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Journal of Computer and Mathematical Sciences, Vol.8(1), 1-7, January 2017 (An International Research Journal), www.compmath-journal.org

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

Some Topological Indices of Certain Nanotubes V. R. Kulli Department of Mathematics, Gulbarga University, Gulbarga 585106, INDIA. email : [email protected] (Received on: January 17, 2017) 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, forgotten topological index, harmonic index and augmented Zagrab index of TUC4[m,n] and TUC4C8[m,n] nanotubes. Mathematics Subject Classification: 05C05, 05C07, 05C35. Keywords: modified Zagreb index, forgotten topological index, harmonic index, augmented Zagreb index, nanotube.

1. INTRODUCTION 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. 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, see2. The modified first and second Zagreb indices3 are respectively defined as m

M1  G  



1

2 uV  G  dG  u 

,

m

M 2 G  

1 . d   uvE  G  G u dG  v 



A vertex degree based topological index was defined in2 and that was shown to influence the total -electron energy (). Further, recently it was studied by Furtula and 1

V. R. Kulli, Comp. & Math. Sci. Vol.8 (1), 1-7 (2017)

Gutman in4. They named this index as forgotten topological index or F-index. This index is defined as F G  



uV  G 

dG  u 3 .

The harmonic index of a graph G is defined as H G  

2 . uvE  G  dG  u   dG  v 



This index was studied by Favaron et al.5 and Zhong6. 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

This index was introduced by Furtula et al. in7 and was studied in8. Many other topological indices were studied, for example, in9, 10, 11, 12, 13, 14, 15, 16, 17, 18 19, 20, 21. In this paper, the modified first and second Zagreb indices, F-index, harmonic index and augmented Zagreb index of TUC4[m, n] and TUC4C8[m, n] nanotubes are determined. 2. RESULTS FOR TUC4[m,n] NANOTUBES We consider TUHRC4(S) nanotubes, which is a family of nanostructures. These nanotubes usually symbolized as TUC4[m, n] for any m, n  N, 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 shown in Figure 1, see9,10. 1

2

3

4

.........

m-1

m

2

n

......... Figure 1

Let G be the TUC4[m, n] nanotubes, see Figure 1. By algebraic method, we get |V(G)| = 2m(n+1). From Figure 1, it is easy to see that there are two partitions of the vertex set of G as follows: V2 = {v  V(G) | dG(v) = 2}, |V2| = 2m. V4 = { v  V(G) | dG(v) = 4}, |V4| = 2mn. Also we obtain two partitions of the edge set of G as follows. E6 = {uv  E(G) | dG(u) = 2, dG(v) = 4}, |E6| = 4m E8 = {uv (G) | dG(u) = dG(v) = 4}, |E8| = 4mn – 2m. 2


We compute the modified first Zagreb index of TUC4[m,n] nanotubes. Theorem 2.1. Let G be the TUC4[m, n] nanotubes. Then m

1 1 M1  G   mn  m. 8 2

Proof: To determine mM1(G), we see that m

M1  G  



uV  G 

1 dG  u 

2



d

uV2

1 G

u 

2



d

uV4

 1   1   2m  2   2mn  2  2  4  G u  1

2

1 1  mn  m. 8 2

Theorem 2.2. Let G be the TUC4[m, n] nanotubes. Then F(G) = 128mn + 16m. Proof: To compute F(G), we see that F G  



uV  G 

dG  u 3 

 dG u 3   dG u 3  2m  23   2mn  43 

uV2

uV4

= 128 mn + 16 m. We now compute the modified second Zagreb index of G. Theorem 2.3. Let G be the TUC4[m, n] nanotubes. Then m

M 2 G  

1 3 mn  m. 4 8

Proof: To compute mM2(G), we see that m

M 2 G  

1 1 1     uvE  G  dG  u  dG  v  uvE6 dG  u  dG  v  uvE8 dG  u  dG  v 



1   1   1 mn  3 m.  4m     4mn  2m    8  2 4   4 4  4

Theorem 2.4. Let G be the TUC4[m, n] nanotubes. Then H(G) = mn +

5 m. 6

Proof: To compute H(G), we see that H G  

2 2 2     d    d   d    d   d    dG  v  G v G v uvE  G  G u uvE6 G u uvE8 G u



2   2   mn  5 m.  4m     4mn  2m    6  24  44

Now we compute augmented Zagreb index of TUC4[m,n] nanotubes. 3


Theorem 2.5. Let G be the TUC4[m,n] nanotubes. Then AZI  G  

2048 160 mn  m. 27 27

Proof: To compute AZI(G), we see that 3

AZI  G  

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E8  G  uvE  G   G uvE6  G 3

3

3

3

8  2 4   4 4   4m     4mn  2m     4m  23    4mn  2m     242  442 3 2048 160  mn  m. 27 27

3. RESULTS FOR TUC4C8[m,n] NANOTUBES

2

........

3

m

........

........

........

2

........

m-1

........

1

........

We consider TUC4C8(S) nanotubes which is a family of nanostructures. These structures are made up 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 as shown in Figure 2, see9,10.

........

n

Figure 2

Let G be the TUC4C8[m, n] nanotubes, see Figure 2. By algebraic method, we get |V(G) | = 8mn + 4m. From Figure 2, it is easy to see that there are two partitions of the vertex set of G as follows: V2 = {v  V(G) | dG(V) = 2}, |V2| = 4m V3 = {v  V(G) | dG(V) = 3}, |V3| = 8mn Also we obtain three partitions of the edge set of G as follows: E4 = {uv  E (G)| dG(u) = dG(v) = 2}, |E4| = 2m. E5 = {uv  E (G)| dG(u) = 2, dG(v) = 3}, |E5| = 4m. E6 = {uv  E(G)| dG(u) = dG(v) =3, |E6| = 12mn – 2m. We compute the modified first Zagreb index of TUC4C8[m, n] nanotubes. 4


Theorem 3.1. Let G be the TUC4C8[m, n] nanotubes. Then m

8 M1  G   mn  m. 9

Proof: To compute mM1(G), we see that m

M1  G  

1



uV  G 

dG  u 

2



d

uV2

1 G

u 

2



d

uV3

 1 8  1   4m  2   8mn  2   mn  m . 2  3  9 G u  1

2

Theorem 3.2. Let G be the TUC4C8[m,n] nanotubes. Then F(G) = 216mn + 32m. Proof: To compute F(G), we see that F G  



uV  G 

dG  u 3 

 dG u 3   dG u 3  4m  23   8mn 33 

uV2

uV3

= 216 mn + 32m. We determine the modified second Zagreb index of TUC4C8[m, n] nanotubes. Theorem 3.3. Let G be the TUC4C8[m,n] nanotubes. Then m

M 2 G  

4 17 mn  m . 3 18

Proof: To determine mM2(G), we see that m

M 2 G  

1 1 1 1       uvE  G  dG  u  dG  v  uvE4 dG  u  dG  v  uvE5 dG  u  dG  v  uvE6 dG  u  dG  v 



17 1   1   1  4  2m    4m    12mn  2m     mn  m. 18  2 2   23   3 3  3

Also we determine the harmonic index of TUC4C8[m,n] nanotubes. Theorem 3.4. Let G be the TUC4C8[m,n] nanotubes. Then H(G) = 4mn +

29 m. 15

Proof: To determine H(G), we see that H G  

2 2 2 2       d    d   d    d   d    d   d    dG  v  G v G v G v uvE  G  G u uvE4 G u uvE5 G u uvE6 G u



29 2   2   2   2m    4m    12mn  2m     4mn  m. 15  22  23  33

Now we determine the augmented Zagreb index of TUC4C8[m,n] nanotubes. 5


Theorem 3.5. Let G be the TUC4C8[m,n] nanotubes. Then AZI  G  

2187 807 mn  m. 16 32

Proof: 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E5  G  uvE6  G  uvE4  G 3

3

3

3

 23   3 3   2 2   2m    4m    12mn  2m     222  23 2   33 2  2187 807  mn  m. 16 32

REFERENCES 1. V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012). 2. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535-538 (1972). 3. S. Nikolić, G. Kovaćević, A. Milićević and N. Trinajstić, The Zagreb indices 30 years after, Croatica Chemica Acta, 76(2), 113-124 (2003). 4. B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53, 1184-1190 (2015). 5. O. Favaron, M. Maho and J.F. Sacle, Some eigenvalue properties in graphs (conjectures of Graffiti II), Discrete Math. 111(1-3), 197-220 (1993). 6. L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25(3), 561-566 (2012). 7. B. Furtula, A. Graovac and D.Vukičević, Augmented Zagreb index, J. Math. Chem 48, 370-380 (2010). 8. Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67, 483-494 (2012). 9. V.R. Kulli, General multiplicative Zagreb indices of TUC4C8[m,n] and TUC4[m,n] nanotubes, International Journal of Mathematical Archive, 11(1), 39-43 (2016). 10. V.R. Kulli, Multiplicative connectivity indices of TUC4C8[m,n] and TUC4[m,n] nanotubes, Journal of Computer and Mathematical Sciences, 7(11), 599-605 (2016). 11. V.R. Kulli, The first and second 𝜅a indices and coindices of graphs, International Journal of Mathematical Archive, 7(5), 71-77 (2016). 12. V.R. Kulli, On K edge index and coindex of graphs, International Journal of Fuzzy Mathematical Archive, 10(2), 111-116 (2016). 6


13. V.R. Kulli, On K edge index of some nanostructures, Journal of Computer and Mathematical Sciences, 7(7), 373-378 (2016). 14. 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), 342-347 (2016). 15. V.R. Kulli, On multiplicative connectivity indices of certain nanotubes, Annals of Pure and Applied Mathematics, 12(2), 169-176 (2016). 16. V.R.Kulli, Multiplicative connectivity indices of nanostructures, Journal of Ultra Scientist of Physical Sciencs- A 29(1) 1-10 (2017). DOI:http://dx.doi.org/10.22147/jusps-A/290101. 17. V.R.Kulli, Computation of some topological indices of certain networks, submitted. 18. V.R.Kulli, Computation of general topological indices for titania nanotubes, International Journal of Mathematical Archive, (2016). 19. V.R.Kulli, F-index and reformulated Zagreb index of certain nanostructures, submitted. 20. V.R.Kulli, Some new multiplicative geometric-arithmetic indices, Journal of Ultra Scientist of Physical Sciencs, A, 29(2), 52-57 (2017). DOI:http://dx.doi.org/10.22147/jusps-A/290201. 21. V.R.Kulli, Two new multiplicative atom bond connectivity indices, Annals of Pure and Applied Mathematics,13(1), 1-7 (2017). DOI:http://dx.doi.org/10.22457/apam.v13n1a1.

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