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In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.

The Dutch windmill graph is denoted by

All graphs considered in this paper are finite, connected, loop less and without multiple edges. Let

Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariants.

The atom-bond connectivity index, ABC index was one of the degree-based molecular descripters, which was introduced by Estrada et al. [

graphs can be found in [

Definition 1.1. Let

of G is defined as,

The fourth atom bond connectivity index,

Definition 1.2. Let G be a graph, then its fourth ABC index is defined as,

where

for

The first and oldest degree based topological index was Randic index [

Definition 1.3. For the graph G Randic index is defined as,

Sum connectivity index belongs to a family of Randic like indices. It was introduced by Zhou and Trinajstic [

Definition 1.4. For a simple connected graph G, its sum connectivity index

The Geometric-arithmetic index,

Definition 1.5. Let G be a graph and

The fifth Geometric-arithmetic index,

Definition 1.6. For a Graph G, the fifth Geometric-arithmetic index is defined as

Where

Theorem 2.1. The Atom bond connectivity index of Dutch windmill graph is

Proof. Consider the Dutch windmill graph

We know that

i.e.,

[From

□Theorem 2.2. The Randic Index of Dutch windmill graph is

Edges of the type | Number of edges |
---|---|

(n − 2)m | |

2m |

i.e.,

Theorem 2.3. The Geometric-arithmetic index (GA) of Dutch windmill graph is

Proof. We know that

Theorem 2.4. The Sum connectivity index

Proof. We know that

i.e.,

Theorem 2.5. The fourth atom bond connectivity index of Dutch windmill graph is

Proof. Any Dutch windmill graph

Case (1) If

We know that

i.e.,

[From

Edges of the type | Number of edges |
---|---|

(n − 4)m | |

2m | |

2m |

Case (2) If

We know that

i.e.,

Theorem 2.6. The fifth Geometric-arithmetic index (

Proof. We know that

Case (1) If

2 and

Case (2) If

Edges of the type | Number of edges |
---|---|

m | |

2m |

[From

The problem of finding the general formula for ABC index,

The first author is also thankful to the University Grants Commission, Government of India for the financial support under the grant MRP(S)-0535/13-14/KAMY004/UGC-SWRO.

The authors declare that there are no conflicts of interests regarding the publication of this paper.

M. R. Rajesh Kanna,R. Pradeep Kumar,R. Jagadeesh, (2016) Computation of Topological Indices of Dutch Windmill Graph. Open Journal of Discrete Mathematics,06,74-81. doi: 10.4236/ojdm.2016.62007