Share This Article:

The Edge Version of Degree Based Topological Indices of p NAqp Nanotube

Abstract Full-Text HTML XML Download Download as PDF (Size:553KB) PP. 1445-1453
DOI: 10.4236/am.2017.810105    404 Downloads   752 Views  

ABSTRACT

Chemical graph theory is an important branch of mathematical chemistry which has wide range applications. In chemical graph theory a molecular graph can be recognized by a numerical quantity which is called a Topological index. Topological indices have some major classes but among these classes degree based topological indices have prominent role in chemical graph theory. In this paper we compute the edge version of some important degree based topological indices like Augmented Zagreb Index, Hyper-Zagreb Index, Harmonic Index and Sum-Connectivity Index of NAqp Nanotube.

1. Introduction and Preliminary Results

Graph theory is an important branch of mathematics which is started by Leonhard Euler’s as early as 1736. A hundred year before an important contribution of Kirchhoff had been made for the analysis of electrical networks. Several properties of special types of graphs known as trees were discovered by Cayley and Sylvester. The first book on graph theory was published in 1936 which received more attention. After the Second World War, further books were published on graph theory. Since then graph theory became one of the fastest expanding branches of mathematics.

Graph theory has many applications in engineering and science such as chemical, civil, electrical and mechanical engineering, architecture, management and control, communication, operational research, sparse matrix technology, combinatorial optimisation, physics, biology and computer science. But in chemistry graph theory has wide range applications; it has very important contributions in chemical documentation, structural chemistry, physical chemistry, inorganic chemistry, quantum chemistry, organic chemistry, chemical synthesis, polymer chemistry, medicinal chemistry, genomics, DNA studies and recent date proteomics.

Chemical graph theory is very important branch of mathematical chemistry. Its pioneers are Alexandru Balaban, Ante Graovac, Ivan Gutman, Haruo Hosoya, Milan Randić and Nenad Trinajstić. In chemical graph theory we use algebraic invariants to minimize the structure of a molecule into a single number which denotes the energy of molecule, structural fragments, molecular branching and electronic structures. Physical observations calculated by experiments are used to associate with these graphs theoretic invariants.

A graph G consists of a vertex set V ( G ) and edge set E ( G ) . Two vertices of G connected by an edge, are said to be adjacent. The Degree of a vertex v is the number of vertices adjacent with vertex v which is denoted by d e g ( v ) . The carbon-atom skeleton of an organic molecule is represented with a molecular graph. In molecular graph vertices represents the carbon atoms and edges represents the bonds between the carbon atoms.

A topological index is a structural descriptor which is derived from a molecular graph that represents an efficient way to express in a numerical form the molecular size, shape, cyclicity and branching. The topological indices of molecular graphs are widely used for establishing correlations between the structure of a molecular compound and its physico-chemical properties or biological activity. There are some major classes of topological indices such as distance based topological indices, degree based topological indices, eccentricity based topological indices and counting related polynomials and indices of graphs. Among these classes degree based topological indices are of great importance and play a vital role in chemical graph theory and particularly in chemistry [1] .

The concept of topological indices was given by Wiener while he was working on boiling point of paraffin, named this index as path number. Later on, the path number was renamed as Wiener index [2] .

Let G be a molecular graph. Then the Wiener index of G is defined as

W ( G ) = 1 2 ( u , v ) d ( u , v ) (1)

where ( u , v ) is any ordered pair of vertices in G and d ( u , v ) is the distance between the vertex u and vertex v.

The first degree based topological index is Randic index which was given by Milan Randic in 1975 in his paper On characterization of molecular branching [3] . The Randic index for a molecular graph G is defined as

R 1 2 ( G ) = u v E ( G ) 1 d e g ( u ) d e g ( v ) (2)

In this article we compute the edge version some important degree based topological indices which are Augmented Zagreb Index, Hyper-Zagreb Index, Harmonic Index and Sum-Connectivity Index of N A q p Nanotube. Now we define the edge versions of Augmented Zagreb Index, Hyper-Zagreb Index, Harmonic Index and Sum-Connectivity Index.

1.1. Edge Version of Augmented Zagreb Index

Furtula et al. [4] [5] modify the Atom bond connectivity index and named as Augmented Zagreb Index. The correlating ability among several topological indices possess by Augmented zagreb index. The edge version of Augmented Zagreb Index is defined as

e A Z I ( G ) = e f E ( L ( G ) ) ( d e g L ( G ) ( e ) d e g L ( G ) ( f ) d e g L ( G ) ( e ) + d e g L ( G ) ( f ) 2 ) 3 (3)

As compared to Atom bond connectivity index Augmented Zagreb Index has better correlation potential [6] .

1.2. Edge Version of Hyper-Zagreb Index

The Hyper-zagreb index was introduced by G.H Shirdel, H. Rezapour and A.M. Sayadi [7] which is basically a new version of Zagreb index. The edge version of Hyper-Zagreb Index is defined as

e H M ( G ) = e f E ( L ( G ) ) ( d e g L ( G ) ( e ) + d e g L ( G ) ( f ) ) 2 (4)

1.3. Edge Version of Harmonic Index

Zhang [8] [9] introduced this index in 2012 and called it Harmonic index. The edge version of Harmonic index is defined as [6]

e H ( G ) = e f E ( L ( G ) ) 2 d e g L ( G ) ( e ) + d e g L ( G ) ( f ) (5)

1.4. Edge Version of Sum-Connectivity Index

Bo Zhou and Nenad Trinajstic [10] introduced Sum-connectivity index. In the definition of Randić’s branching index they replaced the product d e g ( u ) × d e g ( v ) of vertex degrees with the sum d e g ( u ) + d e g ( v ) and get Sum-connectivity index. The edge version of Sum-connectivity index is defined as

e S C I ( G ) = e f E ( L ( G ) ) 1 d e g L ( G ) ( e ) + d e g L ( G ) ( f ) (6)

For future research and more history of these degree based topological indices “Augmented Zagreb, Hyper-Zagreb, Harmonic and Sum-Connectivity” readers can see the papers series [11] - [39] .

2. Main Results

2.1 N A q p Nanotube

Carbon nanotubes are the allotropic forms of carbon with a cylindrical nanostructures. Carbon nanotubes form an interesting class of carbon nonmaterial. There are three types of nanotubes namely, armchair, chiral and zigzag structures nanotubes. These carbon nanotubes shows remarkable mechanical properties. Experimental studies have shown that they belong to the stiffest and elastic known materials. Diudea was the first chemist who consider the problem of topological indices of nano-structures. In this paper we continue this program and compute some degree based topological indices of line graph of N A q p nanotube.

We consider the q × p quadrilateral section P q p with q 2 hexagons on the top and bottom sides and q 2 hexagons on the lateral sides cut from the regular hexagonal lattice L. If we identify two lateral sides of P q p such that we identify the vertices u 0 j and u q j , for j = 0 , 1 , 2 , , p then we obtain the nanotube N A q p shown in Figure 1 [40] . Throughout in Figure 2 we consider p = q > 2 .

We now compute the edge version of augmented zagreb index, hyper-zagreb index, harmonic index and sum-connectivity index of N A q p nanotube. Throughout this figure we consider p = q 2 . The line graph of N A q p nanotube has 6 p 2 + p + s + k 5 edges with degree vertices 2, 3 and 4. The first edge partition has s edges with d L ( G ) ( e ) = d L ( G ) ( f ) = 2 the second edge partition has 2 p + 2 edges with d L ( G ) ( e ) = 2 and d L ( G ) ( f ) = 3 , the third edge partition has k edges with d L ( G ) ( e ) = 2 and d L ( G ) ( f ) = 4 , the fourth edge partition has 4 p 6 edges with d L ( G ) ( e ) = d L ( G ) ( f ) = 3 , the fifth edge partition has 8 p 8 edges with d L ( G ) ( e ) = 3 and d L ( G ) ( f ) = 4 and the sixth edge partition has 6 p 2 13 p + 7 edges with d L ( G ) ( e ) = d L ( G ) ( f ) = 4 .

2.2. Edge Version of Augmented Zagreb Index, Hyper-Zagreb Index, Harmonic Index and Sum-Connectivity Index of N A q p Nanotube

Theorem 2.2.1. For every p = q 2 , consider the graph of G N A q p nano-

Figure 1. N A q p Nanotube.

Figure 2. N A q p and L ( N A q p ) for p = q = 3.

tube. Then the e A Z I ( G ) is equal to

e A Z I ( G ) = 3072 27 p 2 4015657 54000 p + 8 s + 8 k 3261061 108000

Proof. Let G be the graph of N A q p nanotube. Since from (3) we have

e A Z I ( G ) = e f E ( L ( G ) ) ( d e g L ( G ) ( e ) d e g L ( G ) ( f ) d e g L ( G ) ( e ) + d e g L ( G ) ( f ) 2 ) 3

By using edge partition in Table 1, we get

e A Z I ( G ) = s × ( 2 × 2 2 + 2 2 ) 3 + ( 2 p + 2 ) × ( 2 × 3 2 + 3 2 ) 3 + k × ( 2 × 4 2 + 4 2 ) 3 + ( 4 p 6 ) × ( 3 × 3 3 + 3 2 ) 3 + ( 8 p 8 ) × ( 3 × 4 3 + 4 2 ) 3 + ( 6 p 2 13 p + 7 ) × ( 4 × 4 4 + 4 2 ) 3

After an easy simplification, we obtain

e A Z I ( G ) = 3072 27 p 2 + ( 2916 64 + 13824 125 6656 27 + 16 ) p + 8 s + 8 k + 3584 27 2187 32 13824 125 + 16

After more simplification, we get

A e Z I ( G ) = 3072 27 p 2 4015657 54000 p + 8 s + 8 k 3261061 108000

Theorem 2.2.2. For every p = q 2 , consider the graph of G N A q p nanotube. Then the e H M ( G ) is equal to

e H M ( G ) = 384 p 2 246 p + 16 s + 36 k 110

Proof. Let G be the graph of N A q p nanotube. Since from (4) we have

e H M ( G ) = e f E ( L ( G ) ) ( d e g L ( G ) ( e ) + d e g L ( G ) ( f ) ) 2

By using edge partition in Table 1, we get

Table 1. Explanation of the terms present in Table 2.

Table 2. Edge partition of L ( N A q p ) based on degrees of end vertices of each edge.

e H M ( G ) = s × ( 2 + 2 ) ) 2 + ( 2 p + 2 ) × ( 2 + 3 ) 2 + k × ( 2 + 4 ) 2 + ( 4 p 6 ) × ( 3 + 3 ) 2 + ( 8 p 8 ) × ( 3 + 4 ) 2 + ( 6 p 2 13 p + 7 ) × ( 4 + 4 ) 2

After an easy simplification, we obtain

H e M ( G ) = 384 p 2 246 p + 16 s + 36 k 110

Theorem 2.2.3. For every p = q 2 , consider the graph of G N A q p nanotube. Then the e H ( G ) is equal to

e H ( G ) = 3 2 p 2 + 491 420 p + 1 2 s + 1 3 k 243 140

Proof. Let G be the graph of N A q p nanotube. Since from (5) we have

e H ( G ) = e f E ( L ( G ) ) 2 d e g L ( G ) ( e ) + d e g L ( G ) ( f )

By using edge partition in Table 1, we get

e H ( G ) = s × 2 2 + 2 + ( 2 p + 2 ) × 2 2 + 3 + k × 2 2 + 4 + ( 4 p 6 ) × 2 3 + 3 + ( 8 p 8 ) × 2 3 + 4 + ( 6 p 2 13 p + 7 ) × 2 4 + 4

After an easy simplification, we obtain

e H ( G ) = 3 2 p 2 + ( 4 5 + 4 3 + 16 7 13 4 ) p + 1 2 s + 1 3 k + 4 5 16 7 + 7 2 2

After more simplification, we get

H e ( G ) = 3 2 p 2 + 491 420 p + 1 2 s + 1 3 k 243 140

Theorem 2.2.4. For every p = q 2 , consider the graph of G N A q p nanotube. Then the e S C I ( G ) is equal to

e S C I ( G ) = 6 8 p 2 + ( 2 5 + 4 6 + 8 7 13 8 ) p + s 2 + k 6 + 2 5 8 7 + 7 8 6

Proof. Let G be the graph of N A q p nanotube. Since from (6) we have

e S C I ( G ) = e f E ( L ( G ) ) 1 d e g L ( G ) ( e ) + d e g L ( G ) ( f )

By using edge partition from Table 2, we get

e S C I ( G ) = s × 1 2 + 2 + ( 2 p + 2 ) × 1 2 + 3 + k × 1 2 + 4 + ( 4 p 6 ) × 1 3 + 3 + ( 8 p 8 ) × 1 3 + 4 + ( 6 p 2 13 p + 7 ) × 1 4 + 4

After doing some calculations, we get

S e C I ( G ) = 6 8 p 2 + ( 2 5 + 4 6 + 8 7 13 8 ) p + s 2 + k 6 + 2 5 8 7 + 7 8 6

3. Conclusion

In this paper, we have discussed the edge version of augmented zagreb index, hyper-zagreb index, harmonic index and sum-connectivity index. We have considered the line graph of N A q p nanotube and we have computed the edge version of augmented zagreb index, hyper-zagreb index, harmonic index and sum-connectivity index for N A q p nanotube.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Zhang, X. , Sajjad, W. , Baig, A. and Farahani, M. (2017) The Edge Version of Degree Based Topological Indices of p NAqp Nanotube. Applied Mathematics, 8, 1445-1453. doi: 10.4236/am.2017.810105.

References

[1] Raut, N.K. (2014) Degree Based Topological Indices of Isomers of Organic Compounds. International Journal of Scientific and Research Publications, 4, 1-4.
[2] Wiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20. https://doi.org/10.1021/ja01193a005
[3] Randic, M. (1975) On Characterization of Molecular Branching. Journal of the American Chemical Society, 97, 6609-6615. https://doi.org/10.1021/ja00856a001
[4] Ali, A., Raza, Z. and Bhatti, A. (2016) On the Augmented Zagreb Index. Kuwait Journal of Science, 43, 48-63.
[5] Furtula, B., Graovac, A. and Vukicevic, D. (2010) Augmented Zagreb Index. Journal of Mathematical Chemistry, 48, 370-380. https://doi.org/10.1007/s10910-010-9677-3
[6] Gutman, I. (2013) Degree Based Topological Indices. Croatica Chemica Acta, 86, 351-361. https://doi.org/10.5562/cca2294
[7] Shirdel, G.H., Rezapour, H. and Sayadi, A.M. (2013) The Hyper-Zagreb Index of Graph Operations. Iranian Journal of Mathematical Chemistry, 4, 213-220.
[8] Zhong, L. (2012) The Harmonic Index for Graphs. Applied Mathematics Letters, 25, 561-566. https://doi.org/10.1016/j.aml.2011.09.059
[9] Zhong, L. (2012) The Harmonic Index on Unicyclic Graphs. Ars Combinatoria, 104, 261-269. http://dblp.org/rec/journals/arscom/Zhong12
[10] Zhou, B. and Trinajstic, N. (2009) On a Novel Connectivity Index. Journal of Mathematical Chemistry, 46, 1252-1270. https://doi.org/10.1007/s10910-008-9515-z
[11] Balaban, A.T. (1985) Applications of Graph Theory in Chemistry. Journal of Chemical Information and Modeling, 25, 334-343. https://doi.org/10.1021/ci00047a033
[12] Basavanagoud, B. and Patil, S. (2016) A Note on Hyper-Zagreb index of Graph Operations. Iranian Journal of Mathematical Chemistry, 7, 89-92.
[13] Diudea, M.V., Gutman, I. and Lorentz, J. (2001) Molecular Topology. Nova, Huntington.
[14] Du, Z., Zhou, B. and Trinajstic, N. (2010) A Note on Generalized Sum-Connectivity Index. Applied Mathematics Letters, 24, 402-405. https://doi.org/10.1016/j.aml.2010.10.038
[15] Du, Z. and Zhou, B. (2012) On Sum-Connectivity Index of Bycyclic Graphs. Bulletin of the Malaysian Mathematical Sciences Society, 35, 101-117.
[16] Ebrahimi, M. and Alaeiyan, M. (2016) Topological Indices and Interpolation of Sequences. Indian Journal of Science and Technology, 9, 1-4.
[17] Essalih, M., Marraki, M. and Hagri, G. (2011) Calculation of Some Topological Indices of Graphs. Journal of Theoretical and Applied Information Technology, 30, 122-127.
[18] Farahani, M.R. (2013) The Edge Version of Atom Bond Connectivity Index of Connected Graph. Acta Universitatis Apulensis, 36, 277-284.
[19] Farahani, M.R. (2013) On the Randic and Sum-Connectivity Index of Nanotubes. Annals of West University of Timisoara-Mathematics and Computer Science, 51, 29-37.
https://doi.org/10.2478/awutm-2013-0014
[20] Farahani, M.R. (2014) The Second Connectivity and Second-Sum-Connectivity Indices of Armchair Polyhex Nanotubes TUAC6[m, n]. International Letters of Chemistry, Physics and Astronomy, 11, 74-80. https://doi.org/10.18052/www.scipress.com/ILCPA.30.74
[21] Farahani, M.R. (2015) The Hyper-Zagreb Index of TUSC4C8(S) Nanotubes. International Journal of Engineering and Technology Research, 3, 1-6.
[22] Farahani, M.R. (2015) Computing the Hyper-Zagreb Index of Hexagonal Nanotubes. Journal of Chemistry and Materials Research, 2, 16-18.
[23] Farahani, M.R. and Kanna, M.R. (2015) Generalized Zagreb Index of V-Phenylenic Nanotubes and Nanotori. Journal of Chemical and Pharmaceutical Research, 7, 241-245.
[24] Fajtlowicz, S. (1987) On Conjectures of Graffiti-II. Congr. Numer., 60, 187-197.
[25] Gutman, I.O. and Polansky, E. (1986) Mathematical Concepts in Organic Chemistry. Springer-Verlag, New York. https://doi.org/10.1007/978-3-642-70982-1
[26] Gutman, I. (2015) Edge-Decomposition of Topological Indices. Iranian Journal of Mathematical Chemistry, 6, 103-108.
[27] Hayat, S. and Imran, M. (2014) Computation of Certain Topological Indices of Nanotubes. Journal of Computational and Theoretical Nanoscience, 12, 1-7.
https://doi.org/10.1166/jctn.2015.3699
[28] Hayat, S. and Imran, M. (2015) Computation of Certain Topological Indices of Nanotubes Covered by C5 and C7. Journal of Computational and Theoretical Nanoscience, 12, 533-541. https://doi.org/10.1166/jctn.2015.3761
[29] Hayat, S. and Imran, M. (2014) Computation of Topological Indices of Certain Networks. Applied Mathematics and Computation, 240, 213-228. https://doi.org/10.1016/j.amc.2014.04.091
[30] Iranmanesh, A. and Pakravish, Y. (2007) Szeged Index of HAC5C6C7 [k, p] Nanotube. Journal of Applied Sciences, 7, 3606-3617.
[31] Ilic, A. Note on the Harmonic Index of a Graph, 1204.3313v1.
[32] Randic, M. (2003) Chemical Graph Theory-Facts and Fiction. Indian Journal of Chemistry A, 42, 1207-1218.
[33] Raut, N.K. and Ipper, S.N. (2015) Computing Some Topological Indices of Nanotubes. International Journal of Scientific and Research Publications, 5, 1-2.
http://www.ijsrp.org/research-paper-0815/ijsrp-p4454.pdf
[34] Shehnaz, A. and Imran, M. (2016) On Molecular Topological Properties of Benzenoid Structures. Canadian Journal of Chemistry, 94, 687-698. https://doi.org/10.1139/cjc-2016-0032
[35] Xing, R., Zhou, B. and Trinajstic, N. (2001) Sum-Connectivity Index of Molecular Trees. Journal of Mathematical Chemistry, 47, 583-591. https://doi.org/10.1007/s10910-010-9693-3
[36] Zhong, L. and Xu, K. (2014) Inequalities between Vertex Degree Based Topological Indices. MATCH Communications in Mathematical and in Computer Chemistry, 71, 627-642.
[37] Zhou, B. and Trinajstic, N. (2010) On General Sum-Connectivity Index. Journal of Mathematical Chemistry, 47, 210-218. https://doi.org/10.1007/s10910-009-9542-4
[38] Zhou, B. and Gutman, I. (2005) Further Properties of Zagreb Indices. MATCH Communications in Mathematical and in Computer Chemistry, 54, 233-239.
[39] Zhou, B. and Trinajstic, N. (2010) Minimum General Sum-Connectivity Index of Unicyclic Graphs. Journal of Mathematical Chemistry, 48, 697-703. https://doi.org/10.1007/s10910-010-9702-6
[40] Baca, M., Horváthová, J., Mokrisová, M. and Suhányiova, A. (2015) On Topological Indices of Fullerenes. Applied Mathematics and Computation, 251, 154-161.
https://doi.org/10.1016/j.amc.2014.11.069

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.