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Volume: 4 Issue: 9 01 – 08

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On Multiplicative Harmonic Index, Multiplicative ISI Index and Multiplicative F Index of 𝑇𝑈𝐶 4 𝐶 8 𝑚, 𝑛 𝑎𝑛𝑑 𝑇𝑈𝐶 4 𝑚, 𝑛 Nanotubes

M. Bhanumathi 1, K . Easu Julia Rani 2.

Associate Professor, PG and Research of Mathematics, Govt. arts college for Women Autonomous , Pudukottai − 622001, India .

Assistant Professor of Mathematics, TRP Engineering college, Tiruchirappalli − 621105, India . 𝑏ℎ𝑎𝑛𝑢_𝑘𝑠𝑝@𝑦𝑎ℎ𝑜𝑜. 𝑐𝑜𝑚, 𝑗𝑢𝑙𝑖𝑎𝑟𝑎𝑛𝑖16@𝑔𝑚𝑎𝑖𝑙. 𝑐𝑜𝑚

Abstract: Chemical graph theory is a branch of graph theory whose focus of interest is to finding topological indices of chemical graphs, which correlate well with chemical properties of the chemical molecules. In this paper, we compute the Multiplicative F index, Multiplicative ISI index and Multiplicative Harmonic index for TUC4C8 m, n and TUC4 m, n Nanotubes.

Mathematics Subject Classification: 05C05, 05C07

Keywords: Multiplicative F index, Multiplicative ISI index and Multiplicative Harmonic index, and Nanotubes.

__________________________________________________*****_________________________________________________

I. INTRODUCTION:

Let 𝐺 = (𝐸, 𝑉) be a simple connected graph with the vertex set 𝑉(𝐺) and the edge set 𝐸(𝐺). In chemical graph theory, the vertices of molecular graph G correspond to the atoms and the edges correspond to the bonds. The degree 𝑑𝐺 𝑣 𝑜𝑟 𝑑𝑣 of a vertex v is the number of verticesadjacent

to v 1 .There exits many topological indices in mathematical chemistry. Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. Chemical graph theory is a branch of mathematical chemistry which applies graph.

The Harmonic index 𝐻(𝐺) is vertex-degree-based topological index. This index first appeared in [2], and was defined as

𝐻 𝐺 = d 2

u+dv

𝒖𝒗∈𝑬𝑮) 1

The inverse sum indeg index, is the descriptor that was selected in [3] as a significant predictor of total surface area of octane isomers and for which the extremal graphs obtained with the help of Math Chem have a particularly simple and elegant structure.

The inverse sum indeg index is defined as 𝐼𝑆𝐼 𝐺 = 11

𝑑𝑢+1 𝑑𝑣

𝒖𝒗∈𝑬𝑮) = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣

𝒖𝒗∈𝑬𝑮) 2

Recently a degree based topological index was introduced by Furtula et al. in [4]. They named this index as forgotten topological index or F-index. This index is defined as

𝑭 𝑮 = 𝒗∈𝑽(𝑮) 𝒅𝑮(𝒗) 𝟑 𝟑 It is easy to see that

𝐹 𝐺 = 𝑢𝑣 ∈𝐸𝐺) 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 4 we already introduced the multiplicative Harmonic index 10 as

Hπ 𝐾𝑛 = 𝟐

𝒅𝒖+𝒅𝒗

𝒖𝒗∈𝑬𝑮) 𝟓 Now we introduce the multiplicative ISI index and the multiplicative F index as follows:

𝐼𝑆𝐼π 𝐺 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣

𝒖𝒗∈𝑬𝑮) 6

𝑭π 𝑮 = 𝒖𝒗∈𝑬𝑮) 𝒅𝑮(𝒖)𝟐+ 𝒅𝑮(𝒗)𝟐 𝟕

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2

II. MULTIPLICATIVE TOPOLOGICAL INDICES OF SOME SPECIAL GRAPHS

In [10] we calculated the multiplicative Harmonic index (Hπ 𝐺 ) of some special graphs. In this paper we calculate the multiplicative Inverese Sum indeg (𝐼𝑆𝐼π 𝐺 ) index, multiplicative F index(Fπ 𝐺 ) of some special graphs .

Lemma 2.1:

Consider the complete graph 𝐾𝑛 of order n.

(i) The multiplicative Inverese Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index 𝐼𝑆𝐼𝜋 𝐾𝑛 = (𝑛−1)2

𝑛 (𝑛 −1)

2 (8) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝐾𝑛 = 2(𝑛 − 1) 𝑛(𝑛−1) (9)

Proof. The degree of all the vertices of a complete graph 𝐾𝑛 of order n is 𝑛 − 1 and the number of edges for 𝐾𝑛 is equal to 1

2 𝑛(𝑛 − 1) i,e., 𝐸(𝐾𝑛) =12 𝑛 𝑛 − 1 . Then

𝒊 𝐼𝑆𝐼𝜋 𝐾𝑛 = 𝑑𝑢𝑑𝑣

𝑑𝑢+𝑑𝑣 𝒖𝒗∈𝑬𝐾𝑛) = (𝑛−1)2(𝑛−1)2

𝑛 (𝑛 −1)

2 = (𝑛−1)2

𝑛 (𝑛 −1)

2

𝑖𝑖 𝐹𝜋 𝐾𝑛 = 𝒖𝒗∈𝑬(𝐾𝑛) 𝒅𝑲𝒏(𝒖)𝟐+ 𝒅𝑲𝒏(𝒗)𝟐

= (𝑛 − 1)2+ (𝑛 − 1)2 𝑛 (𝑛 −1)2 = 2(𝑛 − 1) 𝑛(𝑛−1)

Lemma 2.2:

Suppose 𝐶𝑛 is a cycle of length n labelled by 1,2, … , 𝑛 . Then (i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index

𝐼𝑆𝐼𝜋 𝐶𝑛 = 1 (10) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝐶𝑛 = 8 𝑛 (11) Proof. Here 𝑉(𝐶𝑛) = 𝑛 = 𝐸(𝐶𝑛) and all the vertices have the degrees 2.Hence 𝒊 𝐼𝑆𝐼𝜋 𝐶𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝒖𝒗∈𝑬(𝐶𝑛)

= 2×22+2 𝑛 = 1

𝑖𝑖 𝐹𝜋 𝐶𝑛 = 𝒖𝒗∈𝑬(𝐶𝑛) 𝒅𝐶𝑛(𝒖)𝟐+ 𝒅𝐶𝑛(𝒗)𝟐

= 22+ 22 𝑛= 8 𝑛

Lemma 2.3:

Suppose 𝑆𝑛 is the Star graph on n vertices,

(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index 𝐼𝑆𝐼𝜋 𝑆𝑛 = (𝑛−1)

𝑛 𝑛−1

(12) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝑆𝑛 = 𝑛2− 2𝑛 + 2 𝑛−1 (13)

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Volume: 4 Issue: 9 01 – 08

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Proof :

In a star 𝑆𝑛 all the leaves are of degree 1 and the degree of internal node is 𝑛 − 1.Also the number of edges for 𝑆𝑛 is 𝑛 − 1 i.e., 𝐸(𝑆𝑛) = 𝑛 − 1 . Then

𝒊 𝐼𝑆𝐼𝜋 𝑆𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝒖𝒗∈𝑬(𝑆𝑛)

= 1×(𝑛−1)1+(𝑛−1)𝑛−1= (𝑛−1)𝑛 𝑛−1

𝑖𝑖 𝐹𝜋 𝑆𝑛 = 𝒖𝒗∈𝑬(𝑆𝑛) 𝒅𝑆𝑛(𝒖)𝟐+ 𝒅𝑆𝑛(𝒗)𝟐 = 12+ (𝑛 − 1)2 𝑛−1= 𝑛2− 2𝑛 + 2 𝑛−1

Lemma 2.4:

If 𝑊𝑛, 𝑛 ≥ 5 is the wheel graph on n vertices, then

(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index 𝐼𝑆𝐼𝜋 𝑊𝑛 = 9(𝑛−1)

2 𝑛+2 𝑛−1

(14) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝑊𝑛 = 18 𝑛2− 2𝑛 + 10 𝑛−1 (15) Proof : The wheel graph 𝑊𝑛 may denote the n-vertex graph with an (n − 1)-cycle on its rim. For a wheel graph 𝑊𝑛, 𝑛 ≥

5 , it has 𝑉( 𝑊𝑛) = 𝑛 𝑎𝑛𝑑 𝐸( 𝑊𝑛) = 2 𝑛 − 1 .

Also the degree of end vertices of (n − 1) edges of a wheel graph 𝑊𝑛 is given by 3, (𝑛 − 1) and the remaining (n − 1) edges has the degree of end vertices as 3,3 . Then

𝒊 𝐼𝑆𝐼𝜋 𝑊𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝒖𝒗∈𝑬(𝑊𝑛)

= 3× 𝑛−1 3+ 𝑛−1 𝑛−1 × 3×33+3 𝑛−1 = 3 𝑛−1 𝑛+2 𝑛−1 32 𝑛−1 = 9(𝑛−1)2 𝑛+2 𝑛−1

𝑖𝑖 𝐹𝜋 𝑊𝑛 = 𝑢𝑣 ∈𝐸(𝑊𝑛) 𝑑𝑊𝑛(𝑢)2+ 𝑑𝑊𝑛(𝑣)2

= 32+ 𝑛 − 1 2 𝑛−1 × 32+ 32 𝑛−1 = 18 𝑛2− 2𝑛 + 10 𝑛−1

Lemma 2.5:

If 𝐹𝑛, 𝑛 ≥ 2 is the fan graph on n vertices, then

(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index 𝐼𝑆𝐼𝜋 𝐹𝑛 =36

25 3𝑛 3+𝑛

𝑛−2 2𝑛 2+𝑛

2 3 2

𝑛−3

(16) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝐹𝑛 = 169 18 𝑛−3 𝑛2+ 9 𝑛−2 𝑛2+ 4 2 (17) Proof For a fan graph 𝐹𝑛, 𝑛 ≥ 2 , 𝑉( 𝐹𝑛) = 𝑛 + 1 𝑎𝑛𝑑 𝐸( 𝐹𝑛) = 2𝑛 − 1. Also in 𝐹𝑛 the degrees of end vertices of

𝑛 − 2 edges are 3, 𝑛 , 2 edges have the degrees of the end vertices as 3,2 , 2 edges have the degrees of the end vertices as 𝑛, 2 and the remaining 𝑛 − 3 edges have the degrees of the end vertices as 3,3 . Then

𝑖 𝐼𝑆𝐼𝜋 𝐹𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸(𝐹𝑛)

= 2×32+32× 3×𝑛3+𝑛 𝑛−2× 2×𝑛2+𝑛 2× 3×33+3 𝑛−3

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4 =3625 3+𝑛3𝑛 𝑛−2 2+𝑛2𝑛 2 32 𝑛−3

𝑖𝑖 𝐹𝜋 𝐹𝑛 = 𝑢𝑣 ∈𝐸(𝐹𝑛) 𝑑𝐹𝑛(𝑢)2+ 𝑑𝐹𝑛(𝑣)2

= 22+ 32 2× 32+ 𝑛2 𝑛−2× 22+ 𝑛2 2× 32+ 32 𝑛−3

= 169 18 𝑛−3 𝑛2+ 9 𝑛−2 𝑛2+ 4 2

Lemma 2.6:

If 𝐾𝑚 ,𝑛 , 𝑛, 𝑚 ≥ 2 is the Complete bipartite graph on n vertices, then (i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index

𝐼𝑆𝐼𝜋 𝐾𝑚 ,𝑛 = 𝑚 +𝑛𝑚𝑛 𝑚𝑛 (18) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝐾𝑚 ,𝑛 = 𝑛2+ 4 2 𝑛2+ 9 2𝑛−3 (19)

Proof :For a Complete bipartite graph 𝐾𝑚 ,𝑛 , 𝑛, 𝑚 ≥ 2 , it has 𝑉(𝐾𝑚 ,𝑛) = 𝑚 + 𝑛 𝑎𝑛𝑑 𝐸( 𝐾𝑚 ,𝑛) = 𝑚𝑛. Also the degrees of end vertices of all the edges of 𝐾𝑚 ,𝑛 is given by 𝑛, 𝑚 . Then

𝑖 𝐼𝑆𝐼𝜋 𝐾𝑚 ,𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸(𝐾𝑚 ,𝑛)

= 𝑚×𝑛𝑚+𝑛 𝑚𝑛 = 𝑚 +𝑛𝑚𝑛 𝑚𝑛

𝑖𝑖 𝐹𝜋 𝐾𝑚 ,𝑛 = 𝑢𝑣 ∈𝐸(𝐾𝑚 ,𝑛) 𝑑𝐾𝑚 ,𝑛(𝑢)2+ 𝑑𝐾𝑚 ,𝑛(𝑣)2

= 𝑚2+ 𝑛2 𝑚𝑛

Lemma 2.7:

If 𝑃𝑛, 𝑛 ≥ 2 is the Path on n vertices, then

(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index

𝐼𝑆𝐼𝜋 𝑃𝑛 =49 (20) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝑃𝑛 = 25 . 8 𝑛−3 (21)

Proof : A path graph 𝑃𝑛 has vertices 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 𝑎𝑛𝑑 𝑒𝑑𝑔𝑒𝑠 𝑒1, 𝑒2, 𝑒2, … 𝑒𝑛−1, such that edge 𝑒𝑘 joins vertices 𝑣𝑘 𝑎𝑛𝑑𝑣𝑘+1 .

For a Path graph 𝑃𝑛, 𝑛 ≥ 2 , it has 𝑉( 𝑃𝑛) = 𝑛 𝑎𝑛𝑑 𝐸( 𝑃𝑛) = 𝑛 − 1.

Also in 𝑃𝑛 the degrees of end vertices of 2 edges are 2,1 and the remaining 𝑛 − 3 edges have the degrees of the end vertices as 2,2 . Then

𝑖 𝐼𝑆𝐼𝜋 𝑃𝑛 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸 (𝑃𝑛)

= 2×12+1 2 2×22+2 𝑛−3= 23 2=49

𝑖𝑖 𝐹𝜋 𝑃𝑛 = 𝑢𝑣 ∈𝐸 (𝑃𝑛) 𝑑 𝑃𝑛(𝑢)2+ 𝑑 𝑃𝑛(𝑣)2 = 22+ 12 2 22+ 22 𝑛−3= 25 . 8 𝑛−3

Lemma 2.8:

If G is a regular graph of degree 𝑟 > 0, then

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Volume: 4 Issue: 9 01 – 08

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(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index 𝐼𝑆𝐼𝜋 𝐺 = 𝑟2

𝑛𝑟

2 (22) (ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝐺 = 𝑛2+ 4 2 𝑛2+ 9 2𝑛−3 (23)

Proof. A regular graph G on n vertices, having degree r, possesses nr

2 edges.Thus 𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸(𝐺) = 𝑟×𝑟𝑟+𝑟

𝑛𝑟 2 = 𝑟2

𝑛𝑟

2

𝑖𝑖 𝐹𝜋 𝐺 = 𝑢𝑣 ∈(𝐸𝐺) 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2

= 𝑟2+ 𝑟2 𝑛𝑟2 = 2𝑟2 𝑛𝑟2

III. NANOTUBES

Carbon nanotubes are one of the most commonly mentioned building blocks of nanotechnology. With one hundred times the tensile strength of steel, thermal conductivity better than all but the purest diamond, and electrical conductivity similar to copper, but with the ability to carry much higher currents, they seem to be a wonder material. Carbon nanotubes, long thin cylinders of carbon, were discovered in 1991 by Iijima’s. Carbon nanotubes (CNTs) are allotropes of carbon which are members of the fullerene structural family, which also includes the spherical bucky balls. These are large macromolecules which are unique for there size, shape and remarkable physical properties.In this paper we deal with two categories of Nanotubes namely 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 and 𝑇𝑈𝐶4 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ .

3.1 MULTIPLICATIVE TOPOLOGICAL INDICES FOR 𝑻𝑼𝑪𝟒𝑪𝟖 𝒎, 𝒏 NANOTUBES

Consider the Carbon nanotube TUC4C8 m, n as shown in the following Fig.1.

Fig.1. -Dimensional and 2-Dimensional lattices of the TUSC4C8(S) Nanotubes.

M.V. Diudea denoted 𝑇𝑈𝐶4𝐶8 the number of Octagons C8 in the first row of G by m and the number of Octagons 𝐶8 in the first column of G by n, and he denoted 𝑇𝑈𝐶4𝐶8 Nanotubes by 𝐺 = 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 ∀𝑚, 𝑛 ∈ ℕ.One can see the 3-Dimensional and 2- Dimensional lattices of 𝐺 = 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 in Figure 1 and for historical background see references [5-9]. ∀𝑚, 𝑛 ∈ ℕ.

We now compute the multiplicative Harmonic, ISI and F indices of 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 Nanotubes ∀𝑚, 𝑛 ∈ ℕ, with 2𝑚𝑛 + 2𝑚.

vertices/atoms and 4𝑚𝑛 + 2𝑚 edges\bonds

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6 Theorem 3.1: Let G = TUC4C8 m, n nanotubes. Then

(i) Hπ G = 6254 𝑚× 19 6𝑚𝑛 −𝑚 (24) (ii) 𝐼𝑆𝐼𝜋 𝐺 = 1296625 𝑚× 94 6𝑚𝑛 −𝑚 (25) (iii) 𝐹𝜋 𝐺 = 8 2𝑚× 13 4𝑚× 18 12𝑚𝑛 −2𝑚 (26)

Proof:

Consider the 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 Nanotubes with 8𝑚𝑛 + 4𝑚 vertices/atoms and 12𝑚𝑛 + 4𝑚 edges\bonds. By according to Figure 1, one can see that the degree of a vertex/atom of all Nanotubes is equal to 1 or 2 or 3 and there are two partitions of vertex/atom set 𝑉 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 are equal to

𝑉2= {𝑣 ∈ 𝑉(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑𝑣= 2} → |𝑉2| = 2𝑚 + 2𝑚 𝑉3= {{𝑣 ∈ 𝑉(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑𝑣= 3} → |𝑉3| = 8𝑚𝑛

Also, there are |𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| = ½(2(4𝑚) + 4(8𝑚𝑛)) = 12𝑚𝑛 + 4𝑚 edges\bonds in this Nanotubes. From the structure of 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 in Fig 1, we see that there are three partitions of edge\bond set 𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )with their size are as follows:

𝐸{2,2} = {𝑒 = 𝑢𝑣𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑𝑢= 𝑑𝑣 = 2} → | 𝐸4| = | 𝐸4∗| = ½|𝑉2| = 2𝑚 𝐸{2,3} = {𝑒 = 𝑢𝑣𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑𝑢= 3 & 𝑑𝑣= 2} → | 𝐸5| = | 𝐸6∗| = |𝑉2| = 4𝑚

𝐸{3,3} = {𝑒 = 𝑢𝑣𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 ])| 𝑑𝑢 = 𝑑𝑣= 3} → | 𝐸6| = | 𝐸9∗| = 12𝑚𝑛 − 2𝑚

Where 𝐸4= 𝐸2+2 , 𝐸4∗= 𝐸2×2 and so on . In Fig.1, we marked all members of these edges partitions of 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 (𝐸{2,2}, 𝐸{2,3} 𝑎𝑛𝑑 𝐸{3,3} ) by yellow, red and black colors, respectively. Hence

i Hπ G = Hπ TUC4C8 m, n = 2

du+dv uv ∈E(TU C4C8 m,n ) = d 2

u+dv

uv ∈E4 × d 2

u+dv

uv ∈E5 × d 2

u+dv uv ∈E6 = 2+22 2m× 2+32 4m× 3+32 12mn −2m = 12 2m× 25 4m× 13 12mn −2m

= 4 −m × 62516 m× 19 6mn −m = 6254 𝑚× 19 6𝑚𝑛 −𝑚

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = 𝑑𝑢𝑑𝑣

𝑑𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶4𝐶8 𝑚 ,𝑛 ) = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣

𝑢𝑣 ∈𝐸4=𝐸4∗ × 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣

𝑢𝑣 ∈𝐸5=𝐸6∗ × 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸6=𝐸9∗ = 2×22+2 2𝑚× 2×32+3 4𝑚× 3×33+3 12𝑚𝑛 −2𝑚

= 65 4𝑚× 32 12𝑚𝑛 −2𝑚 = 65 4𝑚× 32 12𝑚𝑛 −2𝑚 = 1296625 𝑚× 94 6𝑚𝑛 −𝑚

(𝑖𝑖𝑖)𝐹𝜋 𝐺 = 𝐹𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶4𝐶8 𝑚 ,𝑛 ) 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2

= 𝑢𝑣∈𝐸4 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 × 𝑢𝑣 ∈𝐸5 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 × 𝑢𝑣 ∈𝐸6 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 = 22+ 22 2𝑚× 22+ 32 4𝑚× 32+ 3212𝑚𝑛 −2𝑚

= 8 2𝑚× 13 4𝑚× 18 12𝑚𝑛 −2𝑚

(7)

Volume: 4 Issue: 9 01 – 08

_______________________________________________________________________________________________

3.2. MULTIPLICATIVE TOPOLOGICAL INDICES FOR 𝑻𝑼𝑪𝟒𝑪𝟖 𝒎, 𝒏 NANOTUBES

Consider the Carbon nanotube 𝑇𝑈𝐶4 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ as shown in the following Fig 2.

Fig 2. 2-D and 3-D Lattice of the 𝑇𝑈𝐶4 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ Nanotubes Theorem 3.1: Let 𝐺 = 𝑇𝑈𝐶4 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ nanotubes. Then

(i) Hπ G = 81 −m× 161 2mn −m (27) (ii) 𝐼𝑆𝐼𝜋 𝐺 = 1296625 𝑚× 94 6𝑚𝑛 −𝑚 (28)

(iii) 𝐹𝜋 𝐺 = 20 4𝑚× 32 4mn −2m (29) Proof:

Consider the 𝐺 = 𝑇𝑈𝐶4 𝑚, 𝑛 Nanotubes has the number of vertices 𝑣𝐺= 𝑉 𝑇𝑈𝐶4 𝑚, 𝑛

= 2𝑚(𝑛 + 1).That is according to Fig. 2, we see Such that there exist 2m numbers of vertices with degree 2 and 2mn numbers of vertices with degree 4.

In other words, we have two partitions of 𝑉(𝑇𝑈𝐶4[𝑚, 𝑛]) as

𝑉2= {𝑣 ∈ 𝑉(𝐺)| 𝑑𝑣= 2} → |𝑉2| = 2𝑚 𝑉4= {𝑣 ∈ 𝑉(𝐺)| 𝑑𝑣= 4} → |𝑉4| = 2𝑚𝑛 This implies that the number of edges𝑒𝐺 = 4𝑚𝑛 + 2𝑚.

By according to Fig.2, one can see that in general form of this Nanotube, we have two partitions of E(TUC4[m,n]) as 𝐸{2,4} = 𝐸6= 𝐸8∗= 𝑒 = 𝑢𝑣 ∈ 𝐸 𝑇𝑈𝐶4 𝑚, 𝑛 │𝑑𝑢 = 2 & 𝑑𝑣= 4 → 𝐸6 = 𝐸8∗ = 2𝑚 + 2𝑚 = 2|𝑉2|

𝐸{4,4} = 𝐸8= 𝐸16∗= 𝑒 = 𝑢𝑣 ∈ 𝐸 𝑇𝑈𝐶4 𝑚, 𝑛 𝑑𝑢 = 𝑑𝑣= 4 → 𝐸8 = 𝐸16∗ = 4𝑚𝑛 − 2𝑚 Hence 𝐺 = 𝑇𝑈𝐶4 𝑚, 𝑛 the Nanotubes, (∀𝑚, 𝑛 ∈ ℕ), with 2𝑚𝑛 + 2𝑚. vertices/atoms and 4𝑚𝑛 + 2𝑚 edges\bonds.

In Fig.2, we marked all members of these edges partitions of 𝑇𝑈𝐶4 𝑚, 𝑛 (𝐸{2,4}, 𝑎𝑛𝑑 𝐸{4,4} ) by red and black colors, respectively. Hence

i Hπ G = Hπ TUC4 m, n = 2

du+dv uv ∈E(TU C4 m,n ) = d 2

u+dv

uv ∈E6 × d 2

u+dv uv ∈E8

= 2+42 4m× 4+42 4mn −2m= 13 4m× 14 4mn −2m = 81 −m× 161 2mn −m

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4 𝑚, 𝑛 = 𝑑𝑢𝑑𝑣

𝑑𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶4 𝑚 ,𝑛 ) = 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣

𝑢𝑣 ∈𝐸6=𝐸8∗ × 𝑑𝑑𝑢𝑑𝑣

𝑢+𝑑𝑣 𝑢𝑣 ∈𝐸8=𝐸16∗ = 2×42+4 4𝑚× 4×44+4 4mn −2m

= 43 4𝑚× 2 4mn −2m = 43 4𝑚× 2 4mn −2m (𝑖𝑖𝑖)𝐹𝜋 𝐺 = 𝐹𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶4 𝑚 ,𝑛 ) 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2

= 𝑢𝑣 ∈𝐸4 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 × 𝑢𝑣 ∈𝐸5 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 × 𝑢𝑣 ∈𝐸6 𝑑𝐺(𝑢)2+ 𝑑𝐺(𝑣)2 = 22+ 42 4𝑚× 42+ 42 4mn −2m

= 20 4𝑚 × 32 4mn −2m

(8)

_______________________________________________________________________________________________

8 REFERENCES

[1] V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012).

[2] S. Fajtlowicz, On conjectures of Graffiti. II, Congr. Numer, 60 (1987) 189–197

[3] D. Vukiˇcevi´c, M. Gaˇsperov, Bond Additive Modelling 1. Adriatic Indices, Croat. Chem. Acta 83 (2010), 243–260 [4] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184-1190.

[5] M. Alaeiyan, A. Bahrami and M.R. Farahani. Cyclically Domination Polynomial of Molecular Graph of Some Nanotubes. Digest Journal of Nanomaterials and Biostructures. 6(1), 143-147 (2011).

[6] M. Arezoomand. Energy and Laplacian Spectrum of C4C8(S) Nanotori and Nanotube. Digest. J. Nanomater. Bios. 4(6), (2009), 899- 905.

[7] J. Asadpour, R. Mojarad and L. Safikhani. Computing some topological indices of nano structure. Digest. J. Nanomater. Bios. 6(3), (2011), 937-941.

[8] A.R. Ashrafi and S. Yousefi. An Algebraic Method Computing Szeged index of TUC4C8(R) Nanotori. Digest. J. Nanomater. Bios.

4(3), (2009), 407-410.

[9] A.R. Ashrafi, M. Faghani and S. M. SeyedAliakbar. Some Upper Bounds for the Energy of TUC4C8(S) Nanotori. Digest. J.

Nanomater. Bios. 4(1), (2009), 59-64.

[10] M.Bhanumathi, K.Easu Julia Rani, On multilplicative Sum connectivity index , multilplicative Randic index and multilplicative Harmonic index of some Nanostar Dendrimers, submitted.

References

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