## Abstract

Topological indices are the fixed numbers associated with the graphs. In recent years, mathematicians used indices to check the pharmacology characteristics and molecular behavior of medicines. In this article the first Zagreb connection number index is computed for the nanotubes *VC*_{5}*C*_{7}[ *p*, *q*] , *HC*_{5}*C*_{7}[ *p*,*q*] and Boron triangular Nanotubes. Also, the same index is computed for the Quadrilateral section

## 1 Introduction

The molecular structure of every chemical drug can also be represented by a graph by considering the atoms as vertices of the graph and the bond between atoms as edges of the graph. The history and the mathematical concepts for graph theory are discussed in [1, 2, 3]. These days, computation of topological indices for different chemical structures is an important task as discussed in [4, 5, 6, 7, 8, 9].

If we denote a graph by *G*(*V* (*G*), *E* (*G*)), then *V* (*G*) denoted the set of vertices of the graph *G* and *E*(*G*) denotes the set of edges of graph. The cardinality of *V* (*G*) and *E*(*G*) is called the order and size of the graph. The degree of vertex *v* ∈*V* (*G*) is the number of adjacent vertices to *v* and is denoted by *d*(*v*) . The distance between the two vertices *u*,*v*∈*V* (*G*) is denoted by *d*(*u*,*v*) and is defined to be the length of the shortest path between *u* and *v*.

A graph can be determined uniquely by the fixed numbers associated with it. Also, we can associate some numerical sequences with graphs. Similarly, some fixed numbers which are distance based and degree based can be associated with every graph. These invariants are called the topological indices.

Topological indices can also be associated with the molecular graphs of medicines. And with the help of these invariants, some pharmacology characteristics can be checked without using the laboratories and expensive materials.

Currently, many topological indices are being studied. The most studied among the degreed based topological indices are the first and second Zagreb indices. The first Zagreb index of graph *G* is denoted and defined by

And the second Zagreb index of graph *G* is denoted and defined by

Mathematical properties of first and second Zagreb indices are studied by Gutman el at. [10,11], Akhtar et al. [12] and Zhao et al. [13]. For further study of topological indices and related results see [14, 15, 16, 17, 18, 19, 20, 21].

Throughout the paper, all the graphs under study are finite and connected. For a vertex *v*∈*V* (*G*) , τ* _{v}* represents the number of vertices in the graph which are at the distance 2 from

*v*. τ

*is called the connection number of*

_{v}*v*. By using the connection number, a topological index named “the Modified first Zagreb connection index” is defined as

## 2 Motivation

According to the *International Academy of Mathematical Chemistry*, in order to identify whether any topological index is useful for prediction of chemical properties, the correlation between the values of that topological index for different octane isomers and parameter values related to certain physicochemical property of them should be considered. Generally octane isomers are convenient for such studies, because the number of the structural isomers of octane is large [24] enough to make the statistical conclusion reliable. Ali et al [22] checked the correlation ability of

## 3 Zagreb Connection Number Index for Nano-tubes

Since the discovery of *C*_{60}, carbon nanotubes as well as graphenes, the precursor of carbon fullerenes and carbon nanotubes have attracted wide attention due to their electronic properties and great potential applications[25,26]. There is a growing interest in exploring the structure and energetic of these pure carbon

clusters, carbon nanotubes with width on the nanometer scale, and carbon containing molecules because they are expected to have wide applications.

There are various types of nano-tubes which are under study. Some topological indices are found for nano-tubes [24]. The nano-tube *VC*_{5}*C*_{7}[ *p*, *q*],( *p*, *q* >1) is a net which is constructed by altering the *C*_{5} and *C*_{7} following the trivalent decoration. Using this type of tiling, we can cover a cylinder or a torus. Here *p* is a number of pentagons and *q* represents the number of periods. *q* is the number of periods in lattice.

In Figure 1, one period of *VC*_{5}*C*_{7} is shown. In one period there are 4 rows. In one period there are 16 *p* vertices. And there are 3 *p* vertices which are joined at the other end of the graph. Hence |*V* (*VC*_{5}*C*_{7} [*p*, *q*]) |⊨16 *pq* + 3*p* . In Figure 2 the graph of *VC*_{5}*C*_{7}[ *p*, *q*],( *p*, *q* >1) is shown when *p* = 3, *q* = 4.

## Theorem 3.1

Let *G*_{=}*VC*_{5}*C*_{7}[ *p*, *q*] , where *p*, *q* >1.

Then

Proof: For *G* =*VC*_{5}*C*_{7}[ *p*, *q*] , out of total 16 *pq* + 3*p* vertices, 3*p* vertices have degree 2, 3*p* have degree 1 and the remaining 16 *pq* ‒ 3*p* have degree 3. The connection number for the different vertices is given in next table

From the Table 1 and definition of modified first Zagreb connection index we have

After simplification

The nano-tube *HC*_{5}*C*_{7}[ *p*, *q*],( *p*, *q* >1) is a net of two dimensional lattice which is constructed by altering the *C*_{5} and *C*_{7} following the trivalent decoration. Using this type of tiling, we can cover a cylinder or a torus. Here *p* is a number of Heptagons and *q* represents the number of periods in lattice. In 2-dimensional lattice of *HC*_{5}*C*_{7}[ *p*,*q*] , *p* is the number of heptagons in one period.

In Figure 3, one period of *HC*_{5}*C*_{7} is shown for better understanding. One period consisting of 4 rows and 16 *p* vertices. 2 *p* vertices are joined at the other end of the graph. Hence, |*V* (*HC*_{5}*C*_{7} [ *p*, *q*]) |⊨16 *pq* + 2 *p* . In Figure 4, *HC*_{5}*C*_{7}[ *p*, *q*],( *p*, *q* >1) is given for *p* = *q* = 3.

## Theorem 3.2

Let *G* = *HC*_{5}*C*_{7}[ *p*, *q*],( *p*, *q* >1) . Then

Proof: For *HC*_{5}*C*_{7}[ *p*,*q*] , out of total number of vertices 16 *pq* + 2 *p* , the 2 *p* vertices have degree 1, 2 *p* vertices have degree 2 and the remaining 16 *pq* ‒ 2 *p* have degree 3. The connection number for different vertices is given in next table

Now by the Table 2 and definition of the modified first Zagreb connection index we have

After simplification we have

d ( v ) | τ_{v} | no. of vertices |
---|---|---|

2 | 4 | 3 p |

1 | 2 | 3 p |

3 | 4 | 6 p |

3 | 6 | 16 pq^{‒} 9 p |

d (v) | τ_{v} | No of vertices |
---|---|---|

1 | 2 | 2 p |

2 | 4 | 2 p |

3 | 4 | 5 p |

3 | 6 | 16 pq ‒7 p |

The Boron Nanotube was originally predicted by Boustani and Quandt et al. [27]. It was proposed that the most suitable structure of *C*_{20} is a double ring tabular structure, which can be considered as the embryo of single walled Boron Nanotubes. In 2004, Ciuparu et al. [28] successfully synthesized pure Boron single walled nanotubular structures with the diameter in the range of *3nm* and thus confirmed the existence of Boron Nanotubes.

A carbon hexagonal Nanotube of order *m* × *n* is a tube obtained from a carbon hexagonal sheet of order *m*× *n* by merging the vertices of last column with respective vertices of first column. (see Figure 5)

A Boron triangular Nanotube of order *m* × *n* is obtained from the hexagonal nanotubes of order *m* × *n* by adding a new vertex in the center of each hexagon and connecting it to all vertices of the hexagon (see Figure 6).

We denote the Boron nanotube of order *m*× *n* by *BT*[*m*,*n*] . Boron nanotubes have an odd number of rows and an even number of columns. This fact can be observed from the following theorem.

**Theorem** [29]:

A Boron triangular nanotube of order *m*× *n* has

In the next theorem, we will calculate the Zagreb connection index of Boron triangular nanotube.

## Theorem 3.3

Let *G* be the graph of Boron nanotube with m, n ≥ 2.. Then the Zagreb connection index is

**Proof**: It can be observed from the molecular graph of Boron nanotubes that the 3*m* vertices of the first

row and last row are of degree 4, while the remaining *m* have connection number 9 and

## 4 Zagreb Connection number index for regular Hexagonal Lattice

There are various types of lattices which are under study. Some topological indices are obtained for hexagonal lattices [30].

If *L* is a regular hexagonal lattice, let *m* × *n* quadrilateral section cut from *L* where *m*,*n* ≥ 2 . Here *m* represents the number of hexagons on the top and bottom sides and *n* is the number of hexagons on the lateral sides. If we identify the two lateral sides of

∀*i* =1, 2,...,*m*, we obtain the toroidal fullerence with *mn* hexagons. Shown in Figure 7.

If

## Theorem 4.1

If

Proof: Since the total vertices are 2*mn*. From the structure it is clear that the degree of all vertices is 3. The connection number for all the vertices is 6. Hence, by definition of the modified first Zagreb connection index

Let *L* be a regular hexagonal lattice. Let *L*.

Where *m* ≥1, *n* ≥≥ 2 . Here *n* represents the no of hexagons on the lateral sides. If we identify the top and bottom sides of *j* ==1, 2,...,*n* ‒‒1 to obtain the Klien bottle. Shown in Figure 8. If

## Theorem 4.2

Let

**Proof**: Since the total vertices are

## 5 Conclusion

Topological indices of molecular graphs are helpful to study the properties of drugs and manufacture the medicines. Therefore, they have been widely studied [1, 2, 3,5, 6, 10, 11, 12, 13, 14, 15, 29, 30, 31]. The main aim of this paper is to study the modified first Zagreb index of some famous nanotubes and quadrilateral section

## Acknowledgement

This work is supported by National Key Research and Development Program under Grant 2016YFB0800600.

The authors (Muhammad Imran Qureshi and Asfand Fahad) gratefully acknowledge ORIC, COMSATS University Islamabad, Pakistan for supporting this research under the grant of project number 16-52/CRGP/CIIT/VEH/17/1141.

**Ethical approval**: The conducted research is not related to either human or animal use.**Conflict of interest**: Authors declare no conflict of interest.

## Reference

[1] Gutman I., Polansky O. E., Mathematical Concepts in organic chemistry, Berlin: Springer, 1986.10.1007/978-3-642-70982-1Search in Google Scholar

[2] Goodman J. E., Orourke J., Handbook of Discrete and Computational Geometry, CRC Press LLC, 1997, 225-242.Search in Google Scholar

[3] Trinajstic N., Chemical graph theory, 2nd revised ed., Florida: CRC Press, Boca Raton, 1993.Search in Google Scholar

[4] Furtula B., Gutman I., A forgotten topological index, J. Math. Chem., 2015, 53(4), 1184-1190.10.1007/s10910-015-0480-zSearch in Google Scholar

[5] Goubko M., Minimizing degree-based topological indices for trees with given number of pendent vertices, MATCH Commun. Math. Comput. Chem, 2014, 71, 33-46.Search in Google Scholar

[6] Munir M., Nazeer W., Kang S. M., Qureshi M. I., Nizami A. R., Kwun Y. C., Some Invariants of Jahangir Graphs, Symmetry, 2017, 9(17), 1-15.10.3390/sym9010017Search in Google Scholar

[7] Gao W., Siddiqui M. K., Imran M., Jamil M. K., Farahani M. R., Forgotten topological index of chemical structure in drugs, Saudi Pharm. Journal, 2017, 25, 280-286.10.1016/j.jsps.2016.04.012Search in Google Scholar PubMed PubMed Central

[8] Bashir Y., Aslam A., Kamran M., Qureshi M. I., Jahangir A., Rafiq M., et al., On Forgotten Topological Indices of Some Dendrimers Structure, Molecules, 2017, 22(867), 1-8.10.3390/molecules22060867Search in Google Scholar PubMed PubMed Central

[9] Shao Z., Wu P., Zhang X., Dimitrov D., Liu J., On the maximum ABC index of graphs with prescribed size and without pendent vertices, IEEE Access, 2018, 6, 27604--27616.10.1109/ACCESS.2018.2831910Search in Google Scholar

[10] Gutman I., Das K. C., The first zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 2004, 50, 83-92.Search in Google Scholar

[11] Gutman I., Jamil M. K., Akhtar N., Graph with fixed number of pendent vertices and minimal first zagreb index, Transactions on Combinatorics, 2015, 4(1).Search in Google Scholar

[12] Akhtar N., Jamil M. K., Tomescu I., External first and second Zagreb indices of apex trees, U. P. B. Sci. Bull. Series A, 2016, 78(4), 221-230.Search in Google Scholar

[13] Zhao Q., Li S., Sharp biunds for the Zagreb indices of bicyclic graphs with k-pendent vertices, Discrete Appl. Math., 2010, 158, 1953-1962.10.1016/j.dam.2010.08.005Search in Google Scholar

[14] Aslam A., Jamil M. K., Gao W., Nazeer W., Topological Aspects of Some dendrimer structures, Nanotechnology Reviews, 2018, 7(2), 123-128.10.1515/ntrev-2017-0184Search in Google Scholar

[15] Aslam A., Ahmad S., Gao W., On Certain Topological Indices of Boron Triangular Nanotubes, Z. Naturforsch, 2017, 72(8), 711-716.10.1515/zna-2017-0135Search in Google Scholar

[16] Aslam A., Bashir Y., Ahmad S., Gao W., On Topological Indices of Certain Dendrimer Structures, Z. Naturforsch, 2017, 72(6), 559–566.10.1515/zna-2017-0081Search in Google Scholar

[17] Sardar M. S., Zafar S., Farahani M. R., The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series C a k, Open J. Math. Sci.,2017, 1(1), 44-51.10.30538/oms2017.0005Search in Google Scholar

[18] Noreen S., Mahmood A., Zagreb Polynomials and Redefined Zagreb Indices for the line graph of Carbon Nanocones, Open J. Math. Anal., 2018, 2(1), 67-76.10.30538/psrp-oma2018.0012Search in Google Scholar

[19] Gao W., Asif M., Nazeer W., The Study of Honey Comb Derived Network via Topological Indices. Open J. Math. Anal., 2018, 2(2), 10-26.10.30538/psrp-oma2018.0014Search in Google Scholar

[20] Aslam A., Guirao J. L. G., Ahmad S., Gao W., Topological indices of line graph of subdivision graph of complete bipartite graphs, Applied Mathematics and Information sciences, 2017, 11(6), 1631-1636.10.18576/amis/110610Search in Google Scholar

[21] De N., Hyper Zagreb Index of Bridge and Chain Grpahs, Open J. Math. Sci., 2018, 2(1), 1-17.10.30538/oms2018.0013Search in Google Scholar

[22] Ali A., Trinsjstic N., A novel/old modification of the first Zagreb index, arXiv:1705-10430v1.Search in Google Scholar

[23] Wang X. L., Liu J. B., Jamil M. K., Qureshi M. I., Fahad A., Farahani M. R., Zagreb Connection index of Drugs related chaemical structures, Turkish Journal of Medical Sciences (In Press).Search in Google Scholar

[24] Hayat S., Imran M., Computation of Certain Topological Indices of Nanotubes Covered by C5 and C7, J. Computational and Theoretical Nanoscience, 2015, 12, 1-9.10.1166/jctn.2015.3761Search in Google Scholar

[25] Lijima S., Helical Microtubules of Graphitic Carbon, Nature, 1991, 354, 56-58.10.1038/354056a0Search in Google Scholar

[26] Baughman R. H., Zakhidov A. A., De Heer W. A., Carbon Nanotubes--the Route toward Applications, Science, 2002, 297(5582), 787-792.10.1126/science.1060928Search in Google Scholar PubMed

[27] Boustani I., Quandt A., Hernandez E., Rubio A., New boron based nanostructured materials, The Journal of Chemical Physics, 1999, 110(3176).10.1063/1.477976Search in Google Scholar

[28] Ciuparu D., Klie R. F., Zhu Y., Pfefferle L., Synthesis of Pure Boron Single-Wall Nanotubes, The Journal of Physical Chemistry, 2004, 108(13), 3967-3969.10.1021/jp049301bSearch in Google Scholar

[29] Manuel P., Computational Aspects of Carbon and Boron Nanotubes, Molecules, 2010, 15(12), 8709-8722.10.3390/molecules15128709Search in Google Scholar PubMed PubMed Central

[30] Baca M., Horvathova J., Mokrisova M., Suhanyiova A., On topological indices of fullerenes, Applied Mathematics and Computation, 2015, 251, 154-161.10.1016/j.amc.2014.11.069Search in Google Scholar

[31] Shao Z., Wu P., Gao Y., Gutman I., Zhang X., On the maximum ABC index of graphs without pendent vertices, Applied Mathematics and computation, 2017, 315, 298-312.10.1016/j.amc.2017.07.075Search in Google Scholar

**Received:**2018-10-08

**Accepted:**2018-11-08

**Published Online:**2019-02-22

© 2019 Ansheng Ye et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.