Operations of Nanostructures via SDD, ABC4 and GA5 indices

Open access

Abstract

Recently, nanostructures have opened new dimensions in industry, electronics, and pharmaceutical and biological therapeutics. The topological indices are numerical tendencies that often depict quantitative structural activity/property/toxicity relationships and correlate certain physico-chemical properties such as boiling point, stability, and strain energy, of respective nanomaterial. In this article, we established closed forms of various degree-based topological indices of semi-total line graph of 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s].

1 Introduction and Preliminaries

In this article, we will consider only the simple graphs G, that are without loops and multiple edges, with vertex set V (G) and edge set E(G). The degree du of a vertex u is the number of edges that are incident to it and Su = ΣνNu dν where Nu = {νV (G)|E(G)}. Nu is also known as the set of neighbor vertices of the vertex u or the neighborhood of u. The semi-total (line) graph T1(G) of G is the graph whose vertex set is V (G)∪E(G) where two vertices of T1(G) are adjacent if and only if (i) they are adjacent edges of G or (ii) one is a vertex of G and the other is an edge of G incident to that vertex (see also [19]).

The idea of a topological index first appears in the work of H. Wiener in 1947, [27], in which he was working on boiling points of paraffins. He called this index as path number, and later it was called as Wiener index. Since then, the theory of topological indices has begun to have great importance as the topological indices are the mathematical measures which correspond to the structure of any simple finite graph. They are invariant under the graph isomorphisms. The significance of topological indices is usually associated with quantitative structures property relationship (QSPR)/quantitative structure activity relationship (QSAR) [23].

In the study of QSAR/QSPR, [3], topological indices such as Shultz index, generalized Randic index, [14], Zagreb index, general sum-connectivity index, atom-bond connectivity (ABC) index, [2,4], geometric-arithmetic (GA) index, [14], and harmonic index, [15, 18, 22], are exploited to estimate the bioactivity of chemical compounds. A topological index attaches a chemical structure with a numeric number. There are numerous applications of graph theory in this field of research called molecular or chemical graph theory, [28].

Recently [25, 26], D. Vukicevic revealed the set of 148 discrete Adriatic indices. They were analyzed on the testing sets provided by the International Academy of Mathematical Chemistry and it had been shown that they have good predictive properties in many cases. There was a vast research regarding various properties of these topological indices.

Muhammad Faisal Nadeem et. al., [17], computed generalized Randic, general Zagreb, general sum-connectivity, ABC, GA, ABC4, and GA5 indices of the line graphs of 2D-lattice, nanotube and nanotorus of TUC4 C8[p, q] by using the concept of subdivision.

Sunil Hosamani [12], worked on computing sanskruti index of certain nanostructures. Computed the expressions for the Sanskruti index of the line graph of subdivision graph of the 2D-lattice, nanotube and nanotorus of TUC4 C8[p, q].

Recently, C. K. Gupta and et. al., [7, 8], worked on symmetric division deg index for bounds and operations are discussed in detail.

Symmetric division deg index is one of the discrete Adriatic indices, [25, 26]. It is a good predictor of total surface area for polychlorobiphenyls and is defined as

SDD(G)=uvE(G)du·dvdu+dv.

The fourth member of the class of ABC indices was introduced by Ghorbani and Hosseinzadeh in [5]:

ABC4(G)=uvE(G)Su+Sv2Su·Sv.

The fifth member of geometric-arithmetic (GA5) index was introduced by Graovac et. al. in [6] as

GA5(G)=uvE(G)2Su·SvSu+Sv.

The aim of this paper is to compute the SDD index, fourth member of atom-bond connectivity (ABC4) indices and fifth member of geometric-arithmetic (GA5) indices of semi-total (line) graph of the 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s], where r and s denote the number of squares in a row and the number of rows of squares respectively. The construction of nanostructure is shown in Figure 1 and the 2D-lattice, nanotube and nanotorus of the TUC4 C8[r, s] are shown in Figure 2.

Fig. 1
Fig. 1

Nanostructure

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00014

Figure 2
Figure 2

2D-Lattice, Nanotube, Nanotorus of TUC4 C8

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00014

This paper is organised as follows. Section 1 consists of a brief introduction which is essential for the development of main results. Section 2 will consist of the SDD index of the 2D-lattice, nanotube and nanotorus of the TUC4 C8[r, s] using semi-total (line) graph operator and final section concentrates on the results about the neighborhood degree based indices such as ABC4 and GA5 of 2D-lattice, nanotube and nanotorus of the TUC4 C8[r, s] using the semi-total (line) graph operator.

2 Symmetric division deg index of semi-total(line) graph of 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s]

In this section, we computed the general expressions for the SDD index of 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s] using semi-total (line) graph operator and the structure of the graph depicted in Figure 3.

Figure 3
Figure 3

Semi-total(line) graph of 2D-Lattice, Nanotube, Nanotorus

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00014

In 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s], the number of vertices are 4rs and the number of edges are 6rsrs, 6rss and 6rs, respectively. The following Tables 1(a,b), 2(a,b) and 3(a,b) indicates the algebraic method, i.e., the partitioned edges will be helpful to compute the results.

Table 1

Edge partition of Semi-total(line) graph of 2D-Lattice

(a) r > 1 and s = 1
Edges Partitioned(2,4)(2,5)(3,5)(3,6)(4,4)(4,5)(5,5)(5,6)
No of Edges84(r-1)4(r-1)2(r-1)242(2r-3)4(r-1)
(b) r > 1 and s > 1
Edges Partitioned(2,4)(4,5)(2,5)(3,5)(3,6)(5,5)(5,6)(6,6)
No of Edges884(r+s-2)4(r+s-2)2[s(r+s) +5(r+s-4)]2(r+s-4)8(r+s-2)2[(s+3) (r+s)-16]

Table 2

Semi-total line graph of nanotube

(a) r > 1 and s = 1
Edges Partitioned(2,5)(3,3)(3,5)(5,5)(5,6)
No of Edges4r24(r+1)2(r-1)4r4(r-1)
(b) r > 1 and s > 1
Edges Partitioned(2,5)(3,3)(3,5)(3,6)(5,5)(5,6)(6,6)
No of Edges4r2s4(r+1)14(r-2)+26(s-2)+2r(s-2)(r+s-4)+282r4(2r-1)10(r-2)+20(s-2)+2r(s-2)(r+s-4)+16

Table 3

Edge Partition for Semi-total(line) graph of Nanotorus

(a) r > 1 and s = 1
Edges Partitioned(3,3)(3,6)(6,6)
No of Edges2(r + 1)2(7r + 1)4(2r − 1)
(b) r > 1 and s > 1
Edges Partitioned(3,3)(3,6)(6,6)
No of Edges2(r + s)[2r(s − 2) + 26](r + s − 4) + 56[2r(s − 2) + 20](r + s − 4) + 32

Theorem 1.

Let G=TUC4 C8[r, s] be the semi-total (line) graph of 2D-lattice graph of TUC4 C8[r, s], then

lSDD(H)=5.933r+60.93s+rs(9s+42)41s2103.467whenr>1ands>141.8r13.6whenr>1ands=1

Proof. Case (i): Let r > 1 and s > 1.

Let G be the semi-total (line) graph of the 2D-lattice graph of TUC4 C8[r, s]. The number of vertices and edges of the 2D-Lattice graph G is equal to|V| = 2(s − 2)(r + s − 3) + 3rs + 8rs and|E| = 2(2s + 17)(r + s).

Hence the edge partition on the degree sum of vertices of each vertex is obtained, as shown in Table 1(b). We apply the topological indices to the edge partitions to get the required results.

Case (ii): Let r > 1 and s = 1.

It can be observed that in G,|V| = (9r − 1) and|E| = 6(3rs). Hence utilizing Table 1(a) and SDD index, we can obtain the expressions of SDD index of G.

Theorem 2.

Let H be the semi-total (line) graph of the TUC4 C8[r, s] nanotube, then

SDD(H)=40.93r+4s+55(r2)+105(s2)+9r(s2)(r+s4)+102.93whenr>1ands>127.733r+4.933whenr>1ands=1

Proof. Let H be the semi-total (line) graph of nanotube of TUC4 C8[r, s] nanotube. Vertices and edges of H graph are equal to|V|=(9r + 1) and|E|=18r for r > 1, s = 1 and|V|=9rs + r + s + (s − 2)(r + s − 3) and |E|=4r(r(s − 2) + s(s − 6)) + 74r + 48s − 56 for the case and r > 1, s > 1. Hence computing the results using the Table 2(a) and 2(b) and SDD index, we get the required result.

Theorem 3.

Let K be the semi-total (line) graph of the TUC4 C8[r, s] nanotorus. Then

SDD(K)=4(r+s)+(r+s4)[9r(s2)+105]+204whenr>1ands>155r+1whenr>1ands=1

Proof. Let K be semi-total (line) graph of the nanotorus of TUC4 C8[r, s]. The number of vertices and edges of the graph K are equal to|V| = (11r +1),|E| = 32r; and|V| = 10rs+ r + s and|E| = 4r(s − 2)+134 respectively, in the cases of r > 1, s = 1; and r > 1, s > 1. Hence computing the results using the Table 3(a) and 3(b) and SDD definition, we get the required result.

3 ABC4 and GA5 indices of semi-total (line) graph of 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s]

In this section, ABC4 and GA5 by using semi-total (line) graph of the 2D-lattice, nanotube and nanotorus of TUC4 C8[r, s] are determined by means of the neighborhood vertex degree indices.

Theorem 4.

Let G1 be the semi-total (line) graph of the 2D-lattice, then

ABC4(G1)={8.99(r+s)+3.316(r+s4)[1+r/3(s2)]11.754when r>1 and s>15.751r2.119    when r>1 and s=1GA5(G1)={29.347(r+s)+11.822(r+s4)[1+r/3(s2)]43.101when r>1 and s>117.622r5.873    when r>1 and s=1

Proof. To compute the ABC4(G1) and GA5(G1) indices of the 2D-lattice of TUC4 C8[r, s], we need an edge partition of the 2D-lattice of TUC4 C8[r, s], based on the degree sum of neighbors of the two end vertices of each edge. We presented these partitions with their cardinalities in Tables 4(a) and 4(b). Hence using the definitions of the ABC4 and GA5, and Table 4(a) and 4(b), we obtained required results.

Table 4

Edge Partition of Semi-total(line) graph of 2D-Lattice for Neighborhood vertices

(a) r > 1 and s = 1
Edges PartitionedNo of Edges
(8,13) (9,13) (13,20) (9,20) (16,20) (20,26)4
(13,13) (20,20)2
(16,26)2(r-1)
(16,21) (10,21) (21,21) (21,26)4(r-2)
(b) r > 1 and s > 1
Edges PartitionedNo of Edges
(9,14) (9,21) (14,21) (17,21) (28,28)8
(21,28)16
(17,28)4(r + s)
(17,22) (22,29) (10,22) (22,28) (17,29) (28,29)4(r + s − 4)
(22,22)2(r + s − 4)
(29,29) (18,30)2(r + s − 4)[1 + r/3(s − 2)]
(18,29) (29,30)4(r + s − 4)[1 + r/3(s − 2)]

Theorem 5.

Let H1 be the semi-total (line) graph of the TUC4 C8[r, s] nanotube, then

ABC4(H1)={11.7r+7.151s+3.048(s2)(r+s4)+2(r+s2)+3.502(s2)(r+s4)14.84 when r>1 and s>15.749r0.638    when r>1 and s=1GA5(H1)={39.21r+23.447s+12(s2)(r+s4)+2(r+s2)+11.616(s2)(r+s4)66.539 when r>1 and s>117.614r+0.095    when r>1 and s=1

Proof. To compute the ABC4(H1) and GA5(H1) indices of the nanotube of the TUC4 C8[r, s], we need an edge partition of the nanotube of the TUC4 C8[r, s], based on the degree sum of the neighbors of the two end vertices of each edge. We presented these partitions together with their cardinalities in Tables 5(a) and 5(b). Hence using the ABC4 and GA5 formulae together with Table 5(a) and 5(b), we obtain the required results.

Table 5

Semi-total(line) graph of Nanotube for Neighborhood vetices

(a) r > 1 and s = 1
Edges PartitionedNo of Edges
(13,13) (18,18)2
(10,18) (18,21)4
(13,18)8
(10,21) (16,21) (21,26)4(r − 1)
(16,26)2(r − 1)
(21,21)2(2 r − 3)
(b) r > 1 and s > 1
Edges PartitionedNo of Edges
(14,13)2s
(19,26) (26,29) (26,18) (26,30) (19,10) (13,26) (14,19) (14,26) (13,19) (19,22)4
(10,22) (22,17) (17,29) (18.29) (22,29) (17,28) (29,30) (22,28) (28,29)4(r − 1)
(22,22) (29,29)2(r − 2)
(13,27) (14,27) (18,27)4(s − 2)
(27,27)2(s − 2)
(18,30)2(r + s − 2) + 12(s − 2)(r + s − 4)
(30,30)[12(s − 2)(r + s − 4)] − 2

Theorem 6.

Let K1 be the semi-total (line) graph of the TUC4 C8[r, s] nanotorus, then

ABC4(K1)=21.109(r+s)+2(rs)(r+s4)[0.508(r+s)1.371]26.17whenr>1ands>17.103r+0.568whenr>1ands=1GA5(K1)=47.123(r+s)+2(s2)(r+s4)[2(r+s)4.128]94.314whenrgt1ands>1.17.6r+5.961whenr>1ands=1

Proof. To compute the ABC4(K1) and GA5(K1) indices of the nanotorus of the TUC4 C8[r, s], we need an edge partition of the nanotorus of TUC4 C8[r, s], based on the degree sum of the neighbors of the two end vertices of each edge. We presented these partitions with their cardinalities in Tables 6(a) and 6(b). Hence using the ABC4 and GA5 formula together with Tables 6(a) and 6(b), we obtain required results.

Table 6

Semi-total(line) graph of Nanotorous for Neighborhood vertices

(a) r > 1 and s = 1
Edges PartitionedNo of Edges
(15,24)16
(24,27)4
(24,24)2
(15,15)2(r + 1)
(15,27)8(r − 1)
(27,27)2(2 r − 3)
(18,30)2(r − 1)
(27,30) (18,27)4(r − 1)
(b) r > 1 and s > 1
Edges PartitionedNo of Edges
(15,24)16
(24,27)8
(15,15)2(r + s)
(15,27) (27,30)8(r + s − 2)
(18,30)14(r + s) + 8(s − 2)(r + s − 4) − 40
(18,27)4(r + s − 8)
(27,27)2(r + s − 4)
(30,30)10(r + s) + 2(s − 2)(r + s − 4)[2(r + s − 4) − 2] − 32

4 Conclusions

In the field of chemical graph theory, the studies under the framework of semi-total (line) graph operator is a new direction in the field of structural chemistry. In this article, we computed the closed form of the 2D-lattice, nanotubes and nanotorus of some degree-based topological indices for these structures. These indices can help us to understand the physical features, chemical and biological activities of these structures such as the boiling point, the heat of formation, the fracture toughness, the strength, the conductivity, and the hardness. From this point of view, a topological index can be regarded as a score function that maps each molecular structure to a real number and is used as a descriptor of the molecule under testing. These results can also play a vital part in the study of the nanostructures in electronics and industry which can be used in the preparation of armor due to their strength.

Communicated by Wei Gao

References

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    N. Trinajstic (1992) Chemical Graph Theory CRC Press Boca Raton.

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    D. Vukicevic and M. Gasperov (2010) Bond additive modeling 1. Adriatic indices Croatica Chemica Acta 83(3) 243 - 260.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    V. Alexander (2014) Upper and lower bounds of symmetric division deg index Iranian Journal of Mathematical Chemistry 5(2) 91 - 98.

  • [2]

    K. C. Das I. Gutman and B. Furtula (2011) On atom-bond connectivity index Chemical Physics Letters 511(4) 452 - 454.

    • Crossref
    • Export Citation
  • [3]

    J. Devillers and A. T. Balaban (Eds.) (1999) Topological Indices and Related Descriptors in QSAR and QSPR Gordon and Breach Amsterdam.

  • [4]

    E. Estrada L. Torres L. Rodriguez and I. Gutman (1998) An Atom-bond connectivity index: Modelling the enthalpy of formation of alkanes Indian Journal of Chemistry. Sect. A: Inorganic physical theoretical and analytical 37(10) 849 - 855.

  • [5]

    M. Ghorbani M. A. Hosseinzadeh (2010) Computing ABC4 index of nanostar dendrimers Optoelectron. Adv. Matter-Rapid Commun. 4(9) 1419 - 1422.

  • [6]

    A. Graovac M. Ghorbani M. A. Hosseinzadeh (2011) Computing fifth geometric-arithmetic index for nanostar dendrimers J. Math. Nanosci 1(1) 33 - 42.

  • [7]

    C. K. Gupta V. Lokesha S. B. Shetty (2016) On the Symmetric division deg index of graph South East Asian Journal Of Mathematics 41(1) 59 - 80.

  • [8]

    C. K. Gupta V. Lokesha S. B. Shetty and P. S. Ranjini (2017) Graph Operations on Symmetric Division Deg Index of Graphs Palestine Journal Of Mathematics 6(1) 280 - 286.

  • [9]

    W. Gao M. Farahani S. Wang and M. Husin (2017) On the edge-version atom-bond connectivity and geometric arithmetic indices of certain graph operations Appl. Math. and Comput. 308 11 - 17.

  • [10]

    F. Harary (1994) Graph Theory Reading MA: Addison-Wesley.

  • [11]

    S. M. Hosamani (2016) Computing Sanskruti index of certain nanostructures J. Appl. Math. Comput. 1(9) ISSN: 1598-5865.

  • [12]

    S. M. Hosamani V. Lokesha I. N. Cangul K. M. Devendraiah (2016) On Certain Topological Indices of the Derived Graphs of Subdivision Graphs TWMS J. Appl. Eng. Math. 6(2) 324 - 332.

  • [13]

    S. Hayat M. Imran (2014) Computation of topological indices of certain networks Appl. Math. Comput. 240 213 28.

  • [14]

    V. Lokesha S. B. Shetty P. S. Ranjini I. N. Cangul and A. S. Cevik (2013) New Bounds for Randic and GA Indices Journal of Inequalities and Applications 180(1) 1 - 7.

  • [15]

    V. Lokesha A. Usha P. S. Ranjini and T. Deepika (2015) Harmonic index of cubic polyhedral graphs using bridge graphs App. Math. Sci. 9 4245-4253.

  • [16]

    J. Liu S. Wang C. Wang and S. Hayat Further results on computation of topological indices of certain networks IET Control Theory & Applications

    • Crossref
    • Export Citation
  • [17]

    M. F. Nadeema S. Zafarb Z. Zahid (2016) On topological properties of the line graphs of subdivision graphs of certain nanostructures Applied Mathematics Computation 273 125 - 130.

    • Crossref
    • Export Citation
  • [18]

    P. S. Ranjini A. Usha V. Lokesha and T. Deepika (2016) Harmonic index redefined Zagreb indices of dragon graph with complete graph Asian J. of Math. and Comp. Research 9 161 - 166.

  • [19]

    Ranjini P.S and V. Lokesha (2010) Smarandache-Zagreb Index on Three Graph Operators International J.Math. Combin. 3 01 - 10.

  • [20]

    S. B. Shetty V. Lokesha P. S. Ranjini and K. C. Das (2012) Computing Some Topological Indices of Smart Polymer Digest Journal of Nanomaterials and Biostructures 7(3) 1097 - 1102.

  • [21]

    B. S. Shwetha V. Lokesha A. Bayad and P. S. Ranjini (2012) A Comparative Study of Topological Indices and Molecular Weight of Some Carbohydrates Journal of the Indian Academy of Mathematics 34(2) 627 - 636.

  • [22]

    B. S. Shwetha V. Lokesha and P. S. Ranjini (2015) On The Harmonic Index of Graph Operations Transactions on Combinatorics 4(4) 5 - 14.

  • [23]

    Sunilkumar M Hosamani (2016) Correlation of domination parameters with physicochemical properties of octane isomers J. of Appl. Math. and Nonlinear Sciences 1(2) 345 - 352

    • Crossref
    • Export Citation
  • [24]

    N. Trinajstic (1992) Chemical Graph Theory CRC Press Boca Raton.

  • [25]

    D. Vukicevic and M. Gasperov (2010) Bond additive modeling 1. Adriatic indices Croatica Chemica Acta 83(3) 243 - 260.

  • [26]

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