### Edge Version of General Reformulated Zagreb Index of Certain Nanotubes and Nanotori

V. R. Kulli

Department of Mathematics, Gulbarga University, Gulbarga 585106, INDIA.
email: vrkulli @gmail.com
(Received on: November 11, Accepted: November 15, 2017)

ABSTRACT

Chemical graph theory is a branch of Mathematical Chemistry which has an important effect on the development of the Chemical Sciences. In this paper, we define the edge version of the reformulated first Zagreb index, the edge version of the K-edge index and the edge version of the general reformulated Zagreb index and compute exact formulas for certain nanotubes and nanotori.

Mathematics Subject Classification: 05C05, 05C12.

Keywords:reformulated Zagreb index, K-edge index, general reformulated Zagreb index, nanotubes, nanotori.

INTRODUCTION

Let G be a finite, simple, connected graph with vertex set V(G) and edge set E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. The degree of an edge e=uv in G is defined by dG(e) = dG(u) + dG(v) - 2. Any undefined term here may be found in1. A molecular graph is a finite, simple graph such that its vertices correspond to the atoms and edges to the bonds. There are several topological indices that have found some applications in Theoretical Chemistry in QSPR/QSAR research2,3.
In4, Milicevic et al., introduced the reformulated first Zagreb index of a graph G. It is defined as

This index was also studied, for example, in5.
In6, Kulli introduced the K-edge index of a graph G. It is defined as

This index was also studied, for example, in7,8.
In9, Kulli introduced the general reformulated Zagreb index of a graph G. It is defined as

where a is a real number. The line graph L(G) of a graph G is the graph whose vertex set corresponds to the edges of G such that two vertices of L(G) are adjacent if the corresponding edges of G are adjacent. Let dL(G)(e) denote the degree of a vertex e in L(G). We define the edge version of the reformulated first Zagreb index, the edge version of K-edge index and the edge version of the general reformulated first Zagreb index of a graph G as follows: The edge version of the reformulated first Zagreb index of a graph G is defined as

Many other edge version of indices were studied, for example, in10,11,12,13. In this paper, we compute the edge version of the reformulated first Zagreb index, edge version of the K-edge index and edge version of the general reformulated first Zagreb index of TUC4C8(S)[p,q] nanotubes, TUSC4C8(S)[p, q] nanotubes, C4C6C8[p, q] nanotorus and TC4C8(S)[p, q] nanotorus. For more information about nanotubes and nanotorus see14.

2. RESULTS FOR TUC4C8(S)[p,q] NANOTUBES

We consider the graph of 2-D lattice of TUC4C8(S)[p,q] nanotube with p columns and q rows. The graph of TUC4C8(S)[1,1] nanotube and the line graph of TUC4C6C8[1,1] are shown in Figure 1 (a) and Figure 1(b) respectively. Also the graph of 2-D lattice of TUC4C6C8 [4,5] is shown in Figure 1 (c).

Let G be the graph of TUC4C6C8[p,q] nanotube. By calculation, we obtain that the line graph of TUC4C8(S)[p,q] has 18pq - 4p edges. Also by calculation, we obtain that the edge set E(L(G)) can be divided into three partitions based on the sum of degrees of the end vertices of each edge as follows:

In the following theorem, we compute the edge version of the general reformulated first Zagreb index of TUC4C8(S)[p,q] nanotube.

3. RESULTS FOR TUSC4C8(S)[p, q] NANOTUBES

We consider the graph of TUSC4C8(S)[p, q] nanotube with p columns and q rows. The graph of TUSC4C8(S)[1,1] nanotube and the line graph of TUSC4C8(S)[1,1] nanotube are shown in Figure 2(a) and Figure 2(b) respectively. Also the graph of 2-D lattice of TUSC4C8 (S) [5,4] nanotube is shown in Figure 2(c).

Let G be the graph of TUSC4C8(S)[p,q] nanotube. By calculation, we obtain that the line graph of TUSC4C8(S)[p,q] has 24pq + 4p edges. Also by calculation, we obtain that the edge set E(L(G)) can be divided into three partitions based on the sum of degrees of the end vertices of each edge as follows:

In the following theorem, we compute the edge version of the general reformulated first Zagreb index of TUSC4C8(S)[p,q] nanotube.

Corollary 2.2.The edge version of the K-edge index of TUSC4C8(S)[p, q] nanotube is given by

Proof: Put a = 3 in equation (5), we get the desired result.

4. RESULTS FOR C4C6C8[p, q] NANOTORI

Consider the graph 2-D lattice of C4C6C8[p, q] nanotori with p columns and q rows. The graph of C4C6C8[2, 1] nanotori and the line graph of C4C6C8[2, 1] nanotori are shown in Figure 3(a) and Figure 3(b) respectively. Also the graph of 2-D lattice of C4C6C8[4,4] nanotori is shown in Figure 3(c).

Let G be the graph of C4C6C8 [p,q] nanotori. By calculation, we obtain that the line graph of C4C6C8 [p,q] nanotori has 18pq - 2p edges. Also by calculation, we obtain that the edge set E(L(G)) can be divided into four partitions based on the sum of degrees of the end vertices of each edge as follows:

In the following theorem, we compute the edge version of the general reformulated first Zagreb index of C4C6C8[p,q] nanotori.

5. RESULTS FOR TC4C8(S)[p, q] NANOTORI

We consider the graph of 2-D lattice of TC4C8(S)[p, q] nanotori with p columns and q rows. The graph of TC4C8(S)[1, 1] nanotori and the line graph of TC4C8(S)[1, 1] nanotori are shown in Figure 4(a) and Figure 4(b) respectively. Also the graph of 2-D lattice of TC4C8 (S) [5, 3] nanotori is shown in Figure 4(c).

Let G be the graph of TC4C8(S)[p,q] nanotori. By calculation, we obtain the line graph of TC4C8(S)[p,q] has 24pq - 4p edges. Also by calculation, we obtain that the edge set E(L(G)) can be divided into four partitions based on the sum of degrees of the end vertices of each edge as follows:

In the following theorem, we compute the edge version of the general reformulated first Zagreb index of TC4C8(S) [p,q] nanotori.

REFERENCES

1. V.R.Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012).
2. I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin (1986).
3. R.Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley- VCH, Weinheim, (2009).
4. A. Milicevic, S. Nikolic, and N. Trinajstic, On reformulated Zagreb indices, Mol. Divers, 8, 393-399 (2004).
5. V.R.Kulli, F-index and reformulated Zagreb index of certain nanostructures, International Research Journal of Pure Algebra, 7(1), 489-495 (2017).
6. V.R. Kulli, On K edge index and coindex of graphs, International Journal of Fuzzy Mathematical Archive, 10(2), 111-116 (2016).
7. V.R. Kulli, On K edge index of some nanostructures, Journal of Computer and Mathematical Sciences, 7(7), 373-378 (2016).
8. Algebra. 7(5), 748-753 (2017).
9. V.R. Kulli, Computation of general topological indices for titania nanotubes, International Journal of Mathematical Archive, 7(12), 33-38 (2016).
10. Mathematics, 14(3), 449-455 (2017).
11. DOI:http://dx.doi.org/apam.v14n3a11.
12. V.R Kulli, Edge version of multiplicative connectivity indices of some nanotubes and nanotorus, submitted.
13. V.R Kulli, Edge version of multiplicative atom bond connectivity index of certain nanotubes, submitted.
14. V.R Kulli, Edge version of augmented Zagreb indices of certain nanotubes, submitted.
15. M.N. Husin, R. Hasni, M. Imran and H. Kamarulhaili, The edge version of geometric-arithmetic index of nanotubes and nanotori, Optoelectron Adv. Mater.-Rapid Comm. 9 (9-10), 1292-1300 (2015).