# Molecular Descriptors of Nanotube, Oxide, Silicate, and Triangulene Networks

Authors: Wei Gao; Muhammad Kamran Siddiqui
Journal: Journal of Chemistry
 (2017)

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## Abstract

A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity. The concept of hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials was established in chemical graph theory based on vertex degrees. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical networks. In this paper, we study carbon nanotube networks which are motivated by molecular structure of regular hexagonal lattice and also studied interconnection networks which are motivated by molecular structure of a chemical compoundSiO4. We determine hyper Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for some important class of carbon nanotube networks, dominating oxide network, dominating silicate network, and regular triangulene oxide network.


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## Body

## 1. Introduction

Multiprocessor interconnection networks are regularly required to associate a huge number of homogeneously repeated processor-memory matches, each of which is known as a preparing hub. Rather than utilizing a mutual memory, all synchronization and communication between processing nodes for program execution is often done via message passing. Outline and utilization of multiprocessor interconnection systems have as of late attracted extensive consideration due to the accessibility of modest, capable microchips, and memory chips. The work systems have been perceived as adaptable interconnection systems for massively parallel computing; see [1]. Work/torus-like low-dimensional systems have as of late gotten a considerable measure of consideration for their better adaptability to bigger systems, rather than more mind boggling systems, for example, hypercubes. There is a lot of relevant works on interdependent networks which can be reviewed. In particular the failure of cooperation on dependent networks has been studied a lot recently in [2–5].Particles and subnuclear blends are as often as possible showed by subnuclear graphs. A subnuclear outline is a graph in which vertices are particles of a given molecule and edges are its compound bonds. Since the valency of carbon is four, it is customary to consider all graphs with most outrageous degree under 4, as a molecular outline. An outlineG(V,E) with vertex set V and edge set E is related, if there exists a connection between any match of vertices in G. A framework is just a connected graph having no various edges and no circles. All through in this article, the level of vertex v∈V(G) is implied by deg⁡(v) and is the amount of edges incident to v.Atopological index is a numeric amount connected with a graph which describes the topology of diagram and is invariant under chart automorphism. In more exact way, a topological list Top(G) of graph G is a number with the property that, for each diagram H isomorphic to chart G, TopH=Top(G). The idea of topological list originated from work done by Wiener [6] while he was chipping away at breaking point of paraffin. He named this list asway number. Later on, the way number was renamed asWiener index. The Wiener file is the first and the most concentrated topological file, both from hypothetical perspective and applications, and is characterized as the entirety of separation between all sets of vertices in G; see for details [7, 8].One of the oldest topological indices is the first Zagreb index introduced by Gutman and Trinajstić based on degree of vertices ofG in 1972 [4]. The first and second Zagreb indices of a graph G are defined as (1)M1G=∑uv∈EGdeg⁡u+deg⁡v,M2G=∑uv∈EGdeg⁡u×deg⁡v.In 2013, Shirdel et al. [9] introduced a new degree based of Zagreb index named “hyper Zagreb index” as(2)HMG=∑uv∈EGdeg⁡u+deg⁡v2.Ghorbani and Azimi defined two new versions of Zagreb indices of a graph G in 2012 [2]. These indices are the first multiple Zagreb index PM1(G) and the second multiple Zagreb index PM2(G) and these indices are defined as follows:(3)PM1G=∏uv∈EGdeg⁡u+deg⁡v,PM2G=∏uv∈EGdeg⁡u×deg⁡v.The properties of PM1(G) and PM2(G) indices for some chemical structures have been studied in [2, 10].The first Zagreb polynomialM1(G,x) and the second Zagreb polynomial M2(G,x) are defined as follows:(4)M1G,x=∑uv∈EGxdeg⁡u+deg⁡v,M2G,x=∑uv∈EGxdeg⁡u×deg⁡v.The properties of M1(G,x) and M2(G,x) polynomials for some chemical structures have been studied in [3].Nowadays there is an extensive research activity onHM(G), PM1(G), and PM2(G) indices and M1(G,x) and M2(G,x) polynomials and their variants; see also [2, 4, 9–14].For further study of topological indices of various graph families, see [8, 11, 12, 14–27].


## 2. Carbon Nanotube Networks

The disclosure of the fullerene particles and related types of carbon structures, for example, nanotubes, has created an explosion of movement in science, physical science, and materials science. Traditional fullerene is an all-carbon particle in which the iotas are masterminded on a pseudospherical structure made up altogether of pentagons and hexagons. Its subatomic diagram is a finite trivalent chart installed on the surface of a sphere with just hexagonal and (precisely12) pentagonal appearances. Deza et al. [28] considered fullerene’s augmentation to other shut surfaces and demonstrated that exclusive four surfaces are conceivable, in particular, circle, torus, Klein bottle, and projective plane. Not at all like circular fullerenes, toroidal and Klein container’s fullerenes have been viewed as decorations of whole hexagons on their surfaces since they should contain no pentagons; see [17, 18, 28–33].Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. These round and hollow carbon atoms have abnormal properties, which are significant for nanotechnology, hardware, optics, and different fields of materials science and innovation. Nanotubes are individuals from the fullerene auxiliary family. Their name is gotten from their long, empty structure.Let us consider them×n quadrilateral section Pmn with m≥2 hexagons on the top and bottom sides and n≥2 hexagons on the lateral sides cut from the regular hexagonal lattice L; see Figure 1.Figure 1
Quadrilateral sectionPmn cuts from the regular hexagonal lattice.If we identify two lateral sides ofPmn such that we identify vertices u0j and umj, for j=0,1,2,…,n, then we obtainnanotubeNAmn with 2m(n+1) vertices and (3n+2)m edges.Letn be even, n≥2, and m≥2. If we identify the top and bottom sides of the quadrilateral section Pmn in such a way that we identify vertices ui0 and uin, for i=0,1,2,…,m, and vertices vi0 and vin, for i=1,2,…,m, then we obtainnanotubeNCmn of order n(2m+1) and size (3mn+n/2).Recently some well known topological indices, namely, Randić index, atom-bond connectivity index, and geometric arithmetic index for nanotubeNAmn and nanotube NCmn have been studied in [17, 18].Now we compute hyper Zagreb indexHM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for nanotube NAmn.Theorem 1.
Consider nanotubeNAmn for m,n≥2. Then(5)HMNAmn=108mn+28m,PM1NAmn=63mn-2m×54m,PM2NAmn=93mn-2m×64m,M1NAmn,x=3mn-2mx6+4mx5,M2NAmn,x=3mn-2mx9+4mx6.Proof.
NanotubeNAmn has 2m vertices of degree 2 and 2mn vertices of degree 3. The edge set E(NAmn) divides into two edge partitions based on degrees of end vertices. The first edge partition E1(NAmn) contains m(3n-2) edges uv, where deg⁡(u)=deg⁡(v)=3. The second edge partition E2(NAmn) contains 4m edges uv, where deg⁡(u)=2 and deg⁡(v)=3. Now using (2)–(4), we have(6)HMG=∑uv∈EGdeg⁡u+deg⁡v2,HMNAmn=∑uv∈E1NAmndeg⁡u+deg⁡v2+∑uv∈E2NAmndeg⁡u+deg⁡v2=62E1NAmn+52E2NAmn=363mn-2m+254m=108mn+28m,PM1G=∏uv∈EGdeg⁡u+deg⁡v,PM1NAmn=∏uv∈E1NAmndeg⁡u+deg⁡v×∏uv∈E2NAmndeg⁡u+deg⁡v=6E1NAmn×5E2NAmn=63mn-2m×54m,PM2G=∏uv∈EGdeg⁡u×deg⁡v,PM2NAmn=∏uv∈E1NAmndeg⁡u×deg⁡v×∏uv∈E2NAmndeg⁡u×deg⁡v=9E1NAmn×6E2NAmn=93mn-2m×64m,M1G,x=∑uv∈EGxdeg⁡u+deg⁡v,M1NAmn,x=∑uv∈E1NAmnxdeg⁡u+deg⁡v+∑uv∈E2NAmnxdeg⁡u+deg⁡v=∑uv∈E1NAmnx6+∑uv∈E2NAmnx5=E1NAmnx6+E2NAmnx5=3mn-2mx6+4mx5,M2G,x=∑uv∈EGxdeg⁡u×deg⁡v,M2NAmn,x=∑uv∈E1NAmnxdeg⁡u×deg⁡v+∑uv∈E2NAmnxdeg⁡u×deg⁡v=∑uv∈E1NAmnx9+∑uv∈E2NAmnx6=E1NAmnx9+E2NAmnx6=3mn-2mx9+4mx6.Now we compute hyper Zagreb indexHM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for nanotube NCmn.Theorem 2.
Consider nanotubeNCmn for n≥2 even and m≥2. Then(7)HMNCmn=108mn-24n,PM1NCmn=4n×52n×66mn-5n/2,PM2NCmn=4n×62n×96mn-5n/2,M1NCmn,x=nx4+2nx5+6mn-5n2x6,M2NCmn,x=nx4+2nx6+6mn-5n2x9.Proof.
NanotubeNCmn has 2n vertices of degree 2 and n(2m-1) vertices of degree 3. There are three types of edges in E(NCmn) based on degrees of end vertices of each edge; that is, E(NCmn)=E1(NCmn)∪E2(NCmn)∪E3(NCmn). Edge partition E1(NCmn) contains n edges uv, where deg⁡(u)=deg⁡(v)=2; edge partition E2(NCmn) contains 2n edges uv, where deg⁡(u)=2 and deg⁡(v)=3, and edge partition E3(NCmn) contains (6mn-5n)/2 edges uv, where deg⁡(u)=deg⁡(v)=3. Now using (2)–(4), we have(8)HMG=∑uv∈EGdeg⁡u+deg⁡v2,HMNCmn=∑uv∈E1NCmndeg⁡u+deg⁡v2+∑uv∈E2NCmndeg⁡u+deg⁡v2+∑uv∈E3NCmndeg⁡u+deg⁡v2=42E1NCmn+52E2NCmn+62E3NCmn=16n+252n+366mn-5n2=108mn-24n,PM1G=∏uv∈EGdeg⁡u+deg⁡v,PM1NCmn=∏uv∈E1NCmndeg⁡u+deg⁡v×∏uv∈E2NCmndeg⁡u+deg⁡v×∏uv∈E3NCmndeg⁡u+deg⁡v=4E1NCmn×5E2NCmn×6E3NCmn=4n×52n×66mn-5n/2,PM2G=∏uv∈EGdeg⁡u×deg⁡v,PM2NCmn=∏uv∈E1NCmndeg⁡u×deg⁡v×∏uv∈E2NCmndeg⁡u×deg⁡v×∏uv∈E3NCmndeg⁡u×deg⁡v=4E1NCmn×6E2NCmn×9E3NCmn=4n×62n×96mn-5n/2,M1G,x=∑uv∈EGxdeg⁡u+deg⁡v,M1NCmn,x=∑uv∈E1NCmnxdeg⁡u+deg⁡v+∑uv∈E2NCmnxdeg⁡u+deg⁡v+∑uv∈E3NCmnxdeg⁡u+deg⁡v=∑uv∈E1NCmnx4+∑uv∈E2NCmnx5+∑uv∈E3NCmnx6=E1NCmnx4+E2NCmnx5+E3NCmnx6=nx4+2nx5+6mn-5n2x6,M2G,x=∑uv∈EGxdeg⁡u×deg⁡v,M2NCmn,x=∑uv∈E1NCmnxdeg⁡u×deg⁡v+∑uv∈E2NCmnxdeg⁡u×deg⁡v+∑uv∈E3NCmnxdeg⁡u×deg⁡v=∑uv∈E1NCmnx4+∑uv∈E2NCmnx6+∑uv∈E3NCmnx9=E1NCmnx4+E2NCmnx6+E3NCmnx9=nx4+2nx6+6mn-5n2x9.


## 3. Dominating Oxide NetworkDOX(n)

In2015, Baig et al. [34] computed general Randić index, atom-bond connectivity (ABC) index, and geometric arithmetic (GA) index of dominating oxide network DOX(n). In this section, we calculate hyper Zagreb index HM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for dominating oxide network DOX(n).Theorem 3.
Consider the dominating oxide networkDOX(n) for n≥2. Then(9)HMDOXn=3456n2-4128n+1488,PM1DOXn=624n-12×854n2-78n+30,PM2DOXn=824n-12×1654n2-78n+30,M1DOXn,x=24n-12x6+54n2-78n+30x8,(10)M2DOXn,x=24n-12x8+54n2-78n+30x16.Proof.
LetDOX(n) be the dominating oxide network. The number of vertices and edges in DOX(n) are 27n2-21n+6 and 54n2-54n+18, respectively; see Figure 2. The edge set E(DOX(n)) divides into two edge partitions based on degrees of end vertices. The first edge partition E1(DOX(n)) contains 24n-12 edges uv, where deg⁡(u)=2,deg⁡(v)=4. The second edge partition E2(DOX(n)) contains 54n2-78n+30 edges uv, where deg⁡(u)=deg⁡(v)=4.Figure 2
Dominating oxide network (DOX(2)).Now using (2)–(4), we have(11)HMG=∑uv∈EGdeg⁡u+deg⁡v2,HMDOXn=∑uv∈E1DOXndeg⁡u+deg⁡v2+∑uv∈E2DOXndeg⁡u+deg⁡v2=62E1DOXn+82E2DOXn=3456n2-4128n+1488,PM1G=∏uv∈EGdeg⁡u+deg⁡v,PM1DOXn=∏uv∈E1DOXndeg⁡u+deg⁡v×∏uv∈E2DOXndeg⁡u+deg⁡v=6E1DOXn×8E2DOXn=624n-12×854n2-78n+30,PM2G=∏uv∈EGdeg⁡u×deg⁡v,PM2DOXn=∏uv∈E1DOXndeg⁡u×deg⁡v×∏uv∈E2DOXndeg⁡u×deg⁡v=8E1DOXn×16E2DOXn=824n-12×1654n2-78n+30,M1G,x=∑uv∈EGxdeg⁡u+deg⁡v,M1DOXn,x=∑uv∈E1DOXnxdeg⁡u+deg⁡v+∑uv∈E2DOXnxdeg⁡u+deg⁡v=∑uv∈E1DOXnx6+∑uv∈E2DOXnx8=E1DOXnx6+E2DOXnx8=24n-12x6+54n2-78n+30x8,M2G,x=∑uv∈EGxdeg⁡u×deg⁡v,M2DOXn,x=∑uv∈E1DOXnxdeg⁡u×deg⁡v+∑uv∈E2DOXnxdeg⁡u×deg⁡v=∑uv∈E1DOXnx8+∑uv∈E2DOXnx16=E1DOXnx8+E2DOXnx16=24n-12x8+54n2-78n+30x16.


## 4. Dominating Silicate NetworkDSL(n)

In2015, Baig et al. [34] computed general Randić index, atom-bond connectivity (ABC) index, and geometric arithmetic (GA) index of dominating silicate network DSL(n). In this section, we calculate hyper Zagreb index HM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for dominating silicate network DSL(n).Theorem 4.
Consider the dominating silicate networkDSL(n) for n≥2. Then(12)HMDSLn=11070n2-13452n+4746,PM1DSLn=512n-6×824n-12×954n2-66n+24×1254n2-78n+30,PM2DSLn=612n-6×1224n-12×1854n2-66n+24×3654n2-78n+30,M1DSLn,x=12n-6x5+24n-12x8+54n2-66n+24x9+54n2-78n+30x12,M2DSLn,x=12n-6x6+24n-12x12+54n2-66n+24x18+54n2-78n+30x36.Proof.
LetDSL(n) be the dominating silicate network; see Figure 3. The number of vertices and edges in DSL(n) is 45n2-39n+12 and 108n2-108n+36, respectively. The edge set E(DSL(n)) divides into four edge partitions based on degrees of end vertices. The first edge partition E1(DSL(n)) contains 12n-6 edges uv, where deg⁡(u)=2,deg⁡(v)=3. The second edge partition E2(DSL(n)) contains 24n-12 edges uv, where deg⁡(u)=2,deg⁡(v)=6. The third edge partition E3(DSL(n)) contains 54n2-66n+24 edges uv, where deg⁡(u)=3,deg⁡(v)=6. The fourth edge partition E4(DSL(n)) contains 54n2-78n+30 edges uv, where deg⁡(u)=deg⁡(v)=6. Now using (2)–(4), we have(13)HMG=∑uv∈EGdeg⁡u+deg⁡v2,HMDSLn=∑uv∈E1DSLndeg⁡u+deg⁡v2+∑uv∈E2DSLndeg⁡u+deg⁡v2+∑uv∈E3DSLndeg⁡u+deg⁡v2+∑uv∈E4DSLndeg⁡u+deg⁡v2,HMDSLn=52E1DSLn+82E2DSLn+92E3DSLn+122E4DSLn=11070n2-13452n+4746,PM1G=∏uv∈EGdeg⁡u+deg⁡v,PM1DSLn=∏uv∈E1DSLndeg⁡u+deg⁡v×∏uv∈E2DSLndeg⁡u+deg⁡v×∏uv∈E3DSLndeg⁡u+deg⁡v×∏uv∈E4DSLndeg⁡u+deg⁡v=5E1DSLn×8E2DSLn×9E3DSLn×12E4DSLn=512n-6×824n-12×954n2-66n+24×1254n2-78n+30,PM2G=∏uv∈EGdeg⁡u×deg⁡v,PM2DSLn=∏uv∈E1DSLndeg⁡u×deg⁡v×∏uv∈E2DSLndeg⁡u×deg⁡v×∏uv∈E3DSLndeg⁡u×deg⁡v×∏uv∈E4DSLndeg⁡u×deg⁡v=6E1DSLn×12E2DSLn×18E3DSLn×36E4DSLn=612n-6×1224n-12×1854n2-66n+24×3654n2-78n+30,M1G,x=∑uv∈EGxdeg⁡u+deg⁡v,M1DSLn,x=∑uv∈E1DSLnxdeg⁡u+deg⁡v+∑uv∈E2DSLnxdeg⁡u+deg⁡v+∑uv∈E3DSLnxdeg⁡u+deg⁡v+∑uv∈E4DSLnxdeg⁡u+deg⁡v=∑uv∈E1DSLnx5+∑uv∈E2DSLnx8+∑uv∈E3DSLnx9+∑uv∈E4DSLnx12=E1DSLnx5+E2DSLnx8+E3DSLnx9+E4DSLnx12=12n-6x5+24n-12x8+54n2-66n+24x9+54n2-78n+30x12,M2G,x=∑uv∈EGxdeg⁡u×deg⁡v,M2DSLn,x=∑uv∈E1DSLnxdeg⁡u×deg⁡v+∑uv∈E2DSLnxdeg⁡u×deg⁡v+∑uv∈E3DSLnxdeg⁡u×deg⁡v+∑uv∈E4DSLnxdeg⁡u×deg⁡v=∑uv∈E1DSLnx6+∑uv∈E2DSLnx12+∑uv∈E3DSLnx18+∑uv∈E4DSLnx36=E1DSLnx6+E2DSLnx12+E3DSLnx18+E4DSLnx36=12n-6x6+24n-12x12+54n2-66n+24x18+54n2-78n+30x36.Figure 3
Dominating silicate network (DSL(2)).


## 5. Regular Triangulene Oxide NetworkRTOX(n)

In2015, Baig et al. [34] computed general Randić index, atom-bond connectivity (ABC) index, and geometric arithmetic (GA) index of regular triangulene oxide network RTOX(n). Now we compute hyper Zagreb index HM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for regular triangulene oxide network RTOX(n).Theorem 5.
Consider the regular triangulene oxide networkRTOX(n) for n≥2. Then(14)HMRTOXn=192n2+216n-96,PM1RTOXn=42×66n×83n2-2,PM2RTOXn=42×86n×163n2-2,M1RTOXn,x=2x4+6nx6+3n2-2x8,M2RTOXn,x=2x4+6nx8+3n2-2x16.Proof.
LetRTOX(n) be the regular triangulene oxide network; see Figure 4. The number of vertices and edges in RTOX(n) is (3n2+9n+2)/2 and 3n2+6n, respectively. There are three types of edges in E(RTOX(n)) based on degrees of end vertices of each edge; that is, E(RTOX(n))=E1(RTOX(n))∪E2(RTOX(n))∪E3(RTOX(n)). The edge partition E1(RTOX(n)) contains 2 edges uv, where deg⁡(u)=deg⁡(v)=2; the edge partition E2(RTOX(n)) contains 6n edges uv, where deg⁡(u)=2 and deg⁡(v)=4, and the edge partition E3(RTOX(n)) contains 3n2-2 edges uv, where deg⁡(u)=deg⁡(v)=4.Figure 4
Regular triangulene oxide network (RTOX(5)).Now using (2)–(4), we have(15)
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T
O
X
n
x
8
=
E
1
R
T
O
X
n
x
4
+
E
2
R
T
O
X
n
x
6
+
E
3
R
T
O
X
n
x
8
=
2
x
4
+
6
n
x
6
+
3
n
2
-
2
x
8
,
M
2
G
,
x
=
∑
u
v
∈
E
G
x
deg
⁡
u
×
deg
⁡
v
,
M
2
R
T
O
X
n
,
x
=
∑
u
v
∈
E
1
R
T
O
X
n
x
deg
⁡
u
×
deg
⁡
v
+
∑
u
v
∈
E
2
R
T
O
X
n
x
deg
⁡
u
×
deg
⁡
v
+
∑
u
v
∈
E
3
R
T
O
X
n
x
deg
⁡
u
×
deg
⁡
v
=
∑
u
v
∈
E
1
R
T
O
X
n
x
4
+
∑
u
v
∈
E
2
R
T
O
X
n
x
8
+
∑
u
v
∈
E
3
R
T
O
X
n
x
16
=
E
1
R
T
O
X
n
x
4
+
E
2
R
T
O
X
n
x
8
+
E
2
R
T
O
X
n
x
16
=
2
x
4
+
6
n
x
8
+
3
n
2
-
2
x
16
.


## 6. Conclusion

In this paper we determined hyper Zagreb indexHM(G), first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G), and Zagreb polynomials M1(G,x) and M2(G,x) for nanotubes NAmn and NCmn, dominating oxide network DOX(n), dominating silicate network DSL(n), and regular triangulene oxide network RTOX(n).In the future, we are interested in designing some new architectures/networks and then studying their topological indices which will be quite helpful to understand their underlying topologies.



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*Source: 6540754-2017-02-05.xml*