Journal of Computer and Mathematical Sciences, Vol.7(11), 599-605, November 2016 (An International Research Journal), www.compmath-journal.org

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)

Multiplicative Connectivity Indices of TUC4C8[m, n] and TUC4[m, n] Nanotubes V. R. Kulli Department of Mathematics, Gulbarga University, Gulbarga 585 106, INDIA. email: [email protected] (Received on: November 12, 2016) ABSTRACT The topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. In this paper, we compute the multiplicative product connectivity, multiplicative sum connectivity, multiplicative atom bond connectivity, multiplicative geometric- arithmetic indices for TUC4C8[m, n] and TUC4[m, n] nanotubes. Mathematics Subject Classification: 05C05, 05C07 Keywords: multiplicative product connectivity index, multiplicative sum connectivity index, multiplicative atom bond connectivity index, multiplicative geometric-arithmetic index, nanotubes.

1. INTRODUCTION A molecular graph is a simple graph, representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. Thus the vertices of a molecular graph represent the carbon atoms and its edges the carbon-carbon bonds. Chemical graph theory is a branch of mathematical chemistry which has an important effect on the development of the chemical sciences. In Chemical Science, the physico-chemical properties of chemical compounds are often modeled by means of a molecular graph based structure descriptors, which are referred to as topological indices. In this paper, we consider finite simple undirected graphs. Let G be a graph with a vertex set V(G) and an edge set E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. We refer to1 for undefined term and notation. One of the best known and widely used topological index is the product connectivity index or Randić index introduced by Randić in2. The product connectivity index of a graph G is defined as 599

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

G

1

uvE G

dG u dG v

.

Motivated by the definition of the product connectivity index, the multiplicative product connectivity index, multiplicative sum connectivity index, multiplicative atom bond connectivity index and multiplicative geometric-arithmetic index were very recently proposed in3. They are defined as follows: The multiplicative product connectivity index of a graph G is defined as 1 II G . dG u dG v uvE G The multiplicative sum connectivity index of a graph G is defined as 1 XII G . dG u dG v uvE G The multiplicative atom bond connectivity index of a graph G is defined as ABCII G

dG u dG v 2

uvE G

dG u dG v

.

The multiplicative geometric-arithmetic index of a graph G is defined as GAII G

2 dG u dG v

. dG u dG v In3, Kulli introduced the general multiplicative geometric-arithmetic index of a graph G. This topological index is defined as follows: uvE G

a

2 dG u dG v . GA II G uvE G d G u d G v Recent development of molecular descriptors may be found in4, 5, 6, 7, 8, 9, 10, 11, 12. a

In this paper, we determine the multiplicative product connectivity, multiplicative sum connectivity, multiplicative atom bond connectivity, general multiplicative geometricarithmetic indices for TUC4C8 [m, n] and TUC4[m, n] nanotubes. 2. RESULTS FOR TUSC4C8 [S] Nanotubes We discuss TUSC4C8[S] nanotubes which are consisting of cycles C4 and C8. These nanotubes usually symbolized as TUC4C8 [m, n] for any m, n N, in which m is the number of octagons C8 in the first row and n is the number of octagons C8 in the first column. The 2dimensional lattice of TUC4C8 [m, n] is shown in Figure 1. We compute the multiplicative connectivity indices of TUC4C8[m, n] nanotubes. 600

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

2

........

3

m

........

........

........

2

........

m-1

........

1

........

Theorem 2.1. Let G = TUC4C8 [m, n] nanotubes. Then XII G 4 m 52m 6m6mn.

........

n

Figure 1. 2-D graph lattice of TUC4C8[m, n]

Proof: Let G = TUC4C8 [m, n]. By algebraic method, we get |V(G)| = 8mn + 4m and |E(G)| = 12mn + 4m. From Figure 1, it is easy to see that there are three partitions of the edge set of G as follows: E4 E4* uv E G / dG u dG v 2, E4 E4* 2m E5 E6* uv E G / dG u 2, dG v 3, E5 E6* 4m E6 E9* uv E G / dG u dG v 3, E6 E9* 12mn 2m.

We now determine XII(G), we see that 1 XII G dG u dG v uvE G

uvE4

1 dG u dG v 2m

uvE5

1 1 22 23 4 m 52m 6m6mn.

4m

1 dG u dG v

uvE6

1 dG u dG v

12 mn 2 m

1 33

Theorem 2.2. Let G = TUC4C8[m, n] nanotubes. Then II G 4 m 62m 9m6mn. Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have 1 II G dG u dG v uvE G 601

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

1

dG u dG v

uvE4*

1

dG u dG v

uvE6*

2m

1 1 2 2 23 4 m 62m 9m6mn.

4m

uvE9*

1 dG u dG v

12 mn 2 m

1 3 3

Theorem 2.3. Let G = TUC4C8 [m, n] nanotubes. Then 12 mn 6 m

1 2 . . 23m 3 Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have, ABCII G

ABCII G

dG u dG v 2 dG u dG v

uvE G

dG u dG v 2 dG u dG v

uvE4

222 2 2

2m

dG u dG v 2

dG u dG v

uvE5

232 2 3

4m

dG u dG v 2

uvE6

dG u dG v

12 mn 6 m

332 3 3

12 mn 6 m

1 2 3m . 2 3

.

Theorem 2.4. Let G = TUC4C8 [m, n]. Then 4 am

2 6 GA II G . 5 Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have a

2 dG u dG v GAa II G uvE G d G u d G v

a

a

a

2 dG u dG v 2 dG u dG v 2 dG u dG v uvE4 d G u d G v uvE5 d G u d G v uvE6 d G u d G v 2 2 2 22

a 2m

2 23 23

a 4m

2 3 3 33 602

a 12 mn 6 m

2 6 5

4 am

.

a

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

An immediate corollary is the multiplicative geometric-arithmetic index of TUC4C8 [m, n] nanotubes. 4m

2 6 Corollary 2.5. Let G = TUC4C8 [m, n] be the nanotubes. Then GAII G . 5

3. RESULTS FOR TUC4 [m, n] NANOTUBES In this section, we focus on the structures of a family of nanostructures and they are called TUHRC4(S) nanotubes. These nanotubes usually symbolized as TUC4 [m, n], for any m, nN, in which m is the number of cycles C4 in the first row and n is the number of cycles C4 in the first column as depicted in Figure 2. 1

2

3

4

m-1

.........

m

2

n

.........

Figure 2. 2-D graph lattice of G= TUC4[m, n]

We determine the multiplicative connectivity indices of TUC4[m, n] nanotubes. Theorem 3.1. Let G be the TUC4[m, n] nanotubes. Then XII G 62 m 8m2mn. Proof: Let G be the TUC4[m, n] nanotubes as depicted in Figure 2. By algebraic method, we get |E(G)| = 4mn + 2m. From Figure 2, it is easy to see that there are two partitions of the edge set of G as follows: E6 E8* uv E G | dG u 2, dG v 4, E6 E8* 4m. E8 E16* uv E G | dG u dG v 4, E8 E16* 4mn 2m.

We determine XII(G), we see that 1 XII G dG u dG v uvE G

uvE6

1 dG u dG v

uvE8

1 dG u dG v

603

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016) 4m

1 1 24 44 62m 8m2mn.

4 mn 2 m

Theorem 3.2. Let G be the TUC4[m, n] nanotubes. Then 8 mn 2 m

1 . Proof: Let G be the TUC4[m, n] nanotubes. By using the results from the proof of Theorem 3.1, we have 1 II G dG u dG v uvE G

II G 2

uvE8*

1 dG u dG v

1 2 4

4m

1

dG u dG v

* uvE16

1 4 4

4 mn 2 m

1 2

8 mn 2 m

.

Theorem 3.3. Let G be the TUC4[m, n] nanotubes. Then 2 mn m

2m

1 3 ABCII G . 2 8 Proof: Let G = TUC4[m, n]. By using the results from the proof of Theorem 3.1, we have

ABCII G

dG u dG v 2 dG u dG v

uvE G

uvE6

dG u dG v 2 dG u dG v

242 2 4

4m

uvE8

442 4 4

dG u dG v 2 dG u dG v

4 mn 2 m

1 2

2m

3 8

2 mn m

.

Theorem 3.4. Let G be the TUC4[m, n] nanotubes. Then 4 am

2 2 GA II G . 3 Proof: Let G = TUC4[m, n]. By using the results from the proof of Theorem 3.1, we have a

2 dG u dG v GAa II G uvE G d G u d G v

a

604

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016) a

2 dG u dG v 2 dG u dG v uvE6 d G u d G v uvE8 d G u d G v a 4m

a 4 mn 2 m

a

4 am

2 2 2 2 4 2 44 . 3 24 44 An immediate corollary is the multiplicative geometric-arithmetic index of TUC4 [m, n] nanotubes. 4m

2 2 Corollary 3.5. Let G be the TUC4[m, n] nanotubes. Then GAII G . 3

REFERENCES 1.

V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012). 2. M. Randić, On characterization of molecular branching, Journal of the American Chemical Society, 97(23), 6609-6615 (1975). 3. V.R. Kulli, Multiplicative connectivity indices of certain nanotubes, submitted. 4. M.Eliasi, A.Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68, 217-230 (2012). 5. I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka, 18, 1723 (2011). 6. V.R. Kulli, First multiplicative K Banhatti index and coindex of graphs, Annals of Pure and Applied Mathematics, 11(2), 79-82 (2016). 7. V.R. Kulli, Second multiplicative K Banhatti index and coindex of graphs, Journal of Computer and Mathematical Sciences, 7(5), 254-258 (2016). 8. V.R. Kulli, On K hyper-Banhatti indices and coindices of graphs, International Journal of Mathematical Archive, 7(6), 60-65 (2016). 9. V.R. Kulli, On multiplicative K Banhatti and multiplicative K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus, Annals of Pure and Applied Mathematics, 11(2), 145-150 (2016). 10. V.R. Kulli, Multiplicative hyper-Zagreb indices and coindices of graphs, International Journal of Pure Algebra, 6(7), 342-347 (2016). 11. H. Wang and H. Bao, A note on multiplicative sum Zagreb index, South Asian J. Math. 2(6), 578-583 (2012). 12. S. Wang and B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Applied Math. 180, 168-175 (2015).

605

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)

Multiplicative Connectivity Indices of TUC4C8[m, n] and TUC4[m, n] Nanotubes V. R. Kulli Department of Mathematics, Gulbarga University, Gulbarga 585 106, INDIA. email: [email protected] (Received on: November 12, 2016) ABSTRACT The topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. In this paper, we compute the multiplicative product connectivity, multiplicative sum connectivity, multiplicative atom bond connectivity, multiplicative geometric- arithmetic indices for TUC4C8[m, n] and TUC4[m, n] nanotubes. Mathematics Subject Classification: 05C05, 05C07 Keywords: multiplicative product connectivity index, multiplicative sum connectivity index, multiplicative atom bond connectivity index, multiplicative geometric-arithmetic index, nanotubes.

1. INTRODUCTION A molecular graph is a simple graph, representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. Thus the vertices of a molecular graph represent the carbon atoms and its edges the carbon-carbon bonds. Chemical graph theory is a branch of mathematical chemistry which has an important effect on the development of the chemical sciences. In Chemical Science, the physico-chemical properties of chemical compounds are often modeled by means of a molecular graph based structure descriptors, which are referred to as topological indices. In this paper, we consider finite simple undirected graphs. Let G be a graph with a vertex set V(G) and an edge set E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. We refer to1 for undefined term and notation. One of the best known and widely used topological index is the product connectivity index or Randić index introduced by Randić in2. The product connectivity index of a graph G is defined as 599

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

G

1

uvE G

dG u dG v

.

Motivated by the definition of the product connectivity index, the multiplicative product connectivity index, multiplicative sum connectivity index, multiplicative atom bond connectivity index and multiplicative geometric-arithmetic index were very recently proposed in3. They are defined as follows: The multiplicative product connectivity index of a graph G is defined as 1 II G . dG u dG v uvE G The multiplicative sum connectivity index of a graph G is defined as 1 XII G . dG u dG v uvE G The multiplicative atom bond connectivity index of a graph G is defined as ABCII G

dG u dG v 2

uvE G

dG u dG v

.

The multiplicative geometric-arithmetic index of a graph G is defined as GAII G

2 dG u dG v

. dG u dG v In3, Kulli introduced the general multiplicative geometric-arithmetic index of a graph G. This topological index is defined as follows: uvE G

a

2 dG u dG v . GA II G uvE G d G u d G v Recent development of molecular descriptors may be found in4, 5, 6, 7, 8, 9, 10, 11, 12. a

In this paper, we determine the multiplicative product connectivity, multiplicative sum connectivity, multiplicative atom bond connectivity, general multiplicative geometricarithmetic indices for TUC4C8 [m, n] and TUC4[m, n] nanotubes. 2. RESULTS FOR TUSC4C8 [S] Nanotubes We discuss TUSC4C8[S] nanotubes which are consisting of cycles C4 and C8. These nanotubes usually symbolized as TUC4C8 [m, n] for any m, n N, in which m is the number of octagons C8 in the first row and n is the number of octagons C8 in the first column. The 2dimensional lattice of TUC4C8 [m, n] is shown in Figure 1. We compute the multiplicative connectivity indices of TUC4C8[m, n] nanotubes. 600

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

2

........

3

m

........

........

........

2

........

m-1

........

1

........

Theorem 2.1. Let G = TUC4C8 [m, n] nanotubes. Then XII G 4 m 52m 6m6mn.

........

n

Figure 1. 2-D graph lattice of TUC4C8[m, n]

Proof: Let G = TUC4C8 [m, n]. By algebraic method, we get |V(G)| = 8mn + 4m and |E(G)| = 12mn + 4m. From Figure 1, it is easy to see that there are three partitions of the edge set of G as follows: E4 E4* uv E G / dG u dG v 2, E4 E4* 2m E5 E6* uv E G / dG u 2, dG v 3, E5 E6* 4m E6 E9* uv E G / dG u dG v 3, E6 E9* 12mn 2m.

We now determine XII(G), we see that 1 XII G dG u dG v uvE G

uvE4

1 dG u dG v 2m

uvE5

1 1 22 23 4 m 52m 6m6mn.

4m

1 dG u dG v

uvE6

1 dG u dG v

12 mn 2 m

1 33

Theorem 2.2. Let G = TUC4C8[m, n] nanotubes. Then II G 4 m 62m 9m6mn. Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have 1 II G dG u dG v uvE G 601

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

1

dG u dG v

uvE4*

1

dG u dG v

uvE6*

2m

1 1 2 2 23 4 m 62m 9m6mn.

4m

uvE9*

1 dG u dG v

12 mn 2 m

1 3 3

Theorem 2.3. Let G = TUC4C8 [m, n] nanotubes. Then 12 mn 6 m

1 2 . . 23m 3 Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have, ABCII G

ABCII G

dG u dG v 2 dG u dG v

uvE G

dG u dG v 2 dG u dG v

uvE4

222 2 2

2m

dG u dG v 2

dG u dG v

uvE5

232 2 3

4m

dG u dG v 2

uvE6

dG u dG v

12 mn 6 m

332 3 3

12 mn 6 m

1 2 3m . 2 3

.

Theorem 2.4. Let G = TUC4C8 [m, n]. Then 4 am

2 6 GA II G . 5 Proof: Let G = TUC4C8 [m, n] be the nanotubes. By using the results from the proof of Theorem 2.1, we have a

2 dG u dG v GAa II G uvE G d G u d G v

a

a

a

2 dG u dG v 2 dG u dG v 2 dG u dG v uvE4 d G u d G v uvE5 d G u d G v uvE6 d G u d G v 2 2 2 22

a 2m

2 23 23

a 4m

2 3 3 33 602

a 12 mn 6 m

2 6 5

4 am

.

a

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016)

An immediate corollary is the multiplicative geometric-arithmetic index of TUC4C8 [m, n] nanotubes. 4m

2 6 Corollary 2.5. Let G = TUC4C8 [m, n] be the nanotubes. Then GAII G . 5

3. RESULTS FOR TUC4 [m, n] NANOTUBES In this section, we focus on the structures of a family of nanostructures and they are called TUHRC4(S) nanotubes. These nanotubes usually symbolized as TUC4 [m, n], for any m, nN, in which m is the number of cycles C4 in the first row and n is the number of cycles C4 in the first column as depicted in Figure 2. 1

2

3

4

m-1

.........

m

2

n

.........

Figure 2. 2-D graph lattice of G= TUC4[m, n]

We determine the multiplicative connectivity indices of TUC4[m, n] nanotubes. Theorem 3.1. Let G be the TUC4[m, n] nanotubes. Then XII G 62 m 8m2mn. Proof: Let G be the TUC4[m, n] nanotubes as depicted in Figure 2. By algebraic method, we get |E(G)| = 4mn + 2m. From Figure 2, it is easy to see that there are two partitions of the edge set of G as follows: E6 E8* uv E G | dG u 2, dG v 4, E6 E8* 4m. E8 E16* uv E G | dG u dG v 4, E8 E16* 4mn 2m.

We determine XII(G), we see that 1 XII G dG u dG v uvE G

uvE6

1 dG u dG v

uvE8

1 dG u dG v

603

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016) 4m

1 1 24 44 62m 8m2mn.

4 mn 2 m

Theorem 3.2. Let G be the TUC4[m, n] nanotubes. Then 8 mn 2 m

1 . Proof: Let G be the TUC4[m, n] nanotubes. By using the results from the proof of Theorem 3.1, we have 1 II G dG u dG v uvE G

II G 2

uvE8*

1 dG u dG v

1 2 4

4m

1

dG u dG v

* uvE16

1 4 4

4 mn 2 m

1 2

8 mn 2 m

.

Theorem 3.3. Let G be the TUC4[m, n] nanotubes. Then 2 mn m

2m

1 3 ABCII G . 2 8 Proof: Let G = TUC4[m, n]. By using the results from the proof of Theorem 3.1, we have

ABCII G

dG u dG v 2 dG u dG v

uvE G

uvE6

dG u dG v 2 dG u dG v

242 2 4

4m

uvE8

442 4 4

dG u dG v 2 dG u dG v

4 mn 2 m

1 2

2m

3 8

2 mn m

.

Theorem 3.4. Let G be the TUC4[m, n] nanotubes. Then 4 am

2 2 GA II G . 3 Proof: Let G = TUC4[m, n]. By using the results from the proof of Theorem 3.1, we have a

2 dG u dG v GAa II G uvE G d G u d G v

a

604

V. R. Kulli, Comp. & Math. Sci. Vol.7 (11), 599-605 (2016) a

2 dG u dG v 2 dG u dG v uvE6 d G u d G v uvE8 d G u d G v a 4m

a 4 mn 2 m

a

4 am

2 2 2 2 4 2 44 . 3 24 44 An immediate corollary is the multiplicative geometric-arithmetic index of TUC4 [m, n] nanotubes. 4m

2 2 Corollary 3.5. Let G be the TUC4[m, n] nanotubes. Then GAII G . 3

REFERENCES 1.

V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012). 2. M. Randić, On characterization of molecular branching, Journal of the American Chemical Society, 97(23), 6609-6615 (1975). 3. V.R. Kulli, Multiplicative connectivity indices of certain nanotubes, submitted. 4. M.Eliasi, A.Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68, 217-230 (2012). 5. I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka, 18, 1723 (2011). 6. V.R. Kulli, First multiplicative K Banhatti index and coindex of graphs, Annals of Pure and Applied Mathematics, 11(2), 79-82 (2016). 7. V.R. Kulli, Second multiplicative K Banhatti index and coindex of graphs, Journal of Computer and Mathematical Sciences, 7(5), 254-258 (2016). 8. V.R. Kulli, On K hyper-Banhatti indices and coindices of graphs, International Journal of Mathematical Archive, 7(6), 60-65 (2016). 9. V.R. Kulli, On multiplicative K Banhatti and multiplicative K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus, Annals of Pure and Applied Mathematics, 11(2), 145-150 (2016). 10. V.R. Kulli, Multiplicative hyper-Zagreb indices and coindices of graphs, International Journal of Pure Algebra, 6(7), 342-347 (2016). 11. H. Wang and H. Bao, A note on multiplicative sum Zagreb index, South Asian J. Math. 2(6), 578-583 (2012). 12. S. Wang and B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Applied Math. 180, 168-175 (2015).

605