Computation of Topological Indices of Dutch Windmill Graph

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.


Introduction
The Dutch windmill graph is denoted by ( )    u V G ∈ is denoted by u d and is the number of vertices that are adjacent to u.The edge connecting the vertices u and v is denoted by uv.Using these terminologies, certain topological indices are defined in the following manner.
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. [1] in late 1990's.Some upper bounds for the atom-bond connectivity index of     graphs can be found in [2], The atom-bond connectivity index of chemical bicyclic graphs and connected graphs can be seen in [3] [4].For further results on ABC index of trees, see the papers [5]- [8] and the references cited there in.
Definition 1.1.Let ( ) = be a molecular graph and u d is the degree of the vertex u, then ABC index of G is defined as, ( ) The fourth atom bond connectivity index, ( )

4
ABC G index was introduced by M. Ghorbani et al. [9] in 2010.Further studies on ( )

4
ABC G index can be found in [10] [11].Definition 1.2.Let G be a graph, then its fourth ABC index is defined as, ( ) , where u S is sum of the degrees of all neighbours of vertex u in G.In other words, ( ) The first and oldest degree based topological index was Randic index [12] denoted by ( ) . Sum connectivity index belongs to a family of Randic like indices.It was introduced by Zhou and Trinajstic [13].Further studies on Sum connectivity index can be found in [14] [15].Definition 1.4.For a simple connected graph G, its sum connectivity index ( ) S G is defined as, ( ) ( ) The Geometric-arithmetic index,

( )
GA G index of a graph G was introduced by D. Vukicevic et al. [16].Further studies on GA index can be found in [17]- [19].

Definition 1.5. Let G be a graph and e uv
= be an edge of G then, ( ) .
The fifth Geometric-arithmetic index, ( ) GA G was introduced by A.Graovac et al. [20] in 2011.
Definition 1.6.For a Graph G, the fifth Geometric-arithmetic index is defined as Where u S is the sum of the degrees of all neighbors of the vertex u in G, similarly for v S .

Main Results
Theorem 2.1.The Atom bond connectivity index of Dutch windmill graph is Proof.Consider the Dutch windmill graph ( ) m n D .We partition the edges of ( ) where uv is an edge.In ( ) we get edges of the type ( ) E are colored in red and black respectively as shown in the figure [18].The number of edges of these types are given in the Table 1.
We know that ( ) [From Table 1 and Figure 4] ( ) Proof.We know that ( ) ( )  i.e., ( ) Proof.We know that ( ) ( ) ( ) Proof.We know that ( ) ( )    E + are colored in red, green and black respectively as shown in the figure [1].The number of edges of these types are given in the Table 2.
We know that ( )    .The number of edges of these types are given in the Table 3.

Conclusion
The problem of finding the general formula for ABC index, 4 ABC index, Randic connectivity index, Sum connectivity index, GA index and 5 GA index of Dutch Windmill Graph is solved here analytically without using computers.
the graph obtained by taking m copies of the cycle n C with a vertex in common.The Dutch windmill graph is also called as friendship graph if 3 n = .i.e., friendship graph is the graph obtained by taking m copies of the cycle 3 C with a vertex in common.Dutch windmill graph mn edges as shown in the Figures 1-3.All graphs considered in this paper are finite, connected, loop less and without multiple edges.Let ( ) , G V E = be a graph with n vertices and m edges.The degree of a vertex ( )

Theorem 2 . 5 .
The fourth atom bond connectivity index of Dutch windmill graph is

E
mn edges.Let u d denote the degree of the vertex u.We partition the edges of ( ) where uv is an edge and u S is the sum of the degrees of all neighbours of vertex u in G.In other words,

Table 1 .
Edge partition based on degrees of end vertices of each edge. ()

Table 1 and
Figure 4]

Table 2 and
Figure 5]

Table 3 .
Edge partition based on degree sum of neighbors of end vertices of each edge.