Multiplicate Hyper-Zagreb Indices and coindices of

International Journal of Mathematical Archive-8(2), 2017, 99-106

Available online through www.ijma.info ISSN 2229 – 5046 COMPUTATION OF SOME TOPOLOGICAL INDICES OF CERTAIN NETWORKS V. R. KULLI* Department of Mathematics, Gulbarga University, Gulbarga 585106, India. (Received On: 20-01-17; Revised & Accepted On: 18-02-17)

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 modified first and second Zagreb indices, harmonic index and augmented Zagreb index for certain networks like silicate networks and honeycomb networks. Keywords: silicate network, hexagonal network, oxide network, honeycomb network, modified first and second Zagreb indices, harmonic index, augmented Zagreb index. Mathematics Subject Classification: 05C05, 94C15.

1. INTRODUCTION In this paper, we consider finite simple, undirected graphs. Let G = (V, E) be a graph. The degree dG(v) of a vertex v is the number of vertices adjacent to v. The edge connecting the vertices u and v will be denoted by uv. We refer to [1] for undefined term and notation. A molecular graph or a chemical graph is a simple graph related to the structure of a chemical compound. Each vertex of this graph represents an atom of the molecule and its edges to the bonds between atoms. A topological index is a numerical parameter mathematically derived from the graph structure. These indices are useful for establishing correlation between the structure of a molecular compound and its physico-chemical properties. The modified first and second Zagreb indices [2] are respectively defined as 1 1 m , m M 2 (G ) = ∑ . M1 ( G ) = ∑ 2 uv∈E ( G ) d G ( u ) d G ( v ) u∈V ( G ) d G ( u ) Many other topological indices were studied, for example, in [3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15]. The harmonic index of a graph G is defined as 2 . H (G ) = ∑ d dG ( v ) ( ) + uv∈E ( G ) G u This index was studied by Favaron et al. [16] and Zhong [17]. The augmented Zagreb index of a graph G is defined as 3

AZI ( G ) =

 d (u ) d (v )  ∑  d (Gu ) + d G( v ) − 2  . G  uv∈E ( G )  G

The augmented Zagreb index was introduced by Furtula et al. in [18] and was studied, for example, in [19]. In this paper, the modified first and second Zagreb indices, harmonic index and augmented Zagreb index for certain network. For Figures see [20].

Corresponding Author: V. R. Kulli* International Journal of Mathematical Archive- 8(2), Feb. – 2017

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V. R. Kulli* / Computation of Some Topological Indices of Certain Networks / IJMA- 8(2), Feb.-2017.

2 RESULTS FOR SILICATE NETWORKS Silicates are obtained by fusing metal oxides or metal carbonates with sand. A silicate network is symbolized by SLn, where n is the number of hexagons between the center and boundary of SLn. A silicate network of dimension two is depicted in Figure 1. We compute the exact values of mM1(SLn), mM2(SLn), H(SLn) and AZI(SLn) for silicate networks. Theorem 2.1: Let SLn be the silicate networks. Then 11 2 7 3 2 2 (2) m M 2 ( SL (1) m M1 ( SL = n + n. n + n. = n) n) 12 12 2 3

4 2 (3) H ( SL= n ) 7 n + n. 3

3 3 3   1 1 (4) AZI ( SLn )  3 + 3 184 n 2 +  9  +  18  − 2  18   6n. = 7 5  5   4   7 

Figure-1: Silicate network of dimension two Proof: Let G be the graph of silicate network SLn with |V(SLn)| = 15n2 + 3n and |E(SLn)| = 36n2. From Figure 1, it is easy to see that there are two partitions of the vertex set of SLn as follows: V3 = {u∈V(G) | dG(u) = 3}, |V3| = 6n2 + 6n. V6 = {u∈V(G) | dG(u) = 6}, |V6| = 9n2 – 3n. By algebraic method, in SLn there are three types of edges based on the degree of the vertices of each edge as follows: E6 = {uv∈E(G) | dG(u) = dG(v) = 3}, |E6| = 6n. E9 = {uv∈E(G) | dG(u) = 3, dG(v) = 6}, |E9| = 18n2 + 6n. E12 = {uv∈E(G) | dG(u) = dG(v) = 6}, |E12| = 18n2 – 12n. 1) Now to compute mM1(G), we see that 1 1 1 1 1 11 2 7 m M1(G) = ∑= = 2 ( 6n 2 + 6n ) + 2 ( 9n 2 − 3n ) = +∑ n + n. ∑ 2 2 2 12 12 3 6 u∈V ( G ) d G ( u ) u∈V3 d G ( u ) u∈V6 d G ( u ) m 2) To compute M2(G), we see that 1 1 1 1 m M2(G) = ∑ = ∑ + ∑ + ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d d d d d d G v G v G v uv∈E ( G ) G u uv∈E6 G u uv∈E9 G u uv∈E12 G u d G ( v )

3 2 2  1   1   1  =  n + n.  6n +   (18n 2 + 6n ) +   (18n 2 − 12n ) = 2 3  3× 3   3× 6   6× 6  3) To compute H(G), we see that 2 2 2 2 H(G) = ∑ = ∑ + ∑ + ∑ uv∈E ( G ) d G ( u ) + d G ( v ) uv∈E6 d G ( u ) + d G ( v ) uv∈E9 d G ( u ) + d G ( v ) uv∈E12 d G ( u ) + d G ( v ) 4  2   2   2  =  7 n 2 + n.  6n +   (18n 2 + 6n ) +   (18n 2 − 12n ) = 3  3+3  3+ 6  6+6 4) To compute AZI(G), we see that  dG ( u ) dG ( v )  AZI(G) = ∑   uv∈E ( G )  d G ( u ) + d G ( v ) − 2  3

3

3

 dG ( u ) dG ( v )   dG ( u ) dG ( v )   dG ( u ) dG ( v )  = ∑   + ∑  d ( )+d ( )−2 + ∑  d ( )+d ( )−2 d d + − 2 ( ) ( ) G v G v G v  uv∈E9  G u  uv∈E12  G u  uv∈E6  G u © 2017, IJMA. All Rights Reserved

3

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V. R. Kulli* / Computation of Some Topological Indices of Certain Networks / IJMA- 8(2), Feb.-2017. 3

=

3

3

3× 3 3× 6 6×6 ∑  3 + 3 − 2  6n + ∑  3 + 6 − 2  (18n2 + 6n ) + ∑  6 + 6 − 2  (18n2 − 12n ) uv∈E6 uv∈E6 uv∈E12

3 3 3   1 1 =  3 + 3 184 n 2 +  9  +  18  − 2  18   6n. 7 5  5   4   7 

3. RESULTS FOR CHAIN SILICATE NETWORKS We now consider a family of chain silicate networks. This network is symbolized by CSn and is obtained by arranging n tetrahedral linearly, see Figure 2.

Figure-2: Chain silicate network We compute the values of mM1(CSn), mM2(CSn), H(CSn), and AZI(CSn) for chain silicate networks. Theorem 3.1: Let CSn be the chain silicate networks. Then 1 7 13 5 (2) m M 2 ( CS= (1) m M1 ( CS= n+ n+ n) n) 4 36 36 18 (3) H ( CS = n)

3 3 3 3  3 3   25 5 (4) AZI ( CSn )  9  +  18  4 −  18   n +  9  4 −  18  2 −  18  2  . n= + . 18 9 5  7 5   4   7   4 

Proof: Let G be the graph of chain silicate networks CSn with |V(CSn)| = 3n +1 and |E(CSn)| = 6n. From Figure 2, it is easy to see that there are two partitions of the vertex set of CSn as follows: V3 = {u∈V(G) | dG(u) = 3}, |V3| = 2n + 2. V6 = {u∈V(G) | dG(u) = 6}, |V6| = n – 1. By algebraic method, in CSn, n ≥ 2, there are three types of edges based on the degree of the vertices of each edge as follows: E6 = {uv∈E(G) | dG(u) = dG(v) = 3}, |E6| = n + 4. E9 = {uv∈E(G) | dG(u) = 3, dG(v) = 6}, |E9| = 4n – 2. E12 = {uv∈E(G) | dG(u) = dG(v) = 6}, |E12| = n – 2. (1) Now to compute mM1(G) we see that 1 1 1 1 1 7 m = 1 M1(G) = ∑= . +∑ n+ ( ) + 2 ( n − 1) = ∑ 2 2 2 2 2n + 2 4 36 3 6 u∈V ( G ) d G ( u ) u∈V3 d G ( u ) u∈V6 d G ( u ) (2) To compute mM2(G), we see that 1 1 1 1 m M2(G) = ∑ = ∑ + ∑ + ∑ uv∈E ( G ) d G ( u ) d G ( v ) uv∈E6 d G ( u ) d G ( v ) uv∈E9 d G ( u ) d G ( v ) uv∈E12 d G ( u ) d G ( v )

13 5  1   1   1  = n+ .  ( n + 4) +   ( 4n − 2 ) +   ( n − 2) = 36 18  3× 3   3× 6   6×6  (3) To compute H(G), we see that 2 2 2 2 H(G) = ∑ = ∑ + ∑ + ∑ uv∈E ( G ) d G ( u ) + d G ( v ) uv∈E6 d G ( u ) + d G ( v ) uv∈E9 d G ( u ) + d G ( v ) uv∈E12 d G ( u ) + d G ( v )

25 5  2   2   2  = n+ .  ( n + 4) +   ( 4n − 2 ) +   ( n − 2) = 18 9  3+3  3+ 6  6+6 (4) To compute AZI(G), we see that AZI(G) =

 d (u ) d (v )  ∑  d (Gu ) + d G( v ) − 2  G  uv∈E ( G )  G 3

=

3

3

 d (u ) d (v )   d (u ) d (v )   d (u ) d (v )  ∑  d (Gu ) + d G( v ) − 2  + ∑  d (Gu ) + d G( v ) − 2  + ∑  d (Gu ) + d G( v ) − 2  G G G  uv∈E9  G  uv∈E12  G  uv∈E6  G

© 2017, IJMA. All Rights Reserved

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V. R. Kulli* / Computation of Some Topological Indices of Certain Networks / IJMA- 8(2), Feb.-2017. 3

3

3

 3× 3   3× 6   6×6  =   ( n + 4) +   ( 4n − 2 ) +   ( n − 2)  3+3− 2   3+ 6− 2  6+6−2 3 3 3 3  3 3   =  9  +  18  4 −  18   n +  9  4 −  18  2 −  18  2  5  7 5   4   7   4 

Theorem 3.2: Let CSn (n = 1) be the chain silicate network. Then 4 2 (2) m M 2 ( CS1 ) = . (1) m M1 ( CS1 ) = . 9 3 2187 (3) H ( CS1 ) = 2 . (4) AZI ( CS1 ) = . 32 Proof: Let CS1 be the graph of a chain silicate network. Then CS1 = K4. Clearly |V(CS1) | = 4 and |E(CS1)| = 6. Also dCS1 ( u ) = 3 for every u ∈ V(CS1). To compute mM1(CS1), mM2(CS1), H(CS1), and AZI(CS1), we see that 1 1 4 1) m M1 (CS1 ) = ∑ d ( )2 = 32 × 4 = 9 . u u∈V ( CS1 )

) M 2 (CS1=

2)

m

3)

) H (CS1=

4)

= AZI (CS1 )

1 1 2 6 . = ×= 3× 3 3 uv∈E ( CS1 ) d ( u ) d ( v )

∑

2 2 6 2. = ×= 3+3 uv∈E ( CS1 ) d ( u ) + d ( v )

∑

3

2187  d (u ) d (v )   3× 3  = .   ×6  d ( ) + d (=  32 u v) − 2   3 + 3 − 2  uv∈E ( CS1 ) 

∑

3

4. RESULTS FOR HEXAGONAL NETWORKS It is known that there exist three regular plane tilings with composition of same kind of regular polygons such as triangular, hexagonal and square. Triangular tiling is used in the construction of hexagonal networks. This network is symbolized by HXn, where n is the number of vertices in each side of hexagon. A hexagonal network of dimension six in shown in Figure 3.

Figure-3: Hexagonal network of dimension six Now we compute the values of mM1(HXn), mM2(HXn), H(HXn) and AZI(HXn) for hexagonal networks. Theorem 4.1: Let HXn be the hexagonal networks. Then 1 1 2 1 17 2) m M 2 ( HX n ) = 1) m M1 ( HX n )= n + n+ . ( 6n2 − n + 1) . 24 12 8 18 3 2 8 97 3) H ( HX n ) = 4) AZI ( HX n ) ≈ 419.904n 2 − 1101.870n + 678.252. . n − n+ 2 5 210 © 2017, IJMA. All Rights Reserved

102


Proof: Let G be the graph of hexagonal network HXn with |V(HXn)|=3n2 – 3n + 1 and |E(HXn)|=9n2 – 15n + 6. From Figure 3, it is easy to see that there are three partitions of the vertex set of HXn as follows: V3 = {u∈V(G) | dG(u) = 3}, |V3| = 6. V4 = {u∈V(G) | dG(u) = 4}, |V4| = 6n – 2. V6 = {u∈V(G) | dG(u) = 6}, |V6| = 3n2 – 9n + 7. In HXn, by algebraic method, there are five types of edges based on the degree of the vertices of each edge as follows: E7 = {uv∈E(G) | dG(u) = 3, dG(v) = 4}, |E7| = 12. E9 = {uv∈E(G) | dG(u) = 3, dG(v) = 6}, |E9| = 6. E8 = {uv∈E(G) | dG(u) = dG(v) = 4}, |E8| = 6n – 18. E10 = {uv∈E(G) | dG(u) = 4, dG(v) = 6}, |E10| = 12n – 24. E12 = {uv∈E(G) | dG(u) = dG(v) = 6}, |E12| = 9n2 – 33n + 30. (1) Now compute mM1(HXn), we see that 1 1 1 1 m M1(G) = ∑ = ∑ +∑ +∑ 2 2 2 2 d d d d ( ) ( ) ( ) u∈V ( G ) G u u∈V3 G u u∈V4 G u u∈V6 G ( u )

1 1 1 1 2 1 17 n + n+ . ( 6 ) + 2 ( 6n − 12 ) + 2 ( 3n 2 − 9n + 7 )= 12 8 18 32 4 6 (2) To compute mM2(G), we see that 1 m M2(G) = ∑ d dG ( v ) ( ) u uv∈E ( G ) G =

∑

=

uv∈E7

1 1 1 1 1 + ∑ + ∑ + ∑ + ∑ dG ( u ) dG ( v ) uv∈E9 dG ( u ) dG ( v ) uv∈E8 dG ( u ) dG ( v ) uv∈E10 dG ( u ) dG ( v ) uv∈E12 dG ( u ) dG ( v )

 1   1   1   1   1  =  ( 6n − 18 ) +   (12 ) +   ( 6) +   (12n − 24 ) +   ( 9n 2 − 33n + 30 )  4× 4   3× 4   3× 6   4×6   6×6  1 = ( 6n 2 − n + 1) . 24 (3) To compute H(G), we see that 2 ‘ H(G)= ∑ uv∈E ( G ) d G ( u ) + d G ( v ) 2 2 2 2 + ∑ + ∑ + ∑ uv∈E7 d G ( u ) + d G ( v ) uv∈E9 d G ( u ) + d G ( v ) uv∈E8 d G ( u ) + d G ( v ) uv∈E10 d G ( u ) + d G ( v ) 2 + ∑ d dG ( v ) ( ) + u uv∈V12 G =

∑

 2   2   2   2   2  = =  ( 6n − 18 ) +   (12 ) +   ( 6) +   (12n − 24 ) +   ( 9n 2 − 33n + 30 )  4+4  3+ 4   3+ 6   4+6 6+6 3 2 8 97 = n − n+ . 2 5 210 (4) To compute AZI(G), we see that AZI(G) =

 d (u ) d (v )  ∑  d (Gu ) + d G( v ) − 2  G  uv∈E ( G )  G 3

‘

3

3

3

3

 3× 4   3× 6   4× 6   4× 4   ( 6n − 18 ) +    (12 ) +   ( 6) +   (12n − 24 ) + − + − 3 4 2 3 6 2  + −  4 4 2      4+6−2 = 3  6×6  +  ( 9n 2 − 33n + 30 ) 6+6−2

≈ 419.904n 2 − 1101.870n + 678.252. 5. RESULTS FOR OXIDE NETWORKS The oxide networks are of vital importance in the study of silicate networks. An oxide network of dimension n is denoted by OXn. A 5-dimensional oxide network is shown in Figure 4.


103


Figure-4: Oxide network of dimension 5 We compute the values of mM1(OXn), mM2(OXn), H(OXn) and AZI(OXn) for oxide networks. Theorem 5.1: Let OXn be the oxide network. Then 3 2 21 9 2 3 (2) m M 2 ( OX= (1) m M1 ( OX= n + n. n + n n) n) 4 16 8 4 9 2 1024 2 1184 (3) H ( OX= (4) AZI = n + n. n − n. ( OX n ) n) 2 3 9 Proof: Let G be the graph of oxide network OXn with |V(OXn) | = 9n2 + 3n and |E(OXn)| = 18n2. From Figure 4, it is easy to see that there are two partitions of the vertex set of OXn as follows: V2 = {u∈V(G) | dG(u) = 2}, |V2| = 6n. V4 = {u∈V(G) | dG(u) = 4}, |V4| = 9n2 – 3n. In OXn, by algebraic method, there are two types of edges based on the degree of the vertices of each edge as follows; E6 = {uv∈E(G) | dG(u) = 2, dG(v) = 4}, |E6| = 12n. E8 = {uv∈E(G) | dG(u) = dG(v) = 4}, |E8| = 18n2 – 12n. Now to compute mM1(OXn), we see that 3 2 21 1 1 1  1   1  m M1(G) = ∑= =  2  6n +  2  ( 9n 2 − 3n ) = n + n. +∑ ∑ 2 2 2 4 16 2  4  u∈V ( G ) d G ( u ) u∈V2 d G ( u ) u∈V4 d G ( u ) 1.

To compute mM2(OXn), we see that 1 1 1 m M2(G) = ∑ = ∑ + ∑ uv∈E ( G ) d G ( u ) d G ( v ) uv∈E6 d G ( u ) d G ( v ) uv∈E8 d G ( u ) d G ( v )

9 2 3 1   1  =  n + n. 12n +   (18n 2 − 12n ) = 8 4  2× 4   4× 4  2. To compute H(OXn), we see that 2 2 2 H(G) = + ∑ ∑= ∑ uv∈E ( G ) d G ( u ) + d G ( v ) uv∈E6 d G ( u ) + d G ( v ) uv∈V8 d G ( u ) + d G ( v )

3.

9 2  2  n+ 2  = n + n.  12   (18n 2 − 12n ) = 2  2+4  4+4 To compute AZI(OXn), we see that AZI(G) =

3

 dG ( u ) dG ( v )    = uv∈E ( G )  d G ( u ) + d G ( v ) − 2 

∑

3

3

 dG ( u ) dG ( v )   dG ( u ) dG ( v )   d ( )+d ( )−2 + ∑  d ( )+d ( )−2 G v G v  uv∈E8  G u  uv∈E6  G u

∑

3

3

1024 2 1184  2× 4   4× 4  =+ n − n   12n   (18n 2 − 12n ) = 3 9  2+4−2  4+4−2 6. RESULTS FOR HONEYCOMB NETWORKS If we recursively use hexagonal tiling in a particular pattern, honeycomb networks are formed. These networks are very useful in chemistry and also in computer graphics. A honeycomb network of dimension n is denoted by HCn, where n is the number of hexagons between central and boundary hexagon. A 4-dimensional honeycomb network is shown in Figure 5. © 2017, IJMA. All Rights Reserved

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Figure-5: Honeycomb network of dimension four In the following theorem, we compute the values of mM1(HCn), mM2(HCn), H(HCn) and AZI(HCn) for honey comb networks. Theorem 5.1: Let HCn be the honeycomb networks. Then 1 1 3 2 5 (2) m M 2 ( HCn ) = n 2 + n + . (1) m M1 ( HC = n + n. n) 3 6 2 6 1 1 1 (3) H ( HCn ) = 3n 2 − n + . (4) AZI ( HCn ) = ( 6561n 2 − 4791n + 1302 ) 3 . 5 5 4 Proof: Let G be the graph of honeycomb network HCn with |V(HCn)|=6n2 and |E(HCn)| = 9n2 – 3n. From Figure 5, it is easy to see that there are two partitions of the vertex set of HCn as follows: V2 = {u∈V(G) | dG(u) = 2}, |V2| = 6n. V3 = {u∈V(G) | dG(u) = 3}, |V3| = 6n2 – 3n. In HCn, by algebraic method, there are three types of edges based on the degree of the vertices of each edge as follows: E4 = {uv∈E(G) | dG(u) = dG(v) = 2}, |E4| = 6. E5 = {uv∈E(G) | dG(u) = 2, dG(v) = 3}, |E5| = 12n – 12 . E6 = {uv∈E(G) | dG(u) = dG(v) = 3}, |E6| = 9n2 – 15n + 6. (1) Now compute mM1(HCn), we see that 3 2 5 1 1 1  1   1  m M1(G) = ∑= =  2  6n +  2  ( 6n 2 − 6n ) = n + n. +∑ ∑ 2 2 2 2 6 2  3  u∈V ( G ) d G ( u ) u∈V2 d G ( u ) u∈V3 d G ( u ) (2) To compute mM2(HCn), we see that 1 1 1 1 m M2(G) = ∑ = ∑ + ∑ + ∑ uv∈E ( G ) d G ( u ) d G ( v ) uv∈E4 d G ( u ) d G ( v ) uv∈E5 d G ( u ) d G ( v ) uv∈E6 d G ( u ) d G ( v )

1 1  1   1   1  2 =  ( 6) +   (12n − 12 ) +   ( 9n 2 − 15n + 6 ) = n + n + . 3 6  2× 2   2×3   3× 3  (3) To compute H(HCn), we see that 2 H(G) = ∑ = uv∈E ( G ) d G ( u ) + d G ( v )

2 2 2 + ∑ + ∑ dG ( u ) + dG ( v ) uv∈E5 dG ( u ) + dG ( v ) uv∈E6 dG ( u ) + dG ( v )

∑

uv∈E4

1 2   2   2  2 1 =  6 +   (12n − 12 ) +   ( 9n 2 − 15n + 6 ) = 3n − n + . 5 5  2+ 2  2+3  3+3 3 2 8 97 = n − n+ . 2 5 210 (4) To compute AZI(G), we see that 3

AZI(G) =

3 3 3  d (u ) d (v )  ∑  d (Gu ) + d G( v ) − 2  =  2 +2 ×2 2− 2  6 +  2 +2 ×3 −3 2  (12n − 12 ) +  3 +3 ×3 −3 2  ( 9n2 − 15n + 6 ) G  uv∈E ( G )  G

= ( 6561n 2 − 4791n + 1302 )


1 43

.

105


REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012). S. Nikolić, G. Kovaćević, A. Milićević and N. Trinajstić, The Zagreb indices 30 years after, Croatica Chemica Acta, 76(2), (2003) 113-124. V.R. Kulli, The first and second κa indices and coindices of graphs, International Journal of Mathematical Archive, 7(5), (2016) 71-77. V.R. Kulli, On K edge index and coindex of graphs, International Journal of Fuzzy Mathematical Archive, 10(2), (2016) 111-116. V.R. Kulli, On K-edge index of some nanostructures, Journal of Computer and Mathematical Sciences, 7(7), (2016) 373-378. V.R.Kulli, Multiplicative hyper-Zagreb indices and coindices of graphs: Computing these indices of some nanostructures, International Research Journal of Pure Algebra, 6(7), (2016) 342-347. V.R.Kulli, Multiplicative connectivity indices of certain nanotubes, Annals of Pure and Applied Mathematics, 12(2) (2016) 169-176. DOI: http://dx.doi.org/10.22457/apam.v12n2a8. V.R. Kulli, General multiplicative Zagreb indices of TUC4C8[m, n] and TUC4[m, n] nanotubes, International Journal of Fuzzy Mathematical Archive, 11(1) (2016) 39-43. DOI. http://dx.doi.org/10.22457/ijfma.v11n1a6. V.R. Kulli, Multiplicative connectivity indices of TUC4C8[m,n] and TUC4[m, n] nanotubes, Journal of Computer and Mathematical Sciences, 7(11) (2016) 599-605. V.R.Kulli, Multiplicative connectivity indices of nanostructures, Journal of Ultra Scientist of Physical Sciencs, A 29(1) (2017) 1-10. http://dx.doi.org/10.22147/jusps-A/290101. V.R.Kulli, Computation of general topological indices for titania nanotubes, International Journal of Mathematical Archive, 7(12) (2016) 33-38. . V.R.Kulli, Some new multiplicative geometric-arithmetic indices, Journal of Ultra Scientist of Physical Sciencs, A, 29(2) (2017) http://dx.doi.org/10.22147/jusps-A/290201. V.R.Kulli, Two new multiplicative atom bond connectivity indices, Annals of Pure and Applied Mathematics, 13(1) (2017) 1-7. DOI: http://dx.doi.org/10.22147/apam.v12n1a1. V.R.Kulli, Some topological indices of certain nanotubes, Journal of Computer and Mathematical Sciences,8(1)(2017),1-7.. V.R.Kulli, F-index and reformulated Zagreb index of certain nanostructures, International Research Journal of Journal of Algebra, 7(1) (2017), 489-495. O. Favaron, M. Maho and J.F. Sacle, Some eigenvalue properties in graphs (conjectures of Graffiti II), Discrete Math. 111(1-3) (1993) 197-220. L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25(3) (2012) 561-566. B. Furtula, A. Graovac and D.Vukičević, Augmented Zagreb index, J. Math. Chem 48(2010) 370-380. Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67(2012) 483-494. S.Wang, J.B.Liu, C.Wang and S. Hayat, Further results on computation of topological indices of certain networks, arXiv: 1605.00253v2 [math. CO] 5 may 2016. Source of support: Nil, Conflict of interest: None Declared.

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