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BY 4.0 license Open Access Published by De Gruyter Open Access November 13, 2019

Computing Topological Indices for Para-Line Graphs of Anthracene

  • Zhiqiang Zhang , Zeshan Saleem Mufti , Muhammad Faisal Nadeem EMAIL logo , Zaheer Ahmad , Muhammad Kamran Siddiqui and Muhammad Reza Farahani
From the journal Open Chemistry

Abstract

Atoms displayed as vertices and bonds can be shown by edges on a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties such as the molar refraction, molar volume, chromatographic behavior, heat of atomization, heat of vaporization, magnetic susceptibility, and the partition coefficient. Today, industry is flourishing because of the interdisciplinary study of different disciplines. This provides a way to understand the application of different disciplines. Chemical graph theory is a mixture of chemistry and mathematics, which plays an important role in chemical graph theory. Chemistry provides a chemical compound, and graph theory transforms this chemical compound into a molecular graphwhich further is studied by different aspects such as topological indices.We will investigate some indices of the line graph of the subdivided graph (para-line graph) of linear-[s] Anthracene and multiple Anthracene.

1 Introduction

Chemical graph theory is a branch of mathematical chemistry that is concerned with analyses of all consequences of a connectivity in a chemical graph. Some physical properties, e.g., breaking point, can be anticipated in view of the structure of the atoms. Numerical and computational systems are viably used to show and predict the structure at an atomic level. The structures of atoms, from anumerical perspective, are graphs. Graph theory is utilized as a part of nearly every field of science, and it is likewise utilized for training, both for recreation and design.

V(G) and E(G) compose the vertex set and edge set of a graph Grespectively; p, q ∈ V(G) are adjacent if p and q are end points of u ∈ E(G)” and u is an edge whose end vertices are p and q. The set of all neighbors of a vertex prepresented by Np, is called the neighborhood of p. The count of edges that occur on a vertex is called the degree of the vertex denoted by ξp and Sp = Σq∈Np ξ p, where Np = {q ∈ V(G) : pq ∈ E(G)}. We can construct the line graph L(G) of any graph G in such a way that the edges of the original graph will become the vertices of line graph i.e. two vertices p and q occur if and only when these vertices have a common end vertex in G. The para-line graph of Gis the line graph of the subdivision of Gi.e. L(S(G)), which will be represented as G*. The subdivision graph is the graph attained from G by replacing each of its edges by a path of length 2. For instance, consider C2H6 as the hydrocarbon named Ethane which is characterized as a molecular structure. The graph of C2H6 is shown in Figure 1 (a) and (b), and the line graph of subdivision C2H6is shown in Figure 1 (c).

Figure 1 (a) molecular construction of C2H6, (b) molecular graph of C2H6, (c) line graph of subdivision of C2H6.
Figure 1

(a) molecular construction of C2H6, (b) molecular graph of C2H6, (c) line graph of subdivision of C2H6.

The topological index which is also referred to as the molecular descriptor, is a real number which describes the properties of a certain chemical compound. The study of topological indices on different chemical structures has been an area of research for all graph theorists. It is a bridge between mathematics and chemistry. The molecular descriptors which are separated into three groups depend on degree-based [16, 24], distance-based [8, 29] and spectrum-based [2, 18, 20, 21] indices. Some other topological indices have also been studied which are centred on both degrees and distances [4, 7, 11, 19].

The 1st general Zagreb index in [17] is the oldest and most used molecular descriptor and is defined as:

(1)Mα(G)=pV(G)(ξp)α.

The general sum-connectivity index χα(G) is defined as [30]:

(2)χα(G)=pqE(G)(ξp+ξq)α.

The ABC is specified by Estrada in [5]. The ABCindex of graph Gis defined as:

(3)ABC(G)=pqE(G)ξp+ξq2ξpξq.

ABC4 index was presented by Ghorbani in [9] is explained as:

(4)ABC4(G)=pqE(G)Sp+Sq2SpSq.

Rα (general Randic connectivity index) of G is proposed as [1]:

(5)Rα(G)=pqE(G)(ξpξq)α.

Where ∝ ∈ R. If α is −0.5, then R−0.5(G) is called Randic connectivity index of G.

Vukicevic and Furtula presented the (GA) index in [28]. The GAindex for graph G is defined as:

(6)GA(G)=pqE(G)2ξpξqξp+ξq.

GA5 was introduced by Graovac et al. in [10] is proposed as:

(7)GA5(G)=pqE(G)2SpSqSp+Sq.

The hyper-Zagreb index is suggested as:

(8)HM(G)=pqE(G)(ξp+ξq)2

In 2012, Ghorbani and Azimi introduced the Zagreb indices in different form as, the 1st and 2nd multiple Zagreb index PM1(G)and PM2(G), 1st and 2nd Zagreb polynomial M1(G, a) and M2(G, a) respectively, are suggested as:

(9)PM1(G)=pqEG(ξp+"ξq)
(10)PM2(G)=pqE(G)(ξp×ξq)
(11)M1(G,a)=pqE(G)a(ξp+ξq)
(12)M2(G,a)=pqE(G)a(ξp×ξq)

2 Applications of Topological Indices

Randic observed the correlation between the Randic index and physio-chemical properties of alkane such as boiling point, entholphies of formation, surface area and so on. The ABC index is a very effective index in heat formation [5]. GA index is a better pridictive index than the Randic Index as GA has much chosen prophetic control on the prophetic energy of the Rα [3]. The 1st and 2nd Zagreb index were very useful in the calculation of the aggregate π-electron energy of the molecule [12]. These molecular descriptorswere suggested for the approximation of streched carbon-skeleton [13].

3 Topological indices of para-line graphs

Para-line graphs are an attractive field of study in chemical graph theory. Ranjini et al. calculated the explicit expression for the Schultz index and Zagreb index of the para-line graphs of the wheel, ladder, helm, tadpole [25, 26]. In 2015, Su et al. [27] investigated χα (general sum-connectivity index )and co-index of the above mentioned graph of wheel, tadpole. To study the para-line graphs of ladder, tadpole and wheel, Nadeem et al. in [22] evaluated the ABC4, GA5 index and calculated the generalized Randic index, general Zagreb index, χα, ABC index, GA index, ABC4 index and GA5 index of TUC4C8[p, q] in [23]. Klein et al. [15] offered few applications and basic properties of the para-line graphs in chemical graph theory. Gutman also shed a light on the application of line graphs see [14]. Estrada showed the application of line graph in [6].

4 Results for para-line graph of linear [s]-Anthracene

In Figure 2, the graph of linear [s]-Anthracene is presented and it is denoted by Ts. Ts has 14s vertices and 18s − 2 edges.

Figure 2 The molecular graph of linear [s]-Anthracene.
Figure 2

The molecular graph of linear [s]-Anthracene.

Suppose the graph of Anthracene contains three hexagon which are connected with a square expands vertically and horizontally. Let the edge e(p,q) represent the count of edges connecting the vertices of degree ξ p and ξ q. The graph of Anthracene holds the following edges shown in Table 2:

Theorem 1

Let G*be the line graph of the subdivided graph( para-line graph) of Ts. Then

Mα(G)=(12s+8)2α+2+(24s12)3α+1

Proof. In G*, the overall count of the vertices is 48s − 10 which is the sum of degree two and degree three vertices 8 + 12s, 24s − 12respectively. As

Mα(G)=(12s+8)2α+2+3α+1(24s12).

Theorem 2

Let G*be the line graph of the subdivided graph( para-line graph) of Ts. Then

  1. Rα(G)=(6s+10)4α+(12s4)6α+(30s16)9α;

  2. χα(G)=(6s+10)4α+(12s4)5α+(30s16)6α;

  3. ABC(G)=92+20s+32323

  4. GA(G)=36+2456s68/56.

Proof. The overall count of edges of G* is 74s−10. The set of edges of G* is represented as E(G*) which is uniformly disjointed three edge sets established based on the degrees of the adjacent(end) vertices such as E1(G*), E2(G*) and E3(G*). The first set E1(G*) contains the edges such as E(2,2) = 6s + 10, the second set E2(G*) holds the edges E(2,3) = 12s−4, and the third set E3(G*) includes the edges E(3,3) = 30s − 16, where (2, 2), (2, 3) and (3, 3) represent the degree of end vertices respectively. From formulas (5), (2), (3) and (6) and using Table 1, the result is proved.

Table 1

The distribution of edges w.r.t. degree of end vertices of every edge.

Sr. NoE(ξ p ,ξ q)Number of Edges
1E(2,2)6s + 10
2E(2,3)12s − 4
3E(3,3)30s − 16

Table 2

The distribution of edges w.r.t. neighbor sum of every vertex the graph of Anthracene.

Sr. NoS(p,q)Number of Edges
1S(4,4)10
2S(4,5)4
3S(5,5)6s − 4
4S(5,8)12s − 4
5S(8,8)4s
6S(8,9)16s − 8
7S(9,9)10s − 8

Theorem 3

Let G*be the line graph of the subdivided graph (para-line graph) of Ts. Then

ABC4G=35110+1252+4330+409+1214s+526+2535852233015110329GA5(G)=20+481310+192172s2+169516131096172
Figure 3 The Para-Line graph of linear [s]-Anthracene.
Figure 3

The Para-Line graph of linear [s]-Anthracene.

Proof. In Table 3, we have

ABC4(G)=pqE(G)Sp+Sq2SpSqABC4G=S4,4×4+424×4+S4,5×4+524×5+S5,5×5+525×5+S5,8×5+825×8+S(8,9)×8+928×9+S(9,9)

After some simplification, we obtain the required results.

ABC4(G)=35110+1252+4330+409+1214s+526+2535852233015110329

Similarly, by using Table 3, we have

Table 3

The distribution of edges of multiple Anthracene w.r.t. degree of end vertices of every edge.

Sr. NoE(ξ p ,ξ q)Number of Edges
1E(2,2)6r + 6s + 4
2E(2,3)4r + 12s − 8
3E(3,3)63rs − 20r − 33s + 4

GA5G=pqEG2SpSqSp+SqGA5G=S4,4×24×44+4+S4,5×24×54+5+S5,5×25×55+5+S5,8×25×85+8S8,8×28×88+8+S8,9×28×98+9+S9,9×29×99+9GA5G=20+481310+192172s2+169516131096172

Theorem 4

Let G*be the line graph of the subdivided graph (para-line graph) of Ts. Then

  1. HM(G)=1404s636

  2. PM1(G)=46s+10512s4630s16

  3. PM2(G)=46s+10612s4930s16

Proof. The edge set E(G*) established on the degrees of the adjacent(end) vertices is represented as the sum of the cardinality of three sets such as E1(G*), E2(G*) and E3(G*). The cardinality of first set E1(G*), second set E2(G*) and third set E3(G*) is denoted as E(2,2), E(2,3) and E(3,3) respectively. The 1st partite edge set holds E(2,2) = 6s + 10 edges, the 2nd partite set holds E(2,3) = 12s − 4 edges and the 3rd partite set holds E(3,3) = 30s − 16 edges. From Table 3, we obtain the results.

HMG=pqEG(ξp+ξq)2HMG=pqE1G[ξp+ξq]2+pqE2G[ξp+ξq]2+pqE3G[ξp+ξq]2HMG=16×E2,2+25×E2,3+36×E3,3HMG=166s+10+2512s4+3630s16HMG=1404636PM1G=pqEG(ξp+ξq)PM1G=pqE1G(ξp+ξq)×pqE2G(ξp+ξq)×pqE3G(ξp+ξq)PM1G=4E2,2×5E2,3×6E3,3PM1G=46s+10×512s4×630s16PM2G=pqEG(ξp×ξq)PM2G=pqE1G(ξp×ξq)×pqE2G(ξp×ξq)×pqE3G(ξp×ξq)PM2G=4E2,2×6E2,3×9E3,3PM2G=46s+10×612s4×930s16

Theorem 5

Let G*be the line graph of the subdivided graph (para-line graph) of Ts. Then

  1. M1(G,a)=6s+10a4+12s4a5+30s16a6

  2. M2(G,a)=6s+10a4+12s4a6+30s16a9

Proof. From Table 3, we obtain the results.

M1G,a=pqEGaξp+ξqM1G,a=pqE1Gaξp+ξq+pqE2Gaξp+ξq+pqE3Gaξp+ξqM1G,a=pqE1Ga4+pqE2Ga5+pqE3Ga6M1G,a=E2,2×a4+E2,3×a5+E3,3×a6M1G,a=6s+10a4+12s4a5+30s16a6M2G,a=pqEGaξp×ξqM2G,a=pqE1Gaξp×ξq+pqE2Gaξp×ξq+pqE3Gaξp×ξqM2G,a=pqE1Ga4+pqE2Ga6+pqE3Ga9M2G,a=E2,2×a4+E2,3×a6+E3,3×a9M2G,a=6s+10a4+12s4a6+30s16a9

5 Results for para-line graph of multiple Anthracene

In Figure 4we presented the graph G*, and it is denoted by Tr,s. Tr,shas 22rs vertices and 33rs − 2r − 5s edges. Also the para-line graph of multiple Anthracene are depicted in Figure 5.

Figure 4 The molecular graph of multiple Anthracene.
Figure 4

The molecular graph of multiple Anthracene.

Figure 5 The para-line graph of multiple Anthracene.
Figure 5

The para-line graph of multiple Anthracene.

Table 4

The distribution of edges w.r.t. neighbor sum of every vertex the graph G*.

Sr. NoS(p,q)Number of Edges
1S(4,4)2r + 8
2S(4,5)4r
3S(5,5)6s − 4
4S(5,8)12s − 4r − 8
5S(8,8)4s
6S(8,9)8r + 16s − 16
7S(9,9)63rs − 28r − 53s + 20

Theorem 6

Let G*be the line graph of the subdivided graph (para-line graph) of Tr,s. Then

Mα(G)=(2r+3s)2α+2+3α+1(14rs4r6s).

Proof. In Figure 5, we presented a graph G*. In G*, the overall count of the vertices is 63rs−10r−15s which is the sum of 8r + 12s (vertices of degree 2) and 42rs − 12r − 18s (vertices of degree 3). Hence proved.

Theorem 7

Let G*be the line graph of the subdivided graph (para-line graph) of Tr,s. Then

  1. Rα(G)=6s+6r+44α+4r+12s86α+63rs20r33s+49α;

  2. χα(G)=6s+6r+44α+4r+12s85α+63rs20r33s+46α;

  3. ABC(G)=9222s+52403r22+42rs+8/3;

  4. GA(G)=27+2456s+8/5614r+63rs+81656.

Proof. The total number of edges and vertices of subdivision graph S(Tr,s) are 63rs−10r−15s and 198rs−20r−50 respectively. The vertices are subdivided in the following way: The count of degree 2 vertices is 8r+12s and the count of degree 3 vertices is 42rs−12r−18s. The overall count of edges of G* are 63rs − 10r − 15s. The set of edges E(G*) bifurcates into 3 partite sets established on degrees of the adjacent(end) vertices such as E1(G*), E2(G*) and E3(G*) The edge partite set E1(G*) contains E(2,2) = 6r + 6s + 4 edges, where E(2,2) represents the edge whose both vertex has degree 2. The edge partite set E2(G*) holds E(2,3) = 4r+12s−8 edges, where E(2,3) represents the edge the vertex of which has degree 2 and second vertex has 3 and the edge partite set E3(G*) contains E(3,3) = 63rs − 20r − 33s + 4 edges, where E(3,3) represents the edge the vertex of which has degree 3. From formulas (5), (2), (3) and (6), the proof was established.

Theorem 8

Let G*be the line graph of the subdivided graph (para-line graph) of Tr,s. Then

  1. ABC4G=28r+1214+1252s+35110+43302129s+126+15110+25351129+2330r+268522/51104/330+809

  2. GA5G=801310+99m+28817269s+26+161310+1695+96172r192172321310+24

Proof. Let the degree sum of neighbors of end vertices pand qbe represented as S(p,q). Suppose the collection of edge sets E(G*)is divided into 7 partite edge sets E4(G*), E5(G*), E6(G*), E7(G*), E8(G*), E9(G*), E10(G*). The edge partite set E4(G*) contains S(p,q) = (4, 4) = 2r+8 edges, the edge partition E5(G*) contains S(p,q) = (4, 5) = 4r edges, the edge partition E6(G*) contains S(p,q) = (5, 5) = 6s − 4 edges, the edge partition E7(G*) contains S(p, q) = (5, 8) = 12s + 4r − 8 edges, the edge partition E8(G*) contains S(p,q) = (8, 8) = 4s edges, the edge partition E9(G*) contains S(p,q) = (8, 9) = 8r+16s−16 edges, and the edge partition E10(G*) contains S(p,q) = (9, 9) = 63rs−28r−53s+20 edges. From Formulas (4) and (7), results were attained.

Theorem 9

Let G*be the line graph of the subdivided graph (para-line graph) of Tr,s. Then

  1. HM(G)=2268rs596r864s40;

  2. PM1(G)=46s+6r+454r+12s8663rs20r33s+4;

  3. PM2(G)=46s+6r+464r+12s8963rs20r33s+4;

  4. M1(G,a)=6s+6r+4a4+4r+12s8a5+63rs20r33s+4a6;

  5. M2(G,a)=6s+6r+4a4+4r+12s8a6+63rs20r33s+4a9;

Proof. The edge set E (G*) on the basis of the degree of end vertices is represented as thesumof the cardinality of three disjointed sets i.e. E1(G*), E2(G*) and E3(G*) . The cardinality of first set E1(G*), second set E2(G*) and third set E3(G*) is denoted as E(2,2), E(2,3) and E(3,3) respectively. The first partite edge set holds E(2,2) = 8r + 12s edges, the second partite edge set holds E(2,3) = 42rs + 12r − 18s edges, and the third partite edge set holds E(3,3) = 63rs −20r −33s +4 edges.

Using Table 1, we obtained the following results.

HMG=pqEG(ξp+ξq)2HMG=pqE1G[ξp+ξq]2+pqE2G[ξp+ξq]2+pqE3G[ξp+ξq]2HMG=16×E2,2+25×E2,3+36×E3,3HMG=166s+6r+4+2512s+4r8HMG=2268rs596r864s40PM1G=pqE1G(ξp+ξq)×pqE2G(ξp+ξq)×pqE3G(ξp+ξq)PM1G=4E2,2×5E2,3×6E3,3PM1G=46s+6r+4×512s+4r8×663rs20r33s+4PM2G=pqEG(ξp×ξq)PM2G=pqE1G(ξp×ξq)×pqE2G(ξp×ξq)×pqE3G(ξp×ξq)PM2G=4E2,2×6E2,3×9E3,3PM2G=46s+6r+4×612s+4r8×963rs20r33s+4M1G,a=pqEGaξp+ξqM1G,a=pqE1Gaξp+ξq+pqE2Gaξp+ξq+pqE3Gaξp+ξqM1G,a=pqE1Ga4+pqE2Ga5+pqE3Ga6M1G,a=E2,2×a4+E2,3×a5+E3,3×a6M1G,a=6s+6r+4a4+4r+12s8a5+63rs20r33s+4a6M2G,a=pqEGaξp×ξqM2G,a=pqE1Gaξp×ξq+pqE2Gaξp×ξq+pqE3Gaξp×ξqM2G,a=pqE1Ga4+pqE2Ga6+pqE3Ga9M2G,a=E2,2×a4+E2,3×a6+E3,3×a9M2G,a=6s+6r+4a4+4r+12s8a6+63rs20r33s+4a9

Hence proved.

6 Conclusion

In this paper, we have comprehended the work on these indices: Rα, ”Mα, χα, ABC, GA, ABC4, GA5, PM1, PM2, M1( G, a), and M1(G, a) of the line graph of the subdivided graph( para-line graph) of linear [s]-Anthracene and multiple Anthracene. Randic index Rα plays a vital role in the extension of the branching of the carbon-atomskeleton of hydrocarbons. PM1(G) and PM2(G) may be expressed in the QSPR study and shows a central role in the analysis of the boiling and melting point of drugs. Chemical graphs are currently being studied with the help of para-line graphs which are very important in the field of chemistry. Our upcoming work will be the emphasis on some new classes of the above mentioned graphs(para-line) of chemical structures with respect to the different topological indices.

  1. Ethical approval

    The conducted research is not related to either human or animal use.

  2. Conflict of interest

    Conflict of Interests: Authors declare no conflict of interest.

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Received: 2018-10-14
Accepted: 2019-05-29
Published Online: 2019-11-13

© 2019 Z. Zhang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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