitrary BID index, it is difficult to distinguish between the graphs D1, D2, and D3).Figure 2
Edge-equivalent graphs. (a)D1. (b) D2. (c) D3.
(a)(b)(c)


## 3. Main Results and Discussion

Multiprocessor interconnection networks are frequently needed to join many identical processor-memory pairs that are repeated, each one is referred to as a processing node. Message passing rather than shared memory is frequently used for synchronization of a communication between the processing nodes for the execution of the program. Multiprocessor interconnection networks are most attracted networks due to the accessibility of powerful and inexpensive microprocessors and memory chips. By repeating regular polygons, periodic plane tessellations can be easily built. For direct interconnection networks, this design is very important as it offers high global performance. Some parallel networks that originate from popular meshes are silicate, chain silicate, hexagonal, honeycomb, and oxide networks. Such networks have very attractive topological properties that have been investigated in various ways in [22–33].


### 3.1. Oxide Networks

Oxide networks are important in investigating silicate networks. Oxide networks are obtained by the deletion of silicon nodes from a silicate network (see Figure3). An oxide network of dimension n is denoted as OXn. The order and size of OXn is 9n2+3n and 18n2, respectively. In oxide networks OXn, the edge set can be partitioned into six sets depending on degree summation of neighbors of end vertices. This partition is presented in Table 1.Figure 3
An oxide networkOX2.Table 1
Edge partitions ofOXn.
SvSw,wherevw∈EOXnCardinality8,121212,141216,1618n2−36n+188,112n−114,1612n−114,1462n−3In the following theorems, we investigate1χ1 and NM2 of OXn.Theorem 1.
The1χ1 index of oxide networks OXn is given by(5)1χ1OXn=98n2+37+314−3928n+36−37+642−314−956.Proof.
We use equation (2) and Table 1 to calculate the expression for 1χ1 index of the oxide networks OXn:(6)1χ1OXn=∑vw∈EG1SvSw=1218×12+12112×14+12n−118×14+62n−3114×14+12n−1114×16+18n2−36n+18116×16=98n2+37+314−3928n+36−37+642−314−956.Theorem 2.
TheNM2 index of oxide networks OXn is given by(7)NM2OXn=4608n2−2832n+216.Proof.
To calculate the expression for neighborhood second Zagreb index of oxide networksOXn, we use equation (3) and Table 1 to get the required result:(8)NM2OXn=∑vw∈EGSvSw=128×12+1212×14+62n−314×14+12n−18×14+18n2−36n+1816×16+12n−114×16=4608n2−2832n+216.


### 3.2. Honeycomb Networks

In material sciences, honeycomb networks have vital role in different fields especially in computer graphics, cellular phone base stations, and image processing and in representing benzenoid hydrocarbons. Honeycomb networks are shaped by the iterative use of hexagonal tiles in a specific design. We denote then-dimensional honeycomb network with n hexagons between central and boundary hexagon by HCn. HCn is constructed by adding layer of hexagon around HCn−1 (see Figure 4). The order and size of HCn is 6n2 and 9n2−3n, respectively. Table 2 shows partition of edges.Figure 4
Honeycomb network.Table 2
Edge partitions ofHCn.
SvSw,wherevw∈EHCnCardinality5,565,712n−17,96n−19,99n2−21n+12By using partition of edges ofHCn, we investigate 1χ1 and NM2 of HCn in the following theorems.Theorem 3.
The1χ1 index of Honeycomb network HCn is given by(9)1χ1HCn=n2+27+1235−73n+3815−1235−27.Proof.
We use equation (2) and Table 2 to calculate the expression for 1χ1 index of the Honeycomb network HCn:(10)χ11HCn=∑vw∈EG1SvSw=615×5+12n−115×7+6n−117×9+9n2−21n+1219×9=n2+27+1235−73n+3815−1235−27.Theorem 4.
TheNM2 index of Honeycomb network HCn is given by(11)NM2HCn=729n2−903n+324.Proof.
To calculate the expression forNM2 index of Honeycomb network HCn, we use equation (3) and Table 2, to get the required result:(12)NM2HCn=∑vw∈EGSvSw=65×5+12n−15×7+6n−17×9+9n2−21n+129×9=729n2+903n+324.


### 3.3. Silicate Networks

Silicates are constituent elements of the common minerals that form the rock and, absolutely, the most important and the most complicated minerals. The elementary unit of silicates’ minerals are tetrahedronSiO4. Silicate is produced by melting metal oxides or metal sand carbonates. SiO4 tetrahedral content is nearly found in all silicate materials. A silicate sheet is formed by different types of sequences of tetrahedrons which are connected by adjoining oxygen vertices to other tetrahedrons in a plane. Cyclic silicates are networks that result in rings of different lengths after being bound by sharing oxygen vertices. Figures 5 and 6 present certain cyclic and sheet silicates, respectively. Other types of silicates are also available and are illustrated in Figure 7. From a chemical perspective, the oxygen atoms are in fact the corner vertices of tetrahedron SiO4 and silicon atom is its central vertex. Graphically, we describe the central atom as silicon vertex, corner atoms as oxygen vertices, and edges are bonds between them. Figure 8 illustrates a tetrahedron of SiO4. We denote a silicate array with n hexagons between center and border of silicate sheet by SLn. Figure 9 illustrates a three-dimensional silicate network. The order and size of SLn is 15n2+3n and 36n2, respectively. In the following theorems, the silicate networks are analyzed through certain graph invariants.Figure 5
Cyclic silicates.Figure 6
Sheet silicates.Figure 7
(a) Ortho, (b) pyro, and (c) chain silicates.
(a)(b)(c)Figure 8
SiO4 tetrahedron.Figure 9
Forn=3, silicate network SLn.In the next theorem, we computed1χ1 and NM2 of the SLn in the following way.Theorem 5.
The1χ1 index of silicate networks is given by(13)χ11SLn=315+35n2+835+436−515+4310−1645n+410−835−436+215+232−4310−115.Proof.
We use equation (2) and Table 3 to calculate the expression for 1χ1 index of the silicate networks:(14)1χ1SLn=∑vw∈EG1SvSw=6n115×15+24115×24+24n−1115×27+12n−1118×27+18n2−30n+12118×30+12124×27+62n−3127×27+12n−1127×30+18n2−36n+18130×30=2n5+410+8n−135+4n−136+6n2−10n+4215+232+22n−39+4n−1310+6n2−12n+610=315+35n2+835+436−515+4310−1645n+410−835−436+215+232−4310−115.Table 3
Edge partition ofLSn.
SvSw,wherevw∈ELSnCardinality15,156n18,2712n−127,3012n−118,3018n2−30n+1215,2724n−124,271215,242427,2762n−330,3018n2−36n+18Theorem 6.
TheNM2 index of silicate networks is given by(15)NM2SLn=25920n2−13230n+702.Proof.
To calculate the expression forNM2 index of silicate networks, we use equation (3) and Table 3 to get the required result:(16)NM2SLn=∑vw∈EGSvSw=6n15×15+12n−118×27+2415×24+12n−127×30+24n−115×27+18n2−30n+1218×30+1224×27+62n−327×27+18n2−36n+1830×30=25920n2−13230n+702.


### 3.4. Chain Silicate Networks

Chain silicate is acquired by linear arrangement of tetrahedron.n-dimensional chain silicate networks are characterized as follows. A network of n-dimensional chain silicate stands for CSn is acquired by linear arrangement of n tetrahedron. The order and size of CSn with n>1 is 3n+1 and and 6n, respectively. A network of n-dimensional chain silicate is illustrated in Figure 10. Tables 4 and 5 suggest such a partition that we use to compute certain topological indices. Now, within the following theorems, we analyzed the chain silicate networks CSn via topological indices.Figure 10
n-dimensional chain silicate network.Table 4
Edge partition ofCSn.
SvSw,wherevw∈ECSnCardinality12,12612,21615,15n−215,21415,244n−1221,24224,24n−4Table 5
Edge partition ofCS3.
SvSw,wherevw∈ECS3Cardinality12,12615,15112,21615,21421,211(17)1χ1CS3=∑vw∈EG1SvSw=6112×12+1115×15+6112×21+1121×21+4115×21=12+115+17+121+4315.The following results provide the investigation of certain topological indices ofCSn.Theorem 7.
The1χ1 index of CSn is given by(18)1χ1CSn=12+16ifn=2,12+115+17+121+4315ifn=3,13120+2310n+15+17−210+4315+1126ifn>3.Proof.
There are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=CS2, the edge partition is of the type S12,S12 and S12,S18 and each of this partition has six edges. Then,(19)1χ1CS2=∑vw∈EG1SvSw=6112×12+6112×18=12+16.
For the casen=3, the edge partition CSn is presented in Table 5. Now, 1χ1CS3 can be computed as follows:
Let us now turn to the third case ofn>3. To calculate the expression for 1χ1 index of the chain silicate networks CSn, we use equation (2) and Table 4 to obtain the required result:(20)1χ1CS3=∑vw∈EG1SvSw=6112×12+1115×15+6112×21+1121×21+4115×21+4n−3115×24+2121×24+n−4124×24=13120+2310n+15+17−210+4315+1126.Theorem 8.
TheNM2 index of the chain silicate networks CSn is given by(21)NM2CSn=2160ifn=2,4302ifn=3,2241n−2430ifn>3.Proof.
There are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=CS2, the edge partition is of the type S12,S12 and S12,S18 and each of this partition has six edges. Then,(22)NM2CS2=∑vw∈EGSvSw=612×12+612×18=2160.
For the casen=3, the edge partition CSn is presented in Table 5. Now, NM2CS3 can be computed as(23)NM2CS3=∑vw∈EGSvSw=612×12+115×15+612×21+121×21+415×21=4302.
Let us now turn to the third case ofn>3. To calculate the expression for the NM2 index of the chain silicate networks CSn, we use equation (3) and Table 4 to obtain the required result:(24)NM2CSn=∑vw∈EGSvSw=612×12+612×21+415×21+n−215×15+4n−315×24+n−424×24+221×24=2241n−2430.


### 3.5. Hexagonal Networks

It is a known fact that tilling a plane by regular polygon can be done only with regular triangles, hexagons, and squares. A hexagonal network can be obtained by using triangles (see Figure11). A hexagonal network with n vertices in each side of the hexagon is denoted by HXn. Let n>1; then, order and size of HXn is 3n2−3n+1 and 9n2−15n+6, respectively. Figure 11 depicts the graph of HX6. Tables 6–8 depict a partition that we use to compute the topological indices of hexagonal networks HXn for n≥3. Now, we investigate aforementioned indices of hexagonal networks HXn.Figure 11
Forn=6, hexagonal network HXn.Table 6
Edge partitions ofHX3.
SvSw,wherevw∈EHX3Cardinality14,181214,29618,291229,3612Table 7
Edge partitions ofHX4.
SvSw,wherevw∈EHX4Cardinality14,191219,19614,29619,291219,321229,321229,36632,361236,3612Table 8
Edge partitions ofHXn.
SvSw,wherevw∈EHXnCardinality14,191219,201220,206n−514,29619,291219,321220,3212n−429,321229,36632,326n−432,3612n−336,369n2−51n+72In the following theorems, we investigate1χ1 and NM2 of HXn.Theorem 9.
The1χ1 index of hexagonal networks HXn is given by(25)1χ1HXn=35+15ifn=2,27+6406+458+229ifn=3,13+619+18+129+338+358+12266+6406+12551ifn=4,14n2+18+3210−669720n−14−38−610+129+338+358+695+12266+6406+12551ifn>4.Proof.
There are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=HX2, the edge partition is of the type S10,S10 and S10,S18 and each of this partition has six edges. Then,(26)1χ1HX2=∑vw∈EG1SvSw=6110×10+6110×18=35+15.
For the casen=3, the edge partition HX3 is presented in Table 6. Now, 1χ1HX3 can be computed as(27)1χ1HX3=∑vw∈EG1SvSw=12114×18+6114×29+12118×29+12129×36=27+6406+458+229.
For the casen=4, the edge partition HX4 is presented in Table 7. Now, 1χ1HX4 can be computed as(28)1χ1HX4=∑vw∈EG1SvSw=12114×19+6119×19+6114×29+12119×29+12119×32+12129×32+6129×36+12132×36+12136×36=13+619+18+129+338+358+12266+6406+12551.
Let us now turn to the case ofn>4. To calculate the expression for 1χ1 index of hexagonal networks HXn, we use equation (2) and Table 8 to obtain the required result:(29)1χ1HXn=∑vw∈EG1SvSw=12114×19+12119×20+6n−5120×20+12119×29+12119×32+12n−4120×32+6114×29+12129×32+6129×36+12n−3132×36+6n−4132×32+9n2−51n+72136×36=14n2+18+3210−669720n−14−38−610+129+338+358+695+12266+6406+12551.Theorem 10.
TheNM2 index of the hexagonal networks HXn is given by(30)NM2HXn=1680,ifn=2,24252,ifn=3,68478,ifn=4,11664n2−36048n+26040,ifn>4.Proof.
There are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=HX2, the edge partition is of the type S10,S10 and S10,S18 and each of this partition has six edges. Then,
For the casen=3, the edge partition HXn is presented in Table 6. Now, NM2HX3 can be computed as(31)NM2HX3=∑vw∈EG1SvSw=1214×18+614×29+1218×29+1229×36=24252.
For the casen=4, the edge partition HXn is presented in Table 7. Now, NM2HX3 can be computed as(32)NM2HX4=∑vw∈EG1SvSw=1214×19+619×19+614×29+1219×29+1219×32+1229×32+629×36+1232×36+1236×36=68478.
Let us now turn to the third case ofn>4. To calculate the expression for NM2 index of hexagonal networks HXn, we use equation (3) and Table 8 to obtain the required result:(33)NM2HXn=∑vw∈EG1SvSw=1214×19+1219×20+6n−520×20+1219×29+1219×32+12n−420×32+614×29+1229×32+629×36+12n−332×36+6n−432×32+9n2−51n+7236×36=11664n2−36048n+26040.


## 3.1. Oxide Networks

Oxide networks are important in investigating silicate networks. Oxide networks are obtained by the deletion of silicon nodes from a silicate network (see Figure3). An oxide network of dimension n is denoted as OXn. The order and size of OXn is 9n2+3n and 18n2, respectively. In oxide networks OXn, the edge set can be partitioned into six sets depending on degree summation of neighbors of end vertices. This partition is presented in Table 1.Figure 3
An oxide networkOX2.Table 1
Edge partitions ofOXn.
SvSw,wherevw∈EOXnCardinality8,121212,141216,1618n2−36n+188,112n−114,1612n−114,1462n−3In the following theorems, we investigate1χ1 and NM2 of OXn.Theorem 1.
The1χ1 index of oxide networks OXn is given by(5)1χ1OXn=98n2+37+314−3928n+36−37+642−314−956.Proof.
We use equation (2) and Table 1 to calculate the expression for 1χ1 index of the oxide networks OXn:(6)1χ1OXn=∑vw∈EG1SvSw=1218×12+12112×14+12n−118×14+62n−3114×14+12n−1114×16+18n2−36n+18116×16=98n2+37+314−3928n+36−37+642−314−956.Theorem 2.
TheNM2 index of oxide networks OXn is given by(7)NM2OXn=4608n2−2832n+216.Proof.
To calculate the expression for neighborhood second Zagreb index of oxide networksOXn, we use equation (3) and Table 1 to get the required result:(8)NM2OXn=∑vw∈EGSvSw=128×12+1212×14+62n−314×14+12n−18×14+18n2−36n+1816×16+12n−114×16=4608n2−2832n+216.


## 3.2. Honeycomb Networks

In material sciences, honeycomb networks have vital role in different fields especially in computer graphics, cellular phone base stations, and image processing and in representing benzenoid hydrocarbons. Honeycomb networks are shaped by the iterative use of hexagonal tiles in a specific design. We denote then-dimensional honeycomb network with n hexagons between central and boundary hexagon by HCn. HCn is constructed by adding layer of hexagon around HCn−1 (see Figure 4). The order and size of HCn is 6n2 and 9n2−3n, respectively. Table 2 shows partition of edges.Figure 4
Honeycomb network.Table 2
Edge partitions ofHCn.
SvSw,wherevw∈EHCnCardinality5,565,712n−17,96n−19,99n2−21n+12By using partition of edges ofHCn, we investigate 1χ1 and NM2 of HCn in the following theorems.Theorem 3.
The1χ1 index of Honeycomb network HCn is given by(9)1χ1HCn=n2+27+1235−73n+3815−1235−27.Proof.
We use equation (2) and Table 2 to calculate the expression for 1χ1 index of the Honeycomb network HCn:(10)χ11HCn=∑vw∈EG1SvSw=615×5+12n−115×7+6n−117×9+9n2−21n+1219×9=n2+27+1235−73n+3815−1235−27.Theorem 4.
TheNM2 index of Honeycomb network HCn is given by(11)NM2HCn=729n2−903n+324.Proof.
To calculate the expression forNM2 index of Honeycomb network HCn, we use equation (3) and Table 2, to get the required result:(12)NM2HCn=∑vw∈EGSvSw=65×5+12n−15×7+6n−17×9+9n2−21n+129×9=729n2+903n+324.


## 3.3. Silicate Networks

Silicates are constituent elements of the common minerals that form the rock and, absolutely, the most important and the most complicated minerals. The elementary unit of silicates’ minerals are tetrahedronSiO4. Silicate is produced by melting metal oxides or metal sand carbonates. SiO4 tetrahedral content is nearly found in all silicate materials. A silicate sheet is formed by different types of sequences of tetrahedrons which are connected by adjoining oxygen vertices to other tetrahedrons in a plane. Cyclic silicates are networks that result in rings of different lengths after being bound by sharing oxygen vertices. Figures 5 and 6 present certain cyclic and sheet silicates, respectively. Other types of silicates are also available and are illustrated in Figure 7. From a chemical perspective, the oxygen atoms are in fact the corner vertices of tetrahedron SiO4 and silicon atom is its central vertex. Graphically, we describe the central atom as silicon vertex, corner atoms as oxygen vertices, and edges are bonds between them. Figure 8 illustrates a tetrahedron of SiO4. We denote a silicate array with n hexagons between center and border of silicate sheet by SLn. Figure 9 illustrates a three-dimensional silicate network. The order and size of SLn is 15n2+3n and 36n2, respectively. In the following theorems, the silicate networks are analyzed through certain graph invariants.Figure 5
Cyclic silicates.Figure 6
Sheet silicates.Figure 7
(a) Ortho, (b) pyro, and (c) chain silicates.
(a)(b)(c)Figure 8
SiO4 tetrahedron.Figure 9
Forn=3, silicate network SLn.In the next theorem, we computed1χ1 and NM2 of the SLn in the following way.Theorem 5.
The1χ1 index of silicate networks is given by(13)χ11SLn=315+35n2+835+436−515+4310−1645n+410−835−436+215+232−4310−115.Proof.
We use equation (2) and Table 3 to calculate the expression for 1χ1 index of the silicate networks:(14)1χ1SLn=∑vw∈EG1SvSw=6n115×15+24115×24+24n−1115×27+12n−1118×27+18n2−30n+12118×30+12124×27+62n−3127×27+12n−1127×30+18n2−36n+18130×30=2n5+410+8n−135+4n−136+6n2−10n+4215+232+22n−39+4n−1310+6n2−12n+610=315+35n2+835+436−515+4310−1645n+410−835−436+215+232−4310−115.Table 3
Edge partition ofLSn.
SvSw,wherevw∈ELSnCardinality15,156n18,2712n−127,3012n−118,3018n2−30n+1215,2724n−124,271215,242427,2762n−330,3018n2−36n+18Theorem 6.
TheNM2 index of silicate networks is given by(15)NM2SLn=25920n2−13230n+702.Proof.
To calculate the expression forNM2 index of silicate networks, we use equation (3) and Table 3 to get the required result:(16)NM2SLn=∑vw∈EGSvSw=6n15×15+12n−118×27+2415×24+12n−127×30+24n−115×27+18n2−30n+1218×30+1224×27+62n−327×27+18n2−36n+1830×30=25920n2−13230n+702.


## 3.4. Chain Silicate Networks

Chain silicate is acquired by linear arrangement of tetrahedron.n-dimensional chain silicate networks are characterized as follows. A network of n-dimensional chain silicate stands for CSn is acquired by linear arrangement of n tetrahedron. The order and size of CSn with n>1 is 3n+1 and and 6n, respectively. A network of n-dimensional chain silicate is illustrated in Figure 10. Tables 4 and 5 suggest such a partition that we use to compute certain topological indices. Now, within the following theorems, we analyzed the chain silicate networks CSn via topological indices.Figure 10
n-dimensional chain silicate network.Table 4
Edge partition ofCSn.
SvSw,wherevw∈ECSnCardinality12,12612,21615,15n−215,21415,244n−1221,24224,24n−4Table 5
Edge partition ofCS3.
SvSw,wherevw∈ECS3Cardinality12,12615,15112,21615,21421,211(17)1χ1CS3=∑vw∈EG1SvSw=6112×12+1115×15+6112×21+1121×21+4115×21=12+115+17+121+4315.The following results provide the investigation of certain topological indices ofCSn.Theorem 7.
The1χ1 index of CSn is given by(18)1χ1CSn=12+16ifn=2,12+115+17+121+4315ifn=3,13120+2310n+15+17−210+4315+1126ifn>3.Proof.
There are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=CS2, the edge partition is of the type S12,S12 and S12,S18 and each of this partition has six edges. Then,(19)1χ1CS2=∑vw∈EG1SvSw=6112×12+6112×18=12+16.
For the casen=3, the edge partition CSn is presented in Table 5. Now, 1χ1CS3 can be computed as follows:
Let us now turn to the third case ofn>3. To calculate the expression for 1χ1 index of the chain silicate networks CSn, we use equation (2) and Table 4 to obtain the required result:(20)1χ1CS3=∑vw∈EG1SvSw=6112×12+1115×15+6112×21+1121×21+4115×21+4n−3115×24+2121×24+n−4124×24=13120+2310n+15+17−210+4315+1126.Theorem 8.
TheNM2 index of the chain silicate networks CSn is given by(21)NM2CSn=2160ifn=2,4302ifn=3,2241n−2430ifn>3.Proof.
There are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=CS2, the edge partition is of the type S12,S12 and S12,S18 and each of this partition has six edges. Then,(22)NM2CS2=∑vw∈EGSvSw=612×12+612×18=2160.
For the casen=3, the edge partition CSn is presented in Table 5. Now, NM2CS3 can be computed as(23)NM2CS3=∑vw∈EGSvSw=612×12+115×15+612×21+121×21+415×21=4302.
Let us now turn to the third case ofn>3. To calculate the expression for the NM2 index of the chain silicate networks CSn, we use equation (3) and Table 4 to obtain the required result:(24)NM2CSn=∑vw∈EGSvSw=612×12+612×21+415×21+n−215×15+4n−315×24+n−424×24+221×24=2241n−2430.


## 3.5. Hexagonal Networks

It is a known fact that tilling a plane by regular polygon can be done only with regular triangles, hexagons, and squares. A hexagonal network can be obtained by using triangles (see Figure11). A hexagonal network with n vertices in each side of the hexagon is denoted by HXn. Let n>1; then, order and size of HXn is 3n2−3n+1 and 9n2−15n+6, respectively. Figure 11 depicts the graph of HX6. Tables 6–8 depict a partition that we use to compute the topological indices of hexagonal networks HXn for n≥3. Now, we investigate aforementioned indices of hexagonal networks HXn.Figure 11
Forn=6, hexagonal network HXn.Table 6
Edge partitions ofHX3.
SvSw,wherevw∈EHX3Cardinality14,181214,29618,291229,3612Table 7
Edge partitions ofHX4.
SvSw,wherevw∈EHX4Cardinality14,191219,19614,29619,291219,321229,321229,36632,361236,3612Table 8
Edge partitions ofHXn.
SvSw,wherevw∈EHXnCardinality14,191219,201220,206n−514,29619,291219,321220,3212n−429,321229,36632,326n−432,3612n−336,369n2−51n+72In the following theorems, we investigate1χ1 and NM2 of HXn.Theorem 9.
The1χ1 index of hexagonal networks HXn is given by(25)1χ1HXn=35+15ifn=2,27+6406+458+229ifn=3,13+619+18+129+338+358+12266+6406+12551ifn=4,14n2+18+3210−669720n−14−38−610+129+338+358+695+12266+6406+12551ifn>4.Proof.
There are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=HX2, the edge partition is of the type S10,S10 and S10,S18 and each of this partition has six edges. Then,(26)1χ1HX2=∑vw∈EG1SvSw=6110×10+6110×18=35+15.
For the casen=3, the edge partition HX3 is presented in Table 6. Now, 1χ1HX3 can be computed as(27)1χ1HX3=∑vw∈EG1SvSw=12114×18+6114×29+12118×29+12129×36=27+6406+458+229.
For the casen=4, the edge partition HX4 is presented in Table 7. Now, 1χ1HX4 can be computed as(28)1χ1HX4=∑vw∈EG1SvSw=12114×19+6119×19+6114×29+12119×29+12119×32+12129×32+6129×36+12132×36+12136×36=13+619+18+129+338+358+12266+6406+12551.
Let us now turn to the case ofn>4. To calculate the expression for 1χ1 index of hexagonal networks HXn, we use equation (2) and Table 8 to obtain the required result:(29)1χ1HXn=∑vw∈EG1SvSw=12114×19+12119×20+6n−5120×20+12119×29+12119×32+12n−4120×32+6114×29+12129×32+6129×36+12n−3132×36+6n−4132×32+9n2−51n+72136×36=14n2+18+3210−669720n−14−38−610+129+338+358+695+12266+6406+12551.Theorem 10.
TheNM2 index of the hexagonal networks HXn is given by(30)NM2HXn=1680,ifn=2,24252,ifn=3,68478,ifn=4,11664n2−36048n+26040,ifn>4.Proof.
There are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result forn=2. In the graph G=HX2, the edge partition is of the type S10,S10 and S10,S18 and each of this partition has six edges. Then,
For the casen=3, the edge partition HXn is presented in Table 6. Now, NM2HX3 can be computed as(31)NM2HX3=∑vw∈EG1SvSw=1214×18+614×29+1218×29+1229×36=24252.
For the casen=4, the edge partition HXn is presented in Table 7. Now, NM2HX3 can be computed as(32)NM2HX4=∑vw∈EG1SvSw=1214×19+619×19+614×29+1219×29+1219×32+1229×32+629×36+1232×36+1236×36=68478.
Let us now turn to the third case ofn>4. To calculate the expression for NM2 index of hexagonal networks HXn, we use equation (3) and Table 8 to obtain the required result:(33)NM2HXn=∑vw∈EG1SvSw=1214×19+1219×20+6n−520×20+1219×29+1219×32+12n−420×32+614×29+1229×32+629×36+12n−332×36+6n−432×32+9n2−51n+7236×36=11664n2−36048n+26040.


## 4. Conclusion

The aim of this paper is to discuss the degree-based topological indices of several nanotube networks. In this paper, we determined the neighborhood second Zagreb index and the first extended first-order connectivity index for oxide networkOXn, silicate network SLn, chain silicate network CSn, and hexagonal network HXn. Also, we determined the neighborhood second Zagreb index and the first extended first-order connectivity index for honeycomb network HCn. In this paper, results achieved demonstrate the promising opportunities to apply for chemical sciences. It is very helpful to know more about nanotube networks that have become the central interest of fundamental sciences and study their topological indices that will be completely useful to understand their underlying topologies.



---

*Source: 2771059-2022-01-04.xml*