 
STINT 2022 : Study of Topological indices of NAn m and NCn m Nanotube  
Link: https://sites.google.com/pondiuni.ac.in/mathscaldam2022  
 
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Study of Topological indices of NAn
m and NCn m Nanotube October 5, 2021 S.Rajeswari1 and N. Parvathi2 1Dept.of Mathematics, SRM Institute of Science and Technology, Kattankulathur603 203,Tamil Nadu, INDIA suraj1719cyr@gmail.com 2Dept.of Mathematics, SRM Institute of Science and Technology, Kattankulathur603 203,Tamil Nadu, INDIA parvathn@srmist.edu.in Abstract A numerical quantity of a molecular graph that characterize is topology and graph invariant is known as topological indices. The concept of First and Second Zagreb M1(G) and M2(G). Redefined First and Second Zagreb index ReZG1(G) and ReZG2(G) HyperZagreb index HM(G) and and Harmonic index H(G) of topological indices of degree base vertex. In this article, M1(G) and M2(G), ReZG1(G) and ReZG2(G), HM(G), H(G) indices are computed for the nanotube NAn m and NCn m. AMS Subject Classification: ... Key Words and Phrases: Topological indices, First and Second Zagreb index, Redefine First and Second Zagreb, HyperZagreb index, Harmonic index and Carbon Nanotube network.. 1 Introduction Mathematical chemistry is a branch of chemistry research in which mathematical methods are utilized to tackle problems in the field. To model 1 molecules and molecular compounds, chemical graphs are frequently used. A molecular graph is a graph with atoms and bonds indicating the vertices and edges, respectively. A “molecular graph” is described as a carbon molecule with a valence of four and a maximum degree of four. [3]. A topological index is a numerical parameter that depicts the topology of a graph. Topological indices, which are integers related to molecular graphs of chemical compounds, are used in quantitative structureproperty relationships (QSPR). The most wellknown invariants of this type are degreebased topological indices. Many aspects of chemical compounds and Nanomaterial’s, like boiling temperatures, strain energy, stiffness, and fracture toughness. Topological indices, which are invariant up to graph isomorphism, is used to predict. While the topology of a molecule, as represented by the molecular graph, is essentially a nonnumerical mathematical object, the measurable properties of molecules are frequently molecular topology. The information contained in the molecular graph must first be converted into a numerical characteristic before the information contained in the molecular graph can be converted. A graph invariant is an integer that can only be determined by a graph. These invariants are popularly known as “Topological Indices.” Nanotechnology is the study of nanostructures of sizes 1 to 100 nanometres. Sumio Iijima, a Japanese scientist, discovered the carbon nanotube in 1991[2]. (CNTs) are initially used it as an additive in a variety of structural materials for electronics, plastics, and other nanotechnology products. Carbon nanotubes are entirely made up of carbon arranged in a tubular structure made up of condensed benzene rings. Allotropes of carbon with a cylindrical nanostructure make carbon nanotubes. (CNTs) are of two types: Single Walled Carbon Nanotube (SWCNT) or multiple sheets of graphene Multi Walled Carbon Nanotube (MWCNT) (MWCNT). Carbon nanotubes and fullerenes are hollow carbon allotropes with exceptional thermal, electrical, and mechanical. Buckyballs are spherical fullerenes, and nanotubes are cylindrical fullerenes. The walls of these structures are made of a single layer of carbon atoms (graphene). These cylindrical carbon particles have distinct properties that are useful in nanotechnology, optics, electronics, and a broad range of materials research and innovation disciplines. [13]. Carbon nanotubes are the stiffest and most durable materials. Carbon nanotubes are well suited for manipulating other Nano scale structures due to their flexibility and strength, implying that they will play an important role in nanotechnology engineering. 2 Nanotubes are good heat conductors and are a good product for electronics and optical. Nanotubes are being developed for a variety of purposes. Highquality nanotube materials are sought for both basic and technological applications. The absence of structural and chemical flaws at a large length scale (e.g. 110 microns) along the tube axes is referred to as high quality. Several approaches have been devised to generate CNTs and MWNTs in laboratory quantities with various structures and morphologies. Carbon nanotube produce three methods: Arc discharge, Laser ablation, and chemical vapour deposition [16] [17]. The discovery of fullerene molecules and related carbon structures such as nanotubes has sparked a surge in chemistry, physics, and materials research. The atoms are organized on a pseudospherical framework composed primarily of pentagons and hexagons. Its chemical graph consists of only hexagonal and (exactly 12) pentagonal faces embedded on the surface of a sphere [19]. Let G be a connected graph with V (G) and E(G) as he vertex set and the edge set, respectively. The distance between two vertices says a and b of G, denoted by d(a, b) is defined as the number of edges in the shortest path connecting a and b. 2 Definitions and Literature Review Let G be a connected graph with V (G) and E(G) as he vertex set and the edge set, respectively. The distance between two vertices says u and v of G, denoted by d(u, v) is defined as the number of edges in the shortest path connecting u and v. Definition 1. An important topological index introduced more than 30 years ago by I. Gutman and N. Trinajistic is the first and second Zagreb indices (M1(G) and M2(G)) [3]. These indices are defined as: M1(G) = X e=uv∈E(G) (du + dv) M2(G) = X e=uv∈E(G) (du.dv) Definition 2. The Hyper Zagreb index (HM(G)) being introduced by Shirdel 3 et al. It is a distance based Zagreb index in 1913 [9]. HM(G) = X e=uv∈E(G) (du + dv) 2 Definition 3. The Sum Connectivity Index (SCI(G)) being introduced by Zhou and Trinajstic[11]. SCI(G) = X e=uv∈E(G) 1 √ du + dv Definition 4. The Redefined First and Second Zagreb index being introduced by Rajini et. al. For a graph G and there are manifested as Re ZG1(G) = X e=uv∈E(G) d(u) + d(v) d(u).d(v) Re ZG2(G) = X e=uv∈E(G) d(u).d(v) d(u) + d(v) Definition 5. For a graph G, the Harmonic index is defined as H(G) = X e=ab∈E(G) 2 du + dv 3 Main Results and Discussion We will take a look at the carbon nanotube network in this section. Consider the mn quadrilateral section P n m, which is cut from the regular hexagonal lattice L and has m ≥ 2 hexagons on the top and bottom sides and n ≥ 2 hexagons on the lateral sides (see figure). If we identify two lateral sides such that the vertices u j 0 and u j m for j = 0, 1, 2, · · · , n are identified, we get the nanotube NAn m with 2m(n+1) vertices and (3n + 2)m edges. Let n be an even number, n ≥ 2 and m ≥ 2. If we identify the top and bottom sides of the quadrilateral section P n m in such a way that we identify the vertices u 0 i and u n i for i = 0, 1, 2, · · · , m and the vertices v 0 i and v n i for i = 0, 1, 2, · · · , m, we get the nanotube NCn m of order n(2m + 1) and size n(3m + 1). 4 Figure 1: Quadrilateral section P n m cuts from the regular hexagonal lattice Theorem 6. Consider the nanotube NAn m for m, n ≥ 2. Then M1(NAn m) = 18mn + 18m M2(NAn m) = 27mn + 6m Proof: The nanotube NAn m has 2m vertices of degree 2 and 2mn vertices of degree 3. The edge set E(NAn m) divides into two edge partitions based on degrees of end vertices. The first edge partition E1(NAn m) contains m (3n2) edges, where deg(a) = deg(b) = 3. The second edge partition E2(NAn m) contains 4m edges, where deg(a) = 2 and deg(b) = 3. By definition of First Zagreb index, we have M1(G) = X e=uv∈E(G) (du + dv) 5 It follows that M1(NAn m) = X e=uv∈E1(NAnm) (du + dv) = 6E1(NAn m) + 5E2(NAn m) = 6m(3n − 2) + 5(4m) = 18mn − 12m + 20m = 18mn + 8m Similarly, definition Second Zagreb index, we have M2(G) = X e=uv∈E(G) (du.dv) It follows that M2(NAn m) = X e=uv∈E2(NAnm) (du.dv) = 9E1(NAn m) + 6E2(NAn m) = 9m(3n − 2) + 6(4m) = 27mn − 18m + 24m = 27mn + 6m Hence proved. Theorem 7. Consider the nanotube NCn m for m ≥ 2, n ≥ 2. Then M1(NCn m) = 18mn − n M2(NCn m) = 54mn − 13n 2 Proof: Let the nanotube NCn m has 2n vertices of degree 2 and n (2m1) vertices of degree 3. Here are three types of edges in E(NCn m) based on the degrees of end vertices of each edge. (i.e.) E(NCn m) = E1(NCn m) ∪ E2(NCn m) ∪ E2(NCn m). The edge partition E1(NCn m) contains n edges where deg(a) = deg(b) = 2. 6 The edge partition E2(NCn m) contains 2n edges where deg(a) = 2 and deg(b) = 3. The edge partition E3(NCn m) contains n(3m − 5/2) edges where deg(a) = deg(b) = 3. By definition of Firs Zagreb index, we have M1(G) = X e=uv∈E(G) (du + dv) It follows that M1(NAn m) = X e=uv∈E1(NCnm) (du + dv) = (2 + 2)E1(NCn m) + (2 + 3)E2(NCn m) + (3 + 3)E3(NCn m) = 4(n) + 5(2n) + 6[n(3m − 5/2)] = 4n + 10n + 18mn − 15n = 18mn − n Also by definition of Second Zagreb index, we have M2(NCn m) = X e=uv∈E(NCnm) (du.dv) It follows that M2(NAn m) = X e=uv∈E2(NCnm) (du.dv) = (2 × 2)E1(NCn m) + (2 × 3)E2(NCn m) + (3 × 3)E3(NCn m) = 4(n) + 6(2n) + 9[n(3m − 5/2)] = 4n + 12n + 27mn − 45n 2 = 54mn − 13n 2 Hence the proof. Theorem 8. Consider the nanotube NAn m for m, n ≥ 2 then the Hyper Zagreb index is HM(NAn m) = 108mn − 28m where m, n ≥ 2. 7 Proof: Let the nanotube NAn m has 2m vertices of degree 2 and 2mn vertices of degree 3. For determining the HyperZagreb index of nanotube NAn m we apply the edge partitions described in the proof of the theorem 6. Then we have HM(G) = X e=uv∈E(G) (du + dv) 2 We obtain HM(NAn m) = (3 + 3)2 E1(NAn m) + (2 + 3)2 E2(NAn m) = 36[m(3n − 2)] + 25(4m) = 108mn − 72m + 100m = 108mn − 28m Hence the proof. Theorem 9. Consider the nanotube NCn m for m ≥ 2 and n ≥ 2. Then the Hyper Zagreb index is HM(NCn m) = 108mn + 21n. Proof: Let the nanotube NCn m has 2n vertices of degree 2 and n(2m − 1) vertices of degree 3. The edge set can be divided into three types of edges based on degrees of end vertices of each edge. For computing, the HyperZagreb index of the nanotube NCn m, we apply the edge values are defined by the proof of the theorem 7. By definition of HyperZagreb index HM(G) = X e=uv∈E(G) (du + dv) 2 8 Then we have HM(G) = X e=uv∈E(NCnm) (du + dv) 2 = (2.2)2 E1(NCn m) + (2.3)2 E2(NCn m) + (3.3)2 E3(NCn m) = 16(n) + 25(2n) + 81[n(3m − 5/2)] = 16n + 50n + 243mn − 405n/2 = 1 2 [243mn − 273n] Hence the proof. Theorem 10. Consider the nanotube NAn m for m, n ≥ 2 then the Harmonic index is H(G) = 1 3 [3mn + 2m] Let us consider the nanotube NAn m listed in the proof theorem 6 we obtain. H(G) = X e=uv∈E(G) 2 du + dv = 2 3 + 3 E1(NAn m) + 2 2 + 3 E2(NAn m) = 1 3 [m(3n − 2)] + 2 5 (4m) = mn − 2 3 m + 8 5 m = 15mn − 10m + 24m 15 H(NAn m) = 1 3 [3mn + 2m] Hence the proof. Theorem 11. Consider the nanotube NCn m for m, n ≥ 2 then the Harmonic index is H(G) = 1 30 [30mn + 14n] Proof: Let the nanotube NCn m has 2n vertices of degree 2 and n(2m − 1) vertices of 9 degree 3. For determining the Harmonic index of nanotube NCn m we apply the edge partitions described in the proof of the theorem 7. Then we have By definition of Harmonic index H(G) = X e=uv∈E(G) 2 du + dv = 2 2 + 2 E1(NCn m) + 2 2 + 3 E2(NCn m) + 2 3 + 3 E3(NCn m) = 1 2 (n) + 2 5 (2n) + 1 3 [n(3m − 5/2)] = n 2 + 4n 5 + mn − 5n 6 = 30mn + 15n + 24n − 25n 30 = 1 30 [30mn + 14n] Hence the proof. Theorem 12. Consider the nanotube NAn m for m, n ≥ 2. Then the Sum Connectivity index is SCI(G) = 1 √ 6 [3mn − 2m] + 1 √ 5 (4m) Proof: Let the nanotube NAn m has 2m vertices of degree 2 and 2mn vertices of degree 3. The edge set E(NAn m) divides into two edge partitions based on degrees of end vertices. The first edge partition E1(NAn m) contains m(3n − 2) edges, where deg(a) = deg(b) = 3. The second edge partition E2(NAn m) contains 4m edges, where deg(a) = 2 and deg(b) = 3. By definition of Sum Connectivity index, we have SCI(G) = X e=uv∈E(G) 1 √ du + dv 10 It follows that SCI(NAn m) = X e=uv∈E(NAnm) 1 √ du + dv = 1 √ 6 E1(NAn m) + 1 √ 5 E2(NAn m) = 1 √ 6 [m(3n − 2)] + 1 √ 5 (4m) = 1 √ 6 [3mn − 2m] + 1 √ 5 (4m) Hence the proof. Theorem 13. Consider the nanotube NCn m for m, n ≥ 2. Then the Sum Connectivity index is SCI(NCn m) = n 2 + 2n √ 5 + 1 √ 6 6mn − 5n 2 Proof: Let us consider the nanotube NCn m has 2n vertices of degree 2 and n(2m−1) vertices of degree 3. The edge set can be divided into three types of edges based on degrees of end vertices of each edge. The edge set E1(NCn m) contains n edges where deg(u) = deg(v) = 2. The edge set E2(NCn m) contains 2n edges where deg(u) = 2 and deg(v) = 3. The edge set E3(NCn m) contains n(3m−5/2) edges where deg(u) = deg(v) = 3. By definition of Sum Connectivity index SCI(G) = X e=uv∈E(G) 1 √ du + dv 11 It follows that SCI(G) = X e=uv∈E(NCnm) 1 √ du + dv = 1 2 E1(NCn m) + 1 √ 5 E2(NCn m) + 1 √ 6 E3(NCn m) = 1 2 (n) + 1 √ 5 (2n) + 1 √ 6 [n(3m − 5/2)] = n 2 + 2n √ 5 + 1 √ 6 6mn − 5n 2 Hence the proof. 4 Conclusion In this article, we determined the degreebased topological indices. We compute the First and Second Zagreb index, Hyper  Zagreb index, Harmonic index, and Sum Connectivity index for the nanotube NAn m and NCn m. References [1] J.A. Bondy, U.S.R. Murthy, Graph Theory with Applications, Macmillan Press, New York. 1976. [2] R. Todeschini, V.Consonni, Handbook of Molecular Descriptors, WileyVCH, Weinheim, 3, 2000. [3] Gutman, N. Trinajstic, Graph Theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons, Chem. Phys. Lett.17 (1972) 535538. [4] Gutman, I. Polansky, O.E. Mathematical concepts in organic chemistry; Springer Verrlag, New York, 1986. [5] Sumio Iijima. “Helical microtubules of graphic carbon”. Nature. (1991). 354 (6348): 5658. [6] A.T. Balaban, I.Motoc, D.Bonchev, O.Mekenyan, Topological indices for structureactivity corrections, Topics Cur. Chem 114 (1983) 2155. 12 [7] B. Zhou, Zagreb indices, MATCH Commun.Math.Comput. Chem.52 (2004) 113118. [8] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 233  239. [9] Nikolic, S., Kovacevic, G., Milicevic, A. & Trinajstic, N. The Zagreb indices 30 years after. Croatica chemica acta 76 (2), 113 124 (2003). [10] Gutman. I. & Das, K.C. The first Zagreb index 30 years after. MATCH Commun. Math.Comput. Chem 50 (1), 83  92 (2004). [11] Das, K.C. & Gutman, I. Some properties of the second Zagreb index. MATCH Commun. Math. Comput. Chem 52 (1), 3  1 (2004). [12] G.H.Shridal, H.Rezapour and A.M.Sayadi, The Hyper Zagreb index graph operaions, Iranian J.Mat.Chem., 4 (2013). 213220. [13] S.Fajtlowicz, On conjectures of GraffitiII. Congr. Numer, 60 (1987), pp.583591. [14] O.Favron, M.Maheo, J.F.Sacle some eigenvalues properties in graphs (Conjectures of GraffitiII). Discrete Math., 111(1993), pp.197220. [15] Xing.R, Zhou B, Trinajstic N.s SumConnectivity index of molecular trees. J Math Chem, 2010, 48: 583 591. [16] Wilder, J.W.G; Venema, L.C; Rinzler, A.G; Smalley, R.E; ekker, C. (1 January 1998). “Electronic structure of atomically resolve carbon nanotubes”. Nature. 391 (6662): 59 62. [17] S.B.Sinnott & R.Andreys (2001). “Carbon Nanotubes: Synhesis, properties an Applications”. Critical Reviews in solid state and Materials science. 26 (3): 145  249. [18] Can. J. Chem. (2015). “On Topological indices of carbon nanotube network”, 1  5. [19] Deza, M; Fowler, P.W.; Rassat,; Rogers, K.M.J. Chem. Inf. Comput. Sci. (2000). 40. 550 558. 13 
