Operations of Nanostructures via ,  and  indices

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 d_u of a vertex u is the number of edges that are incident to it and S_u = Σ_{ν∊N_u}d_ν where N_u = {ν ∊ V (G)|uν ∊ E(G)}. N_u 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 TUC₄C₈[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 TUC₄C₈[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

$S D D (G) = \sum_{u v \in E (G)} \frac{d_{u} \cdot d_{v}}{d_{u} + d_{v}} .$ $$\begin{array}{} \displaystyle SDD(G)=\sum_{uv \in E(G)}\frac{d_u \cdot d_v}{d_{u}+d_{v}}. \end{array}$$

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

$A B C_{4} (G) = \sum_{u v \in E (G)} \sqrt{\frac{S_{u} + S_{v} - 2}{S_{u} \cdot S_{v}}} .$ $$\begin{array}{} \displaystyle ABC_{4}(G)=\sum_{uv \in E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u} \cdot S_{v}}}. \end{array}$$

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

$G A_{5} (G) = \sum_{u v \in E (G)} \frac{2 \sqrt{S_{u} \cdot S_{v}}}{S_{u} + S_{v}} .$ $$\begin{array}{} \displaystyle GA_{5}(G)=\sum_{uv \in E(G)}\frac{2\sqrt{S_{u} \cdot S_{v}}}{S_{u}+S_{v}}. \end{array}$$

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 TUC₄C₈[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 TUC₄C₈[r, s] are shown in Figure 2.

2D-Lattice, Nanotube, Nanotorus of TUC₄C₈

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 TUC₄C₈[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 TUC₄C₈[r, s] using the semi-total (line) graph operator.

Symmetric division deg index of semi-total(line) graph of 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s]

In this section, we computed the general expressions for the SDD index of 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s] using semi-total (line) graph operator and the structure of the graph depicted in Figure 3.

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

In 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s], the number of vertices are 4rs and the number of edges are 6rs − r − s, 6rs − s 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 Edges	8	4(r-1)	4(r-1)	2(r-1)	2	4	2(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 Edges	8	8	4(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 Edges	4r	2	4(r+1)	2(r-1)	4r	4(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 Edges	4r	2s	4(r+1)	14(r-2)+26(s-2)+2r(s-2)(r+s-4)+28	2r	4(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 Edges	2(r + 1)	2(7r + 1)	4(2r − 1)

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

Theorem 1.

Let G=TUC₄C₈[r, s] be the semi-total (line) graph of 2D-lattice graph of TUC₄C₈[r, s], then

$\begin{matrix} l S D D (H) = (\begin{matrix} 5.933 r + 60.93 s + r s (9 s + 42) - 41 s^{2} - 103.467 w h e n r > 1 a n d s > 1 \\ 41.8 r - 13.6 w h e n r > 1 a n d s = 1 \end{matrix}) \end{matrix}$ $$\begin{align}{l} SDD(H) = \left\{ \begin{array}{l}{5.933r + 60.93s+rs(9s+42)-41s^2 -103.467 {\rm{\ }}when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s \gt 1}\\ {41.8r - 13.6 \ {when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s = 1}}\end{array}\right. \end{align}$$

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

Let G be the semi-total (line) graph of the 2D-lattice graph of TUC₄C₈[r, s]. The number of vertices and edges of the 2D-Lattice graph G is equal to|V| = 2(s − 2)(r + s − 3) + 3r − s + 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(3r − s). 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 TUC₄C₈[r, s] nanotube, then

$\begin{matrix} S D D (H) = (\begin{matrix} 40.93 r + 4 s + 55 (r - 2) + 105 (s - 2) + 9 r (s - 2) (r + s - 4) + 102.93 \\ w h e n r > 1 a n d s > 1 \\ 27.733 r + 4.933 w h e n r > 1 a n d s = 1 \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle SDD(H) = \left\{\begin{array}{} {40.93r + 4s + 55(r - 2) + 105(s - 2) + 9r(s - 2)(r + s - 4) + 102.93}\\ {when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s \gt 1}\\ {27.733r + 4.933\,\,\,\,\,\,\,{\rm{ }}when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right. \end{array}$$

Proof. Let H be the semi-total (line) graph of nanotube of TUC₄C₈[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 TUC₄C₈[r, s] nanotorus. Then

$\begin{matrix} S D D (K) = (\begin{matrix} 4 (r + s) + (r + s - 4) [9 r (s - 2) + 105] + 204 w h e n r > 1 a n d s > 1 \\ 55 r + 1 w h e n r > 1 a n d s = 1 \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle SDD(K) = \left\{ \begin{array}{}{4(r + s) + (r + s - 4)[9r(s - 2) + 105] + 204\,\,\,\,\,{\rm{\ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s > 1}\\ {55r + 1\,\,\,\,\,\,\,\,{\rm{ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right. \end{array}$$

Proof. Let K be semi-total (line) graph of the nanotorus of TUC₄C₈[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.

ABC₄ and GA₅ indices of semi-total (line) graph of 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s]

In this section, ABC₄ and GA₅ by using semi-total (line) graph of the 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s] are determined by means of the neighborhood vertex degree indices.

Theorem 4.

Let G₁be the semi-total (line) graph of the 2D-lattice, then

$\begin{matrix} A B C_{4} (G_{1}) = {\begin{matrix} 8.99 (r + s) + 3.316 (r + s - 4) [1 + r / 3 (s - 2)] - 11.754 \\ w h e n r > 1 a n d s > 1 \\ 5.751 r - 2.119 w h e n r > 1 a n d s = 1 \end{matrix} \\ G A_{5} (G_{1}) = {\begin{matrix} 29.347 (r + s) + 11.822 (r + s - 4) [1 + r / 3 (s - 2)] - 43.101 \\ w h e n r > 1 a n d s > 1 \\ 17.622 r - 5.873 w h e n r > 1 a n d s = 1 \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {AB{C_4}\left( {{G_1}} \right) = } \\ {} \\ \end{array}\begin{array}{*{20}{c}} \left\{\begin{array}{}{8.99\left( {r + s} \right) + 3.316\left( {r + s - 4} \right)\left[ {1 + r/3\left( {s - 2} \right)} \right] - 11.754} \\ {when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s > 1} \\ {5.751r - 2.119{\rm{ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right. \\ \end{array}} \\ {G{A_5}\left( {{G_1}} \right) = \begin{array}{*{20}{c}} \left\{\begin{array}{}{29.347\left( {r + s} \right) + 11.822\left( {r + s - 4} \right)\left[ {1 + r/3\left( {s - 2} \right)} \right] - 43.101} \\ {when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s > 1} \\ {17.622r - 5.873{\rm{ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right.\\ \end{array}} \\ \end{array} \end{array}$$

Proof. To compute the ABC₄(G₁) and GA₅(G₁) indices of the 2D-lattice of TUC₄C₈[r, s], we need an edge partition of the 2D-lattice of TUC₄C₈[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 Partitioned	No 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 Partitioned	No 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 H₁be the semi-total (line) graph of the TUC₄C₈[r, s] nanotube, then

$\begin{matrix} A B C_{4} (H_{1}) = {\begin{matrix} 11.7 r + 7.151 s + 3.048 (s - 2) (r + s - 4) + 2 (r + s - 2) \\ + 3.502 (s - 2) (r + s - 4) - 14.84 w h e n r > 1 a n d s > 1 \\ 5.749 r - 0.638 w h e n r > 1 a n d s = 1 \end{matrix} \\ G A_{5} (H_{1}) = {\begin{matrix} 39.21 r + 23.447 s + 12 (s - 2) (r + s - 4) + 2 (r + s - 2) + \\ 11.616 (s - 2) (r + s - 4) - 66.539 w h e n r > 1 a n d s > 1 \\ 17.614 r + 0.095 w h e n r > 1 a n d s = 1 \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} \begin{array}{*{20}{c}} {AB{C_4}\left( {{H_1}} \right) = } \\ {} \\ \end{array} \left\{\begin{array}{}{11.7r + 7.151s + 3.048\left( {s - 2} \right)\left( {r + s - 4} \right) + 2\left( {r + s - 2} \right)} \\ { + 3.502\left( {s - 2} \right)\left( {r + s - 4} \right) - 14.84{\rm{\ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s > 1} \\ {5.749r - 0.638{\rm{ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s = 1} \end{array}\right. \\ \\ G{A_5}\left( {{H_1}} \right) = \left\{\begin{array}{}{39.21r + 23.447s + 12\left( {s - 2} \right)\left( {r + s - 4} \right) + 2\left( {r + s - 2} \right) + } \\ {11.616\left( {s - 2} \right)\left( {r + s - 4} \right) - 66.539{\rm{\ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s > 1} \\ {17.614r + 0.095{\rm{ }}when{\rm{\ }}r > 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right. \\ \\ \end{array} \end{array}$$

Proof. To compute the ABC₄(H₁) and GA₅(H₁) indices of the nanotube of the TUC₄C₈[r, s], we need an edge partition of the nanotube of the TUC₄C₈[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 Partitioned	No 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 Partitioned	No 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 K₁be the semi-total (line) graph of the TUC₄C₈[r, s] nanotorus, then

$\begin{matrix} \begin{matrix} A B C_{4} (K_{1}) = (\begin{matrix} 21.109 (r + s) + 2 (r - s) (r + s - 4) [0.508 (r + s) - 1.371] - 26.17 \\ w h e n r > 1 a n d s > 1 \\ 7.103 r + 0.568 w h e n r > 1 a n d s = 1 \end{matrix}) \\ G A_{5} (K_{1}) = (\begin{matrix} 47.123 (r + s) + 2 (s - 2) (r + s - 4) [2 (r + s) - 4.128] - 94.314 \\ w h e n r g t 1 a n d s > 1. \\ 17.6 r + 5.961 w h e n r > 1 a n d s = 1 \end{matrix}) \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} AB{C_4}({K_1}) = \left\{\begin{array}{} {21.109(r + s) + 2(r - s)(r + s - 4)[0.508(r + s) - 1.371] - 26.17}\\ {{\rm{\ }}when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s \gt 1}\\ {7.103r + 0.568\,\,\,\,\,\,when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s = 1}\end{array}\right. \\ G{A_5}({K_1}) = \left\{\begin{array}{} {47.123(r + s) + 2(s - 2)(r + s - 4)[2(r + s) - 4.128] - 94.314}\\ {{\rm{\ }}when{\rm{\ }}r gt 1{\rm{\ }}and{\rm{\ }}s \gt 1.}\\ {17.6r + 5.961\,\,{\rm{\ }}when{\rm{\ }}r \gt 1{\rm{\ }}and{\rm{\ }}s = 1} \end{array}\right. \\ \end{array} \end{array}$$

Proof. To compute the ABC₄(K₁) and GA₅(K₁) indices of the nanotorus of the TUC₄C₈[r, s], we need an edge partition of the nanotorus of TUC₄C₈[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 Partitioned	No 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 Partitioned	No 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

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.

eISSN:: 2444-8656
Język:: Angielski

Częstotliwość wydawania:: 2 razy w roku
Dziedziny czasopisma:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Kanał RSS czasopisma

Operations of Nanostructures via SDD, ABC4 and GA5 indices

Data publikacji: 16 maj 2017

Zakres stron: 173 - 180

Otrzymano: 07 mar 2017

Przyjęty: 16 maj 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00014

Słowa kluczoweTopological indices, 2-lattice, nanotube, nanotorus, semi-total line operator

© 2017 V. Lokesha, T. Deepika, P. S. Ranjini, I. N. Cangul, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Figure 2

Figure 3

Operations of Nanostructures via SDD, ABC₄ and GA₅ indices

Słowa kluczowe
Topological indices, 2-lattice, nanotube, nanotorus, semi-total line operator