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.

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Figure 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)

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Figure 2. Dandelion graph D(17,8)

Theorem 2.2. Let be the line graph of the subdivision 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)

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Figure 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.

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Figure 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)

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Figure 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)

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Figure 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)

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Figure 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).

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Figure 8. Ladder graphs L₁, L₂, L₃, L₄ and L₅

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.

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