Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures

Imran Nadeem; Hani Shaker; Muhammad Hussain; Asim Naseem

doi:10.1515/math-2019-0020

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Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures

Imran Nadeem , Hani Shaker , Muhammad Hussain and Asim Naseem

Published/Copyright: April 12, 2019

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From the journal Open Mathematics Volume 17 Issue 1

Abstract

The degree-based topological indices are numerical graph invariants which are used to correlate the physical and chemical properties of a molecule with its structure. Para-line graphs are used to represent the structures of molecules in another way and these representations are important in structural chemistry. In this article, we study certain well-known degree-based topological indices for the para-line graphs of V-Phenylenic 2D lattice, V-Phenylenic nanotube and nanotorus by using the symmetries of their molecular graphs.

Keywords: para-line graph; nanostructure; topological index

MSC 2010: 05C90; 92E10

1 Introduction

Chemical graph theory is a field of mathematical chemistry in which we implement the tools from graph theory to model chemical aspects mathematically. It is recorded in [1,2] that the structure of a molecule is strongly related to its chemical properties such as strain energy, boiling point and heat of formation. Molecular graphs can be used to model the molecules and molecular compounds by considering atoms as vertices and the chemical bonds between the atoms as edges. Topological index (TI) is a kind of numerical graph invariant which is used to correlate the physical and chemical properties of a molecular graph. In this sense, topological indices perform a significant role in chemical graph theory.

Consider the molecular graph G having vertex set V_G and edge set E_G. Let I_{G_p} be the set of edges of G that are incident with a vertex p ∈ V_G, then the degree, d_p, of p is defined as the cardinality of set I_{G_p} and S_p = ∑_{q∈N_p} d_q, where d_q is the degree of vertex q and the set N_p consists of all neighbor vertices of p i.e. N_p = {q ∈ V_G|pq ∈ E_G}. For any natural number t, we define V_t = {p ∈ V_G | S_p = t}. The subdivided graph of G is denoted by S(G) and defined by replacing each of its edge with the path having length 2. The line graph of G is symbolized by L(G). This graph is constructed by taking the vertex set V_L(G) = E_G and the edge set E_L(G) which has the property that for two vertices p, q ∈ V_L(G), pq ∈ E_L(G) ⟺ p, q ∈ E_G have a common vertex. The line graph of subdivided graph L(S(G)) is termed as the para-line graph of G.

Para-line graphs are used to understand the structure of a molecular graph and in this sense they receive much attention in structural chemistry. The atomic hybrid orbitals in a molecular graph corresponds to the vertices of its para-line graph and the strong links between the pairs of these orbitals correspond to the edges of its para-line graph. Klein et al. [3] presented some applications and basic properties of the para-line graphs in chemical graph theory.

Generally, topological indices can be categorized in three classes: degree-based, distance-based and spectrum-based indices. Among them, degree-based indices have great applications in chemical graph theory [4,5] and they can be defined in two ways as

TI(G)=∑pq∈E(G)F(dp,dq) (1)

TI(G)=∑pq∈E(G)F(Sp,Sq) (2)

where the sum runs over all pairs of adjacent vertices of G and F = F(x, y) is a suitably selected function.

Milan Randić proposed in 1975 a structural descriptor called the branching index [6] which is applicable for rating the degree of branching of the carbon-atom skeleton of saturated hydrocarbons. This index was renamed as the Randić connectivity index, which is defined as the sum of the Randić weights (dpdq)−12 for all edges. The generalization of the Randić index for any real number α, is termed as the general Randić connectivity index and is defined by taking F = (d_pd_q)^α in equation (1). Li and Zhao presented the first general Zagreb index in [7], which is defined as M_α(G) = ∑_p∈V(G)(d_p)^α. The sum-connectivity index was presented in [8] and was modified to the general sum-connectivity index in [9], which is formulated by selecting F = (d_p + d_q)^α in equation (1). It is recorded in [10] that the general Randić connectivity and sum-connectivity indices correlate greatly with the π-electron energy of benzenoid hydrocarbons. Estrada et al. [11] presented the atom bond connectivity index (ABC). This index is defined by choosing F=(dp+dq−2)/dpdq in equation (1). D. Vukičević and B. Furtula [12] proposed the geometric-arithmetic (GA) index that is defined by setting F=2dpdq/(dp+dq) in equation (1). The fourth ABC index was presented by Ghorbani and Hosseinzadeh [13] and is defined by choosing F=(Sp+Sq−2)/SpSq in equation (2). The fifth GA index (GA₅) was presented by Graovac et al. [14] which is defined by setting F=2SpSq/(Sp+Sq) in equation (2).

The explicit expressions of Zagreb indices for the para-line graphs of ladder, tadpole and wheel graphs, was presented by Ranjini et al. [15]. Su and Xu [16] studied general sum-connectivity indices for these para-line graphs. Nadeem et al. [17] presented the ABC₄ and GA₅ indices for these para-line graphs. In [18], they also studied ABC, ABC₄, GA, GA₅, general Zagreb, generalized Randić and general sum-connectivity indices for the para-line graphs of 2D-lattice TUC₄C₈(R), TUC₄C₈(R) nanotube and TUC₄C₈(R) nanotorus.

Recently, Akhter et al. [19] and Mufti et al. [20] computed ABC, ABC₄, GA, GA₅, first general Zagreb, general sum-connectivity and general Randić connectivity indices for the para-line graphs of certain benzenoid structures. In this paper, we present these indices for the para-line graphs of V-phenylenic 2D-lattice, V-phenylenic nanotube and nanotorus.

2 V-Phenylenic Nanostructures

The Phenylenes belong to the family of polycyclic non-benzenoid alternate conjugated hydrocarbons in which the carbon atoms form hexagons and squares. Each square is adjacent to two detached hexagons. From this, some larger compounds can be formed such as V-phenylenic 2D lattice, V-Phenylenic nanotube and nanotorus.

Let TUC₄C₆C₈[m, n] represents the V-phenylenic nanostructures where m denotes the number of hexagons in a row and n denotes the number of rows of hexagons in V-Phenylenic 2D-lattice, V-Phenylenic nanotube and nanotorus as presented respectively in Figure 1 (a), (b) and (c). The order and size of these nanostructures are given in Table 1.

$Figure 1 (a) The 2D-lattice TUC4C6C8[3, 3]; (b) The TUC4C6C8[3, 3] nanotube; (c) The TUC4C6C8[3, 3]nanotorus.$

Figure 1

(a) The 2D-lattice TUC₄C₆C₈[3, 3]; (b) The TUC₄C₆C₈[3, 3] nanotube; (c) The TUC₄C₆C₈[3, 3]nanotorus.

Table 1

The order and size of V-phenylenic nanostructures.

Graph	Order	Size
2D-lattice TUC₄C₆C₈[m, n]	6mn	9mn − m − 2n
TUC₄C₆C₈[m, n]	6mn	9mn − m
TUC₄C₆C₈[m, n] nanotorus	6mn	9mn

3 Main Results

In this section, we derive the topological indices for the para-line graphs of V-Phenylenic nanostructures by using their symmetric structures. The para-line graphs of these structures are presented in Figure 2 (a), (b) and (c) respectively.

$Figure 2 (a) The para-line graph G; (b) The Para-line graph H; (c) The Para-line graph K.$

Figure 2

(a) The para-line graph G; (b) The Para-line graph H; (c) The Para-line graph K.

3.1 TI’s of the para-line graph of 2D-lattice TUC₄C₆C₈[m, n]

Theorem 1

Consider the graph G of 2D-lattice TUC₄C₆C₈[m, n]. Then

Rα(G)=2.4α(m+3n+2)+4.6α(m+n−2)+9α(27mn−11m−20n+4)Mα(G)=2α+2(m+2n)+2.3α+1(3mn−m−2n)χα(G)=2.4α(m+3n+2)+4.5α(m+n−2)+6α(27mn−11m−20n+4)ABC(G)=2(3m+5n−2)+18mn−223m−403n+83GA(G)=−9m−14n+8+85(m+n−2)6+27mn

Proof

The para-line graph G of 2D-lattice TUC₄C₆C₈[m, n] is presented in Figure 2 (a). It can easily be checked that |V_G| = 2(9mn − m − 2n). Among them, there are 4(m + 2n) and 6(3mn − m − 2n) vertices of degree 2 and 3 respectively. By using the handshaking lemma, we have

4(m+2n)(2)+6(3mn−m−2n)(3)=2|EG| |EG|=27mn−5m−10n

So, we have the following disjoint edge partite subsets of E_G with respect to the degree of the end vertices.

E(2,2)={pq∈EG|dp=dq=2}E(2,3)={pq∈EG|dp=2 and dq=3}E(3,3)={pq∈EG|dp=dq=3}

We use cardinalities of partite sets given in Table 2 and by choosing the corresponding function F(d_p, d_q) in

Table 2

The cardinalities of the edge partite subsets of E_G with respect to degree of end vertices.

E_(p,q)	E_(2,2)	E_(2,3)	E_(3,3)
\|E_(p,q)	2p + 6q + 4	4p + 4q −8	27pq − 11p − 20q + 4

equation (1) to obtain the required results. □

Theorem 2

Consider the graph G of 2D-lattice TUC₄C₆C₈[m, n]. Then for m > 1 and n ≥ 1

ABC4(G)=62(n+4)+2355n+425(m−2)+1105(m+n−2) +2303(m+n−2)+12mn−769m−1129n+809GA5(G)=−17m−26n+1659n+161013(m+n−2)+96217(m+n−2)+27mn+24

Proof

For m > 1 and n ≥ 1, it can easily be checked from Figure 2 (a) that in G |V₄| = 4(n + 2), |V₅| = 4(m + n − 2), |V₈| = 4(m + n − 2) and |V₉| = 2(9mn − 5m − 8n + 4). So, we have the following disjoint edge partite subsets of E_G which consist of edges having end vertices labeled by the degree sum of adjacent vertices and their cardinalities are given in Table 3.

δ(4,4)={pq∈EG|Sp=Sq=4}δ(4,5)={pq∈EG|Sp=4 and Sq=5}δ(5,5)={pq∈EG|Sp=5 and Sq=5}δ(5,8)={pq∈EG|Sp=5 and Sq=8}δ(8,9)={pq∈EG|Sp=8 and Sq=9}δ(9,9)={pq∈EG|Sp=Sq=9}

Table 3

The cardinalities of the edge partite subsets of E_G with respect to degree sum of adjacent vertices.

δ_(p,q)	δ_(4,4)	δ_(4,5)	δ_(5,5)	δ_(5,8)	δ_(8,9)	δ_(9,9)
\|δ_(p,q)\|	2(n + 4)	4n	2(m − 2)	4(m + n −2)	8(m + n − 2)	27mn − 19m −28n + 20

By using Table 3 and choosing the corresponding function F(S_p, S_q) in equation (2), we get the required results. □

3.2 TI’s of the para-line graph of TUC₄C₆C₈[m, n] nanotube

Theorem 3

Consider the para-line graph H of TUC₄C₆C₈[m, n] nanotube. Then for m ≥ 1 and n ≥ 1

Rα(H)=2.4αm+4.6αm+9α(27mn−11m) Mα(H)=2α+2m+2.3α+1(3mn−m)χα(H)=2.4αm+4.5αm+6α(27mn−11m)ABC(H)=32m+73(27mn−11m) GA(H)=−9m+865m+27mn

Proof

The para-line graph H of TUC₄C₆C₈[m, n] nanotube and is presented in Figure 2 (b). One can easily verify that |V_H| = 2(9pq − p). Among them, there are 4m and 6m(3n − 1) vertices of degree 2 and 3 respectively. By using the handshaking lemma, we have

4m(2)+6m(3n−1)(3)=2|EH| |EH|=27mn−5m

Therefore, we get the following disjoint edge partite subsets of E_H and present its cardinalities in Table 4.

E(2,2)={pq∈EH|dp=dq=2}E(2,3)={pq∈EH|dp=2 and dq=3}E(3,3)={pq∈EH|dp=dq=3}

Table 4

The cardinalities of the edge partite subsets of E_H with respect to degree of end vertices.

E_(p,q)	E_(2,2)	E_(2,3)	E_(3,3)
\|E_(p,q)\|	2m	4m	27mn − 11m

We apply equation (1) to the information in Table 4 by choosing the corresponding functions F(d_p, d_q) and get the desired results. □

Theorem 4

Consider the para-line graph H of TUC₄C₆C₈[m, n] nanotube. Then for m ≥ 1 and n ≥ 1

ABC4(H)=12mn+425m+1105m+2303m−769m GA5(H)=27mn−17m+161013m+96217m

Proof

For m ≥ 1 and n ≥ 1, it can easily be checked from Figure 2 (b) that in H, |V₅| = 4m, |V₈| = 4m and |V₉| = 2(9mn − 5m). So, we have the following edge partite subsets of E_H which consist of edges having end vertices labeled by the degree sum of adjacent vertices and their cardinalities are given in Table 5.

δ(5,5)={pq∈EG|Sp=5 and Sq=5}δ(5,8)={pq∈EG|Sp=5 and Sq=8}δ(8,9)={pq∈EG|Sp=8 and Sq=9}δ(9,9)={pq∈EG|Sp=Sq=9}

Table 5

The cardinalities of the edge partite subsets of E_H with respect to degree sum of adjacent vertices.

δ_(p,q)	δ_(5,5)	δ_(5,8)	δ_(8,9)	δ_(9,9)
\|δ_(p,q)\|	2m	4m	8m	27mn − 19n

We apply equation (2) to Table 5 by taking the corresponding function F(S_p, S_q) and get the desired indices. □

3.3 TI’s of the para-line graph of TUC₄C₆C₈[m, n] nanotorus

Theorem 5

Consider the para-line graph K of TUC₄C₆C₈[p, q] nanotorus. Then

Rα(G)=27.9αmn Mα(G)=18.3αmnχα(G)=27.6αmnABC(K)=18pq GA(K)=27pq

Proof

The para-line graph of TUC₄C₆C₈[m, n] nanotorus and its para-line graph K is presented in Figure 2 (c). One can easily check that in K,|V_K| = 18mn and all these vertices are of degree 3. By using the handshaking lemma, we have |E_K| = 27mn. So, we have exactly one edge partition of E_K which is given by

E(3,3)={pq∈EK|dp=dq=3}

and clearly |E_(3,3)| = |E_K| = 27mn.

With this cardinality, we apply equation (1) by setting the corresponding function F(d_p, d_q) and get the desired indices. □

Theorem 6

Consider the para-line graph K of TUC₄C₆C₈[p, q] nanotorus. Then

ABC4(K)=12mn GA5(K)=27mn

Proof

It is easy to see from Figure 2 (c) that |V₉| = 18mn. So, we have exactly one edge partition with respect to end vertices labeled by degree sum of adjacent vertices, given by

δ(9,9)={pq∈EK|Sp=Sq=9}

and clearly |δ_(9,9)| = |E_K| = 27mn.

With this cardinality, we apply equation (2) by choosing the corresponding function F(S_p, S_q) and obtain the required results. □

4 Conclusion

In this article, well-known degree-based topological indices such as first general Zagreb, general Randić connectivity, general sum-connectivity, ABC, ABC₄, GA and GA₅ indices are studied. These indices correlate many chemical properties such as stability, heat of formation, boiling point and strain energy of chemical compounds. By using the symmetric structure property of V-phenylenic nanostructures, we present these indices for their para-line graphs which will help the people to interpret and analyze the underlying topologies of these nanostructures.

Acknowledgement

The authors would like to express their sincere gratitude to the anonymous referees and the editor for many valuable, friendly, and helpful suggestions, which led to a great deal of improvement of the original manuscript. This work was done under the project titled “On Two Dimensional Topological Descriptors of Molecular Graphs” which is supported by the Higher Education Commission, Pakistan via Grant No. 5331/Federal/NRPU/R&D/HEC/2016.

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Received: 2017-08-09

Accepted: 2019-01-29

Published Online: 2019-04-12

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Keywords for this article

para-line graph; nanostructure; topological index

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Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures

Article

Abstract

1 Introduction

2 V-Phenylenic Nanostructures

3 Main Results

3.1 TI’s of the para-line graph of 2D-lattice TUC4C6C8[m, n]

Theorem 1

Proof

Theorem 2

Proof

3.2 TI’s of the para-line graph of TUC4C6C8[m, n] nanotube

Theorem 3

Proof

Theorem 4

Proof

3.3 TI’s of the para-line graph of TUC4C6C8[m, n] nanotorus

Theorem 5

Proof

Theorem 6

Proof

4 Conclusion

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

3.1 TI’s of the para-line graph of 2D-lattice TUC₄C₆C₈[m, n]

3.2 TI’s of the para-line graph of TUC₄C₆C₈[m, n] nanotube

3.3 TI’s of the para-line graph of TUC₄C₆C₈[m, n] nanotorus