**Turkish Journal of Analysis and Number Theory**

## First Zagreb Index, F-index and F-coindex of the Line Subdivision Graphs

**S. Ghobadi**^{1,}, **M. Ghorbaninejad**^{2}

^{1}Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran

^{2}Department of Mathematics, Allame Tabarsi Institute, Qaemshahr, Iran

### Abstract

In this paper we investigate first Zagreb index, F-index and F-coindex of the line graph of some chemical graphs using the subdivision concept.

**Keywords:** chemical graphs, Zagreb index, F-index, F-coindex

Received October 31, 2016; Revised December 20, 2016; Accepted December 28, 2016

**Copyright**© 2017 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- S. Ghobadi, M. Ghorbaninejad. First Zagreb Index, F-index and F-coindex of the Line Subdivision Graphs.
*Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 1, 2017, pp 23-26. http://pubs.sciepub.com/tjant/5/1/5

- Ghobadi, S., and M. Ghorbaninejad. "First Zagreb Index, F-index and F-coindex of the Line Subdivision Graphs."
*Turkish Journal of Analysis and Number Theory*5.1 (2017): 23-26.

- Ghobadi, S. , & Ghorbaninejad, M. (2017). First Zagreb Index, F-index and F-coindex of the Line Subdivision Graphs.
*Turkish Journal of Analysis and Number Theory*,*5*(1), 23-26.

- Ghobadi, S., and M. Ghorbaninejad. "First Zagreb Index, F-index and F-coindex of the Line Subdivision Graphs."
*Turkish Journal of Analysis and Number Theory*5, no. 1 (2017): 23-26.

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### At a glance: Figures

### 1. Introduction

The line graph of a simple graph , denoted by , is the graph whose vertices correspond to the edges of such that two vertices of being adjacent if and only if the corresponding edges of share a common vertex [see ^{[2, 12]}]. The *subdivision graph * of a graph * *is obtained from * *by deleting every edge * *of* ** *and replacing it by a vertex * *of degree 2 that is joined to * *and * *[see p.151 of ^{[3]}]. If is the *subdivision graph *of a graph , then the line subdivision of * *is . Following ^{[16]}, we can construct the *Line Subdivision* of *a *graph* *, as follows:

(i) Replace each vertex by , the complete graph on vertices;

(ii) There is an edge joining a vertex of and a vertex of in * *if and only if there is an edge joining and in ;

(iii) For each vertex * *of , = .

A Hydrocarbon graph and its line subdivision is shown in Figure 1.

**Fig**

**ure**

**1.**Hydrocarbon graph

*G*and its Line subdivision

Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure-activity/property relationships. Topological indices correlate certain Physico-Chemical properties like boiling point, stability, strain energy etc of chemical compounds.

One of the oldest most popular and extremely studied topological indices are well–known Zagreb indices first introduced in 1972 by *Gutman *and* Trinajestic *^{[8]}.

Let be a simple graph and let and be its vertex and edge sets, respectively. The edge connecting the vertices and will be denoted by . The complement of the graph is the graph with vertex set , in which two vertices in are adjacent if and only if they are not adjacent in . The degree of the vertex , denoted by , is the number of first neighbors of in the underlying graph . Then the first and second Zagreb index are defined as

(1.1) |

and

respectively.

There is another expression for the first Zagreb index namely

(1.2) |

In 2008, bearing in mind Eq. (1.2), Doslic in ^{[6]} put forward the first Zagreb coindex, defined as

Recently, *Furtula* and *Gutman* ^{[7]} introduced a new topological index and named this index as forgotten topological index. They showed that the predictive ability of this index is almost similar to that of first Zagreb index. Throughout the present paper we name this index as F-index and denote it by , so

(1.3) |

There is another expression for the F-index namely

Similar to the first Zagreb coindex, the F-coindex of a graph is defined as

For more details on the topological indices and coindices we refer to the articles ^{[1, 4, 9, 10, 11, 13, 18]}.

In 2011, Ranjini et al. calculated the explicit expressions for the Shultz indices of the subdivision graphs of the Tadpole, Wheel, Helm and Ladder graphs ^{[15]}. They also studied the Zagreb indices of the line graphs of the Tadpole, Wheel and Ladder graphs with subdivision in ^{[14]}. In 2015, Su and Xu calculated the general Sum-connectivity indices and coindices of the line graphs of the Tadpole, Wheel and Ladder graphs with subdivision in ^{[17]}. In ^{[12]}, Nadeem et al. computed and indices of the line graphs of the Tadpole, Wheel and Ladder graphs by using the notion of subdivision.

In this paper we compute first Zagreb index, F-index and F-coindex of Dandelion graph , Comet graph , Fence graph , Closed fence graph , Friendship graph , t-fold bristled of and t-fold bristled of , Tadpole graph , Wheel and Ladder graph .

### 2. Main Results

We begin with a lemma used in the proof of our results.

**Lemma 2. 1.**** ****(**^{[5]}**)** *Let ** be a simple graph with ** vertices and ** edges. Then*

Let be a Dandelion graph with vertices consisted of a copy of the star and a copy of the path with vertices ,,…,, where is identified with a star center. (See Figure 2)

**Fig**

**ure**

**2.**Dandelion graph

*D*(17,8)

**Theorem 2.2. ***Let ** be the line graph of the subdivis**ion graph of a Dandelion graph**.** Then*

**Proof**. The number of vertices in are among which vertices are of degree , vertices are of degree 1 and the remaining vertices are of degree 2. Using Eqs.(1.1) and (1.3) and lemma 2.1 we have,

For a positive integer let be a Comet graph with vertices consisted of a copy of the Complete graph and a copy of the Path with vertices ,,…,, where is identified with a vertex from . (See Figure 3)

**Fig**

**ure**

**3.**Comet graph

*C*(7,3)

**Theorem 2.3.** *Let ** be the line graph of the subdivision graph of a Comet graph, then *

**Proof**. For the proof is easy, so we consider the case . The subdivision graph contains edges, so contain vertices among which, vertices are of degree , vertices are of degree , vertices are of degree 2 and one vertex is of degree 1. Now, using Eqs. (1.1) and (1.3) and lemma 2.1 we have the proof.

**Fig**

**ure**

**4.**Fence and Closed fence graph

**Theorem 2.4**. *Let ** be the line graph of the subdivision graph of a Fence graph ** (See Figure 4), then *

**Proof**. The number of vertices in are among which vertices are of degree 5 and 12 vertices are of degree 3. Thus, using Eqs. (1.1) and (1.3) and lemma 2.1 we have the proof.

A Friendship graph (or Dutch windmill graph) is a graph with *2m+1* vertices and *3m* edges constructed by joining *m* copies of the cycle graph with a common vertex.(See Figure 5)

**Fig**

**ure**

**5.**Friendship graph

*F*

_{m}

**Theorem 2.5**. *Let ** be the line graph of the subdivision graph of a Friendship graph** **, **then*

**Proof**. The number of vertices in are among which vertices are of degrees , and vertices are of degree 2. Hence, using Eqs. (1.1) and (1.3) and lemma 2.1 we can get the proof.

For a given graph , its t-fold bristled graph *Brst(G)* is obtained by attaching *t* vertices of degree one to each vertex of .(See Figure 6)

**Fig**

**ure**

**6.**t-fold bristled graph of

*P*

_{n}and

*C*

_{n}

**Theorem 2.6.** *Let ** be the line graph of the subdivision of a t-fold bristled graph of **, then*

**Proof**. The number of vertices of are , among which vertices are of degree one and vertices are of degree . Thus, using Eqs.(1.1) and (1.3) and lemma 2.1 we have the proof.

With reference to the above theorems, the proof of next theorems are easy, so we omit the proofs.

**Theorem 2.7.** *Let ** be the line graph of the subdivision of a Closed fence graph ** (See Figure 4), then*

**Theorem 2.8**. *Let ** be the line graph of the subdivision of a t-fold bristled graph of ** **(See Figure 6), then*

**Theorem 2.9.*** Let ** be the line graph of the subdivision graph of Wheel **, then*

A Tadpole graph is a special type of graph consisting of a Cycle graph with (at least 3) vertices and a Path graph with vertices, connected with a bridge. (See Figure 7)

**Fig**

**ure**

**7.**Tadpole graph

*T*

_{4,3}

**Theorem**** ****2.10**.* Let ** be the line graph of the subdivision graph of a Ttadpole graph **, then*

A Ladder graph is a graph obtained as the Cartesian Product of two path graphs, one of which has only one edge (See Figure 8).

**Fig**

**ure**

**8.**Ladder graphs

*L*

_{1},

*L*

_{2},

*L*

_{3},

*L*

_{4}and

*L*

_{5}

**Theorem**** ****2.11**.* Let ** be the line graph of the subdivision graph of a Ladder graph **, then*

### Acknowledgements

The authors thank the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper.

### References

[1] | Baskar Babujee, J. and Ramakrishnan, S. “Zagreb indices and coindices for compound graphs,” in: R. Nadarajan, R. S. Lekshmi, G. Sai Sundara Krishnan (Eds.), Computational and Mathematical Modeling, Narosa, New Delhi, pp. 357-362. (2012). | ||

In article | |||

[2] | Beineke, L.W.. “Characterizations of derived graphs,” Journal of combinatorial theory, 9. 129-135. (1970). | ||

In article | |||

[3] | Chartrand, G. and Zhang, P.. Introduction to graph theory, Mcgraw-Hill, Kalamazoo, MI, (2004). | ||

In article | PubMed | ||

[4] | K. C. Das, K. C., Gutman, I. and Horoldagva, B.. “Comparing Zagreb indices and coindices of trees,” MATCH Commun. Math. Comput. Chem. 68. 189-198. (2012). | ||

In article | |||

[5] | De, N., Abu Nayeem, Sk.Md. and Anita Pal. “The F-index of some graph operations,” springer plus 5:221. (2016). | ||

In article | |||

[6] | Doslić, T.. “Vertex-weighted Wiener polynomials for composite graphs,” Ars Math.Contemp. 1. 66-80. (2008). | ||

In article | |||

[7] | Furtula, B. and Gutman, I.. “A forgotten topological index,” J. Math. Chem. 53. 1184-1190. (2015). | ||

In article | View Article | ||

[8] | Gutman, I., Trinajstić, N.. “Graph theory and molecular orbitals, total π-electron energy of alternant hydrocarbons,” Chem. Phys. Lett. 17. 535-538. (1972). | ||

In article | View Article | ||

[9] | Hossein Zadeh, S., Hamzeh, A. and Ashrafi, A. R.” Extermal properties of Zagreb coindices and degree distance of graphs,” Miskolc Math. Notes 11. 129-138.( 2010). | ||

In article | |||

[10] | Hua, H. and Zhang, S. “Relations between Zagreb coindices and some distance-based topological indices,” MATCH Commun. Math. Comput. Chem. 68. 199-208. (2012). | ||

In article | |||

[11] | Kovijanić Vukičević, Z. and Popivoda, G. “Chemical trees with extreme values of Zagreb indices and coindices,” Iran. J. Math. Chem. 5(1). 19-29. (2014). | ||

In article | |||

[12] | Nadeem, M.F., Zafar, S. and Zahid, Z. “On certain topological indices of the line graph of subdivision graphs,” Appl. Math. Comput. (2015). | ||

In article | View Article | ||

[13] | Nikolić, S., Kovačević, G., Miličević, A. and Trinajstić, N. “The Zagreb indices 30 years after,” Croat. Chem. Acta 76. 113-124. (2003). | ||

In article | |||

[14] | Ranjini, P.S., Lokesha, V. and Cangl, I.N.. “On the Zagreb indices of the line graphs of the subdivision graphs,” Appl. Math. Comput. 218. 699–702. (2011). | ||

In article | View Article | ||

[15] | Ranjini, P.S., Lokesha, V. and Rajan, M.A.. “On the Shultz index of the subdivision graphs,” Adv. Stud. Contemp. Math. 21(3). 79-290. (2011). | ||

In article | |||

[16] | Shirai, T.. “Spectrum of Infinite Regular Line Graphs” Transactions of the American Mathematical Society. 352, Number 1. 115-132. (1999). | ||

In article | View Article | ||

[17] | Su, G. and Xu, L.. “Topological indices of the line graph of subdivision graphs and their Schur-bounds,” Appl. Math. Comput. 253. 395-401. (2015). | ||

In article | View Article | ||

[18] | Wang, M. and Hua, H. “More on Zagreb coindices of composite graphs,” Int. Math. Forum 7. 669-673. (2012). | ||

In article | |||

[19] | Whintney, H.. “Congruent Graphs and the connectivity of graphs,” American Journal of Mathematics. 54, 150-168. (1932). | ||

In article | View Article | ||