**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 𝑑*𝐺 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 = 1^{1}

𝑑𝑢+^{1}
𝑑𝑣

𝒖𝒗∈𝑬𝑮) = _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

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 𝐾**𝑛 = ^{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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}

𝒖𝒗∈𝑬𝑮) 6

** 𝑭π 𝑮 = **_{𝒖𝒗∈𝑬𝑮)} 𝒅_{𝑮}(𝒖)^{𝟐}+ 𝒅_{𝑮}(𝒗)^{𝟐} ** 𝟕 **

**_______________________________________________________________________________________________**

**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) =^{1}_{2} 𝑛 𝑛 − 1 . Then

𝒊 𝐼𝑆𝐼𝜋 𝐾𝑛 = ^{𝑑}^{𝑢}^{𝑑}^{𝑣}

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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝒖𝒗∈𝑬(𝐶_{𝑛})

= ^{2×2}_{2+2} ^{𝑛} = 1

𝑖𝑖 𝐹𝜋 𝐶𝑛 = 𝒖𝒗∈𝑬(𝐶_{𝑛}) 𝒅𝐶_{𝑛}(𝒖)^{𝟐}+ 𝒅𝐶_{𝑛}(𝒗)^{𝟐} ** **

= 2^{2}+ 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 = 𝑛^{2}− 2𝑛 + 2 ^{𝑛−1}* (13) *

**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 𝑆_{𝑛} is 𝑛 − 1 i.e., 𝐸(𝑆_{𝑛}) = 𝑛 − 1 . Then

** 𝒊 𝐼𝑆𝐼𝜋 𝑆**_{𝑛} = _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

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} ** **
= 1^{2}+ (𝑛 − 1)^{2} ^{𝑛−1}= 𝑛^{2}− 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 𝑛^{2}− 2𝑛 + 10 ^{ 𝑛−1 }* (15) *
** Proof : The wheel graph 𝑊**𝑛 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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝒖𝒗∈𝑬(𝑊_{𝑛})

= ^{3× 𝑛−1 }_{3+ 𝑛−1 } ^{ 𝑛−1 }× ^{3×3}_{3+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}

= 3^{2}+ 𝑛 − 1 ^{2} ^{ 𝑛−1 }× 3^{2}+ 3^{2} ^{ 𝑛−1 }
= 18 𝑛^{2}− 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} 𝑛^{2}+ 9 ^{𝑛−2} 𝑛^{2}+ 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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝑢𝑣 ∈𝐸(𝐹_{𝑛})

= ^{2×3}_{2+3}^{2}× ^{3×𝑛}_{3+𝑛} ^{𝑛−2}× ^{2×𝑛}_{2+𝑛} ^{2}× ^{3×3}_{3+3} ^{𝑛−3}

**_______________________________________________________________________________________________**

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

𝑖𝑖 𝐹𝜋 𝐹_{𝑛} = _{𝑢𝑣 ∈𝐸(𝐹}_{𝑛}_{)} 𝑑_{𝐹}_{𝑛}(𝑢)^{2}+ 𝑑_{𝐹}_{𝑛}(𝑣)^{2}

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

= 169 18 ^{𝑛−3} 𝑛^{2}+ 9 ^{𝑛−2} 𝑛^{2}+ 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 = 𝑛^{2}+ 4 ^{2} 𝑛^{2}+ 9 ^{2𝑛−3}* (19) *

** Proof :For a Complete bipartite graph 𝐾**𝑚 ,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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

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}

= 𝑚^{2}+ 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} =^{4}_{9}* (20) *
(ii) The multiplicative F index(Fπ 𝐺 )

Fπ 𝑃𝑛 = 25 . 8 ^{𝑛−3} (21)

** Proof : A path graph 𝑃**_{𝑛} has vertices 𝑣_{1}, 𝑣_{2}, 𝑣_{3}, … 𝑣_{𝑛} 𝑎𝑛𝑑 𝑒𝑑𝑔𝑒𝑠 𝑒_{1}, 𝑒_{2}, 𝑒_{2}, … 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}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝑢𝑣 ∈𝐸 (𝑃_{𝑛})

= ^{2×1}_{2+1} ^{2} ^{2×2}_{2+2} ^{𝑛−3}= ^{2}_{3} ^{2}=^{4}_{9}

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

**Lemma 2.8: **

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

**Volume: 4 Issue: 9 ** ** 01 – 08 **

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(i) The multiplicative Inverse Sum indeg (𝐼𝑆𝐼π 𝐺 ) Index
𝐼𝑆𝐼𝜋 𝐺 = ^{𝑟}_{2}

𝑛𝑟

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

Fπ 𝐺 = 𝑛^{2}+ 4 ^{2} 𝑛^{2}+ 9 ^{2𝑛−3}* (23) *

**Proof. A regular graph G on n vertices, having degree r, possesses **^{nr}

2** edges.Thus **
𝑖 𝐼𝑆𝐼𝜋 𝐺 = _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝑢𝑣 ∈𝐸(𝐺)
= ^{𝑟×𝑟}_{𝑟+𝑟}

𝑛𝑟
2 = ^{𝑟}_{2}

𝑛𝑟

2

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

= 𝑟^{2}+ 𝑟^{2} ^{𝑛𝑟}^{2} = 2𝑟^{2} ^{𝑛𝑟}^{2}

**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
𝑇𝑈𝐶_{4}𝐶_{8} 𝑚, 𝑛 and 𝑇𝑈𝐶_{4} 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ .

**3.1 MULTIPLICATIVE TOPOLOGICAL INDICES FOR 𝑻𝑼𝑪**𝟒𝑪𝟖** 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 TUSC*_{4}*C*_{8}*(S) Nanotubes. *

* M.V. Diudea denoted 𝑇𝑈𝐶*4𝐶8* the number of Octagons C*_{8 }*in the first row of G by m and the number of Octagons 𝐶*8 in the first
*column of G by n, and he denoted 𝑇𝑈𝐶*4𝐶8 Nanotubes by 𝐺 = 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 ∀𝑚, 𝑛 ∈ ℕ.One can see the 3-Dimensional and 2-
Dimensional lattices of 𝐺 = 𝑇𝑈𝐶_{4}𝐶_{8}* 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 𝑇𝑈𝐶**4𝐶8 𝑚, 𝑛 Nanotubes ∀𝑚, 𝑛 ∈ ℕ, with 2𝑚𝑛 + 2𝑚.

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

**_______________________________________________________________________________________________**

**6 **
**Theorem 3.1: Let G = TUC**_{4}C_{8} m, n nanotubes. Then

(i) Hπ G = _{625}^{4} ^{𝑚}× ^{1}_{9} ^{6𝑚𝑛 −𝑚}** (24) **
(ii) 𝐼𝑆𝐼𝜋 𝐺 = ^{1296}_{625} ^{𝑚}× ^{9}_{4} ^{6𝑚𝑛 −𝑚} (25)
(iii) 𝐹𝜋 𝐺 = 8 ^{2𝑚}× 13 ^{4𝑚}× 18 ^{12𝑚𝑛 −2𝑚} (26)

**Proof: **

Consider the 𝑇𝑈𝐶_{4}𝐶_{8}* 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 𝑇𝑈𝐶_{4}𝐶_{8} 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 𝑇𝑈𝐶_{4}𝐶_{8} 𝑚, 𝑛 in Fig 1, we see that there are three partitions of edge\bond set 𝐸(𝑇𝑈𝐶_{4}𝐶_{8} 𝑚, 𝑛 )with their size are as follows:

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

𝐸{3,3} = {𝑒 = 𝑢𝑣𝐸(𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 ])| 𝑑_{𝑢} = 𝑑𝑣= 3} → | 𝐸6| = | 𝐸9∗| = 12𝑚𝑛 − 2𝑚

Where 𝐸_{4}= 𝐸_{2+2 } , 𝐸_{4}^{∗}= 𝐸_{2×2 }*and so on . In Fig.1, we marked all members of these edges partitions of *𝑇𝑈𝐶_{4}𝐶_{8} 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 C_{4}C_{8} m,n )
= _{d} ^{2}

u+d_{v}

uv ∈E_{4} × _{d} ^{2}

u+d_{v}

uv ∈E_{5} × _{d} ^{2}

u+d_{v}
uv ∈E_{6}
= _{2+2}^{2} ^{2m}× _{2+3}^{2} ^{4m}× _{3+3}^{2} ^{12mn −2m}
= ^{1}_{2} ^{2m}× ^{2}_{5} ^{4m}× ^{1}_{3} ^{12mn −2m}

= 4 ^{−m} × _{625}^{16} ^{m}× ^{1}_{9} ^{6mn −m} = _{625}^{4} ^{𝑚}× ^{1}_{9} ^{6𝑚𝑛 −𝑚}

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = ^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑑_{𝑢}+𝑑_{𝑣}
𝑢𝑣 ∈𝐸(𝑇𝑈𝐶_{4}𝐶_{8} 𝑚 ,𝑛 )
= _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}

𝑢𝑣 ∈𝐸_{4}=𝐸_{4}^{∗} × _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}

𝑢𝑣 ∈𝐸_{5}=𝐸_{6}^{∗} × _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝑢𝑣 ∈𝐸_{6}=𝐸_{9}^{∗}
= ^{2×2}_{2+2} ^{2𝑚}× ^{2×3}_{2+3} ^{4𝑚}× ^{3×3}_{3+3} ^{12𝑚𝑛 −2𝑚}

= ^{6}_{5} ^{4𝑚}× ^{3}_{2} ^{12𝑚𝑛 −2𝑚}
= ^{6}_{5} ^{4𝑚}× ^{3}_{2} ^{12𝑚𝑛 −2𝑚}
= ^{1296}_{625} ^{𝑚}× ^{9}_{4} ^{6𝑚𝑛 −𝑚}

(𝑖𝑖𝑖)𝐹𝜋 𝐺 = 𝐹𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶_{4}𝐶_{8} 𝑚 ,𝑛 ) 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2}

= 𝑢𝑣∈𝐸_{4} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2} × 𝑢𝑣 ∈𝐸_{5} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2} × 𝑢𝑣 ∈𝐸_{6} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2}
= 2^{2}+ 2^{2} ^{2𝑚}× 2^{2}+ 3^{2} ^{4𝑚}× 3^{2}+ 3^{2}^{12𝑚𝑛 −2𝑚}

= 8 ^{2𝑚}× 13 ^{4𝑚}× 18 ^{12𝑚𝑛 −2𝑚}

**Volume: 4 Issue: 9 ** ** 01 – 08 **

**_______________________________________________________________________________________________**

**3.2. MULTIPLICATIVE TOPOLOGICAL INDICES FOR 𝑻𝑼𝑪**_{𝟒}𝑪_{𝟖}** 𝒎, 𝒏 NANOTUBES **

Consider the Carbon nanotube 𝑇𝑈𝐶_{4} 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 = 𝑇𝑈𝐶**4 𝑚, 𝑛 , ∀𝑚, 𝑛 ∈ ℕ nanotubes. Then

(i) Hπ G = 81 ^{−m}× _{16}^{1} ^{2mn −m}** (27) **
(ii) 𝐼𝑆𝐼𝜋 𝐺 = ^{1296}_{625} ^{𝑚}× ^{9}_{4} ^{6𝑚𝑛 −𝑚} (28)

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

*Consider the 𝐺 = 𝑇𝑈𝐶*_{4}* 𝑚, 𝑛 Nanotubes has the number of vertices 𝑣*𝐺= 𝑉 𝑇𝑈𝐶_{4} 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(𝑇𝑈𝐶_{4}[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(TUC*_{4}*[m,n]) as *
𝐸{2,4} = 𝐸_{6}= 𝐸_{8}^{∗}= 𝑒 = 𝑢𝑣 ∈ 𝐸 𝑇𝑈𝐶_{4} 𝑚, 𝑛 │𝑑_{𝑢} = 2 & 𝑑_{𝑣}= 4 → 𝐸_{6} = 𝐸_{8}^{∗} = 2𝑚 + 2𝑚 = 2|𝑉_{2}|

𝐸{4,4} = 𝐸8= 𝐸16∗= 𝑒 = 𝑢𝑣 ∈ 𝐸 𝑇𝑈𝐶4 𝑚, 𝑛 𝑑_{𝑢} = 𝑑𝑣= 4 → 𝐸8 = 𝐸_{16}^{∗} = 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 C_{4} m,n )
= _{d} ^{2}

u+d_{v}

uv ∈E_{6} × _{d} ^{2}

u+d_{v}
uv ∈E_{8}

= _{2+4}^{2} ^{4m}× _{4+4}^{2} ^{4mn −2m}= ^{1}_{3} ^{4m}× ^{1}_{4} ^{4mn −2m}
= 81 ^{−m}× _{16}^{1} ^{2mn −m}

𝑖𝑖 𝐼𝑆𝐼𝜋 𝐺 = 𝐼𝑆𝐼𝜋 𝑇𝑈𝐶4 𝑚, 𝑛 = ^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑑_{𝑢}+𝑑_{𝑣}
𝑢𝑣 ∈𝐸(𝑇𝑈𝐶_{4} 𝑚 ,𝑛 )
= _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}

𝑢𝑣 ∈𝐸_{6}=𝐸_{8}^{∗} × _{𝑑}^{𝑑}^{𝑢}^{𝑑}^{𝑣}

𝑢+𝑑_{𝑣}
𝑢𝑣 ∈𝐸_{8}=𝐸_{16}^{∗}
= ^{2×4}_{2+4} ^{4𝑚}× ^{4×4}_{4+4} ^{4mn −2m}

= ^{4}_{3} ^{4𝑚}× 2 ^{4mn −2m} = ^{4}_{3} ^{4𝑚}× 2 ^{4mn −2m}
(𝑖𝑖𝑖)𝐹𝜋 𝐺 = 𝐹𝜋 𝑇𝑈𝐶4𝐶8 𝑚, 𝑛 = 𝑢𝑣 ∈𝐸(𝑇𝑈𝐶_{4} 𝑚 ,𝑛 ) 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2}

= 𝑢𝑣 ∈𝐸_{4} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2} × 𝑢𝑣 ∈𝐸_{5} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2} × 𝑢𝑣 ∈𝐸_{6} 𝑑_{𝐺}(𝑢)^{2}+ 𝑑𝐺(𝑣)^{2}
= 2^{2}+ 4^{2} ^{4𝑚}× 4^{2}+ 4^{2} ^{4mn −2m}

= 20 ^{4𝑚} × 32 ^{4mn −2m}

**_______________________________________________________________________________________________**

**8 **
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