On Multiplicative Harmonic Index, Multiplicative ISI Index and Multiplicative F Index of

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.

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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+d_v

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) = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣

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)} 𝒅_𝑮(𝒗) ^𝟑 𝟑 It is easy to see that

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

Hπ 𝐾𝑛 = ^𝟐

𝒅_𝒖+𝒅_𝒗

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

𝐼𝑆𝐼π 𝐺 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣

𝒖𝒗∈𝑬𝑮) 6

𝑭π 𝑮 = _{𝒖𝒗∈𝑬𝑮)} 𝒅_𝑮(𝒖)^𝟐+ 𝒅_𝑮(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 𝐾_𝑛 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 𝐾_𝑛 of order n is 𝑛 − 1 and the number of edges for 𝐾_𝑛 is equal to ¹

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

𝒊 𝐼𝑆𝐼𝜋 𝐾𝑛 = ^{𝑑𝑢𝑑𝑣}

𝑑_𝑢+𝑑_𝑣 𝒖𝒗∈𝑬𝐾_𝑛) = ^(𝑛−1)_2(𝑛−1)²

𝑛 (𝑛 −1)

2 = ^(𝑛−1)₂

𝑛 (𝑛 −1)

2

𝑖𝑖 𝐹𝜋 𝐾𝑛 = 𝒖𝒗∈𝑬(𝐾_𝑛) 𝒅𝑲_𝒏(𝒖)^𝟐+ 𝒅𝑲_𝒏(𝒗)^𝟐

= (𝑛 − 1)²+ (𝑛 − 1)^{2 𝑛 (𝑛 −1)2} = 2(𝑛 − 1) ^{𝑛(𝑛−1)}

Lemma 2.2:

Suppose 𝐶_𝑛 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 𝐶_𝑛 = 8 ^𝑛 (11) Proof. Here 𝑉(𝐶_𝑛) = 𝑛 = 𝐸(𝐶_𝑛) and all the vertices have the degrees 2.Hence 𝒊 𝐼𝑆𝐼𝜋 𝐶_𝑛 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝒖𝒗∈𝑬(𝐶_𝑛)

= ^2×2₂₊₂ ^𝑛 = 1

𝑖𝑖 𝐹𝜋 𝐶𝑛 = 𝒖𝒗∈𝑬(𝐶_𝑛) 𝒅𝐶_𝑛(𝒖)^𝟐+ 𝒅𝐶_𝑛(𝒗)^𝟐

= 2²+ 2^{2 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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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 𝑆_𝑛 is 𝑛 − 1 i.e., 𝐸(𝑆_𝑛) = 𝑛 − 1 . Then

𝒊 𝐼𝑆𝐼𝜋 𝑆_𝑛 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝒖𝒗∈𝑬(𝑆_𝑛)

= ^{1×(𝑛−1)}_1+(𝑛−1)^𝑛−1= ^(𝑛−1)_𝑛 ^𝑛−1

𝑖𝑖 𝐹𝜋 𝑆𝑛 = 𝒖𝒗∈𝑬(𝑆_𝑛) 𝒅𝑆_𝑛(𝒖)^𝟐+ 𝒅𝑆_𝑛(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 𝑊_𝑛 is given by 3, (𝑛 − 1) and the remaining (n − 1) edges has the degree of end vertices as 3,3 . Then

𝒊 𝐼𝑆𝐼𝜋 𝑊_𝑛 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝒖𝒗∈𝑬(𝑊_𝑛)

= ^{3× 𝑛−1} _{3+ 𝑛−1} ^𝑛−1 × ^3×3₃₊₃ ^𝑛−1 = ^{3 𝑛−1} _𝑛+2 ^{𝑛−1 3}₂ ^𝑛−1 = ^9(𝑛−1)_{2 𝑛+2} ^𝑛−1

𝑖𝑖 𝐹𝜋 𝑊𝑛 = 𝑢𝑣 ∈𝐸(𝑊_𝑛) 𝑑𝑊_𝑛(𝑢)²+ 𝑑𝑊_𝑛(𝑣)²

= 3²+ 𝑛 − 1 ^{2 𝑛−1} × 3²+ 3^{2 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 𝐹_𝑛, 𝑛 ≥ 2 , 𝑉( 𝐹_𝑛) = 𝑛 + 1 𝑎𝑛𝑑 𝐸( 𝐹_𝑛) = 2𝑛 − 1. Also in 𝐹_𝑛 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 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸(𝐹_𝑛)

= ^2×3₂₊₃²× ^3×𝑛_3+𝑛 ^𝑛−2× ^2×𝑛_2+𝑛 ²× ^3×3₃₊₃ ^𝑛−3

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4 =³⁶_{25 3+𝑛}^{3𝑛 𝑛−2} _2+𝑛^{2𝑛 2 3}₂ ^𝑛−3

𝑖𝑖 𝐹𝜋 𝐹_𝑛 = _{𝑢𝑣 ∈𝐸(𝐹𝑛)} 𝑑_𝐹𝑛(𝑢)²+ 𝑑_𝐹𝑛(𝑣)²

= 2²+ 3^{2 2}× 3²+ 𝑛^{2 𝑛−2}× 2²+ 𝑛^{2 2}× 3²+ 3^{2 𝑛−3}

= 169 18 ^𝑛−3 𝑛²+ 9 ^𝑛−2 𝑛²+ 4 ²

Lemma 2.6:

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

𝐼𝑆𝐼𝜋 𝐾_{𝑚 ,𝑛} = _{𝑚 +𝑛}^{𝑚𝑛 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 𝐾_{𝑚 ,𝑛} is given by 𝑛, 𝑚 . Then

𝑖 𝐼𝑆𝐼𝜋 𝐾_{𝑚 ,𝑛} = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸(𝐾_{𝑚 ,𝑛})

= ^𝑚×𝑛_𝑚+𝑛 ^𝑚𝑛 = _{𝑚 +𝑛}^{𝑚𝑛 𝑚𝑛}

𝑖𝑖 𝐹𝜋 𝐾𝑚 ,𝑛 = 𝑢𝑣 ∈𝐸(𝐾_{𝑚 ,𝑛}) 𝑑𝐾_{𝑚 ,𝑛}(𝑢)²+ 𝑑𝐾_{𝑚 ,𝑛}(𝑣)²

= 𝑚²+ 𝑛^{2 𝑚𝑛}

Lemma 2.7:

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

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

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

Fπ 𝑃𝑛 = 25 . 8 ^𝑛−3 (21)

Proof : A path graph 𝑃_𝑛 has vertices 𝑣₁, 𝑣₂, 𝑣₃, … 𝑣_𝑛 𝑎𝑛𝑑 𝑒𝑑𝑔𝑒𝑠 𝑒₁, 𝑒₂, 𝑒₂, … 𝑒_𝑛−1, such that edge 𝑒_𝑘 joins vertices 𝑣_𝑘 𝑎𝑛𝑑𝑣_𝑘+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 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸 (𝑃_𝑛)

= ^2×1₂₊₁ ^{2 2×2}₂₊₂ ^𝑛−3= ²₃ ²=⁴₉

𝑖𝑖 𝐹𝜋 𝑃𝑛 = 𝑢𝑣 ∈𝐸 (𝑃_𝑛) 𝑑 𝑃_𝑛(𝑢)²+ 𝑑 𝑃_𝑛(𝑣)² = 2²+ 1^{2 2} 2²+ 2^{2 𝑛−3}= 25 . 8 ^𝑛−3

Lemma 2.8:

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

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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 = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸(𝐺) = ^𝑟×𝑟_𝑟+𝑟

𝑛𝑟 2 = ^𝑟₂

𝑛𝑟

2

𝑖𝑖 𝐹𝜋 𝐺 = _{𝑢𝑣 ∈(𝐸𝐺)} 𝑑_𝐺(𝑢)²+ 𝑑_𝐺(𝑣)²

= 𝑟²+ 𝑟^{2 𝑛𝑟2} = 2𝑟^{2 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 = ₆₂₅^{4 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 )| 𝑑_𝑣= 2} → |𝑉2| = 2𝑚 + 2𝑚 𝑉3= {{𝑣 ∈ 𝑉(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑_𝑣= 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 )| 𝑑_𝑢= 𝑑𝑣 = 2} → | 𝐸4| = | 𝐸4∗| = ½|𝑉2| = 2𝑚 𝐸{2,3} = {𝑒 = 𝑢𝑣𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 )| 𝑑_𝑢= 3 & 𝑑𝑣= 2} → | 𝐸5| = | 𝐸6∗| = |𝑉2| = 4𝑚

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

Where 𝐸₄= 𝐸₂₊₂ , 𝐸₄^∗= 𝐸_2×2 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

d_u+d_v uv ∈E(TU C₄C₈ m,n ) = _d ²

u+d_v

uv ∈E₄ × _d ²

u+d_v

uv ∈E₅ × _d ²

u+d_v uv ∈E₆ = ₂₊₂^{2 2m}× ₂₊₃^{2 4m}× ₃₊₃^{2 12mn −2m} = ¹₂ ^2m× ²₅ ^4m× ¹₃ ^{12mn −2m}

= 4 ^−m × ₆₂₅^{16 m}× ¹₉ ^{6mn −m} = ₆₂₅^{4 𝑚}× ¹₉ ^{6𝑚𝑛 −𝑚}

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = ^{𝑑𝑢𝑑𝑣}

𝑑_𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶₄𝐶₈ 𝑚 ,𝑛 ) = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣

𝑢𝑣 ∈𝐸₄=𝐸₄^∗ × _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣

𝑢𝑣 ∈𝐸₅=𝐸₆^∗ × _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 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 ) 𝑑_𝐺(𝑢)²+ 𝑑𝐺(𝑣)²

= 𝑢𝑣∈𝐸₄ 𝑑_𝐺(𝑢)²+ 𝑑𝐺(𝑣)² × 𝑢𝑣 ∈𝐸₅ 𝑑_𝐺(𝑢)²+ 𝑑𝐺(𝑣)² × 𝑢𝑣 ∈𝐸₆ 𝑑_𝐺(𝑢)²+ 𝑑𝐺(𝑣)² = 2²+ 2^{2 2𝑚}× 2²+ 3^{2 4𝑚}× 3²+ 3^{212𝑚𝑛 −2𝑚}

= 8 ^2𝑚× 13 ^4𝑚× 18 ^{12𝑚𝑛 −2𝑚}

Volume: 4 Issue: 9 01 – 08

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3.2. MULTIPLICATIVE TOPOLOGICAL INDICES FOR 𝑻𝑼𝑪_𝟒𝑪_𝟖 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× ₁₆^{1 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 │𝑑_𝑢 = 2 & 𝑑_𝑣= 4 → 𝐸₆ = 𝐸₈^∗ = 2𝑚 + 2𝑚 = 2|𝑉₂|

𝐸{4,4} = 𝐸8= 𝐸16∗= 𝑒 = 𝑢𝑣 ∈ 𝐸 𝑇𝑈𝐶4 𝑚, 𝑛 𝑑_𝑢 = 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

d_u+d_v uv ∈E(TU C₄ m,n ) = _d ²

u+d_v

uv ∈E₆ × _d ²

u+d_v uv ∈E₈

= ₂₊₄^{2 4m}× ₄₊₄^{2 4mn −2m}= ¹₃ ^4m× ¹₄ ^{4mn −2m} = 81 ^−m× ₁₆^{1 2mn −m}

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4 𝑚, 𝑛 = ^{𝑑𝑢𝑑𝑣}

𝑑_𝑢+𝑑_𝑣 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶₄ 𝑚 ,𝑛 ) = _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣

𝑢𝑣 ∈𝐸₆=𝐸₈^∗ × _𝑑^{𝑑𝑢𝑑𝑣}

𝑢+𝑑_𝑣 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 ) 𝑑_𝐺(𝑢)²+ 𝑑𝐺(𝑣)²

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

= 20 ^4𝑚 × 32 ^{4mn −2m}

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