### 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 in^{1}.
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 research^{2,3}.

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 TUC_{4}C_{8}(S)[p,q] nanotubes, TUSC_{4}C_{8}(S)[p, q] nanotubes, C4C6C8[p, q] nanotorus and
TC_{4}C_{8}(S)[p, q] nanotorus. For more information about nanotubes and nanotorus see14.

**2. RESULTS FOR TUC _{4}C_{8}(S)[p,q] NANOTUBES**

We consider the graph of 2-D lattice of TUC_{4}C_{8}(S)[p,q] nanotube with p columns and
q rows. The graph of TUC_{4}C_{8}(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 TUC_{4}C_{8}(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 TUC_{4}C_{8}(S)[p,q] nanotube.

**3. RESULTS FOR TUSC _{4}C_{8}(S)[p, q] NANOTUBES**

We consider the graph of TUSC_{4}C_{8}(S)[p, q] nanotube with p columns and q rows. The
graph of TUSC_{4}C_{8}(S)[1,1] nanotube and the line graph of TUSC_{4}C_{8}(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 TUSC_{4}C_{8}(S)[p,q] nanotube. By calculation, we obtain that the
line graph of TUSC_{4}C_{8}(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 TUSC_{4}C_{8}(S)[p,q] nanotube.

**Corollary 2.2.**The edge version of the K-edge index of TUSC_{4}C_{8}(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 C_{4}C_{6}C_{8}[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 TC _{4}C_{8}(S)[p, q] NANOTORI**

We consider the graph of 2-D lattice of TC_{4}C_{8}(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.

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