Abstract 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

M R R Kumar

R P Jagadeesh

CiteSeerX

Open Journal of Discrete Mathematics, 2016, 6, 74-81 
Published Online April 2016 in SciRes. http://www.scirp.org/journal/ojdm 
http://dx.doi.org/10.4236/ojdm.2016.62007  
How to cite this paper: Kanna, M.R.R., Kumar, R.P. and Jagadeesh, R. (2016) Computation of Topological Indices of Dutch 
Windmill Graph. Open Journal of Discrete Mathematics, 6, 74-81. http://dx.doi.org/10.4236/ojdm.2016.62007  
 
 
Computation of Topological Indices of  
Dutch Windmill Graph 
M. R. Rajesh Kanna1, R. Pradeep Kumar2, R. Jagadeesh3 
1Post Graduate Department of Mathematics, Maharani’s Science College for Women,  
Mysore, India 
2Department of Mathematics, The National Institute of Engineering, Mysore, India 
3Research and Development Centre, Bharathiar University, Coimbatore, India 
  
 
Received 11 January 2016; accepted 5 April 2016; published 8 April 2016 
 
Copyright © 2016 by authors and Scientific Research Publishing Inc. 
This work is licensed under the Creative Commons Attribution International License (CC BY). 
http://creativecommons.org/licenses/by/4.0/ 
  
 
 
 
Abstract 
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. 
 
Keywords 
ABC Index, ABC4 Index, Sum Connectivity Index, Randic Connectivity Index, GA Index, GA5 Index 
 
 
1. Introduction 
The Dutch windmill graph is denoted by ( )mnD  and it is the graph obtained by taking m copies of the cycle nC  
with a vertex in common. The Dutch windmill graph is also called as friendship graph if 3n = . i.e., friendship 
graph is the graph obtained by taking m copies of the cycle 3C  with a vertex in common. Dutch windmill graph ( )m
nD  contains ( )1 1n m− +  vertices and 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 ( )u V G∈  is denoted by ud  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  
M. R. R. Kanna et al. 
 
 
75 
 
Figure 1. (4)3D .                               
 
 
Figure 2. (5)5D .                             
 
 
Figure 3. (5)4D .                             
 
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 ( ),G V E=  be a molecular graph and ud  is the degree of the vertex u, then ABC index  
of G is defined as, ( ) 2u vuv E
u v
d dABC G
d d∈
+ −
= ∑ .  
The fourth atom bond connectivity index, ( )4ABC G  index was introduced by M. Ghorbani et al. [9] in 
2010. Further studies on ( )4ABC G  index can be found in [10] [11]. 
Definition 1.2. Let G be a graph, then its fourth ABC index is defined as, ( ) ( )4
2u v
uv E G
u v
S SABC G
S S∈
+ −
= ∑ ,  
where uS  is sum of the degrees of all neighbours of vertex u in G. In other words, ( )u vuv E GS d∈= ∑ , Similarly  
for vS .  
M. R. R. Kanna et al. 
 
 
76 
The first and oldest degree based topological index was Randic index [12] denoted by ( )Gχ  and was 
introduced by Milan Randic in 1975. 
Definition 1.3. For the graph G Randic index is defined as, ( ) ( )
1
uv E G
u v
G
d d
χ
∈
= ∑ .  
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,  
( ) ( )
1
uv E G
u v
S G
d d∈
=
+
∑ .  
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, ( ) ( )
2 u v
uv E G
u v
d d
GA G
d d∈
= ∑ .  
The fifth Geometric-arithmetic index, ( )5GA 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 ( ) ( )5
2 u v
uv E G
u v
S S
GA G
S S∈
=
+∑ ,  
Where uS  is the sum of the degrees of all neighbors of the vertex u in G, similarly for vS .  
2. Main Results 
Theorem 2.1. The Atom bond connectivity index of Dutch windmill graph is ( )( )
2
m
n
mnABC D = .  
Proof. Consider the Dutch windmill graph ( )mnD . We partition the edges of 
( )m
nD  into edges of the type 
( ),u vd dE  where uv is an edge. In 
( )m
nD  we get edges of the type ( )2,2E  and ( )2 ,2nE . Edges of the type ( )2,2E  
and ( )2 ,2nE  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 ( ) ( )
2u v
uv E G
u v
d dABC G
d d∈
+ −
= ∑  
i.e., ( )( ) ( ) ( )( ) ( ) ( )( )2,2 2 ,22,2 2 ,2
2 2
m
m u v u v
n muv E G uv E G
u v u v
d d d dABC D E E
d d d d∈ ∈
+ − + −
= +∑ ∑  
( )( ) ( ) 2 2 2 2 2 22 22 2 2 2
m
n
mABC D n m m
m
+ − + −
= − +
⋅ ⋅
 
[From Table 1 and Figure 4] 
( ) 1 12 2 .
2 2 2
mnn m m= − + =                                    □ 
Theorem 2.2. The Randic Index of Dutch windmill graph is ( )( ) ( )2 22
m
n
n m m
Dχ
− +
=   
Proof. We know that ( ) ( )
1
uv E G
u v
G
d d
χ
∈
= ∑  
 
Table 1. Edge partition based on degrees of end vertices of each edge.                                                         
Edges of the type ( ),u vd dE  Number of edges 
( )2,2E  (n − 2)m 
( )2 ,2mE  2m 
M. R. R. Kanna et al. 
 
 
77 
 
Figure 4. ( )mnD .                             
 
i.e., ( )( ) ( ) ( )( ) ( ) ( )( )2,2 2 ,22,2 2 ,2
1 1
m
m
n muv E G uv E G
u v u v
D E E
d d d d
χ
∈ ∈
= +∑ ∑  
( ) 1 12 2
2 2 2 2
n m m
m
= − +
⋅ ⋅
 [From Table 1 and Figure 4] 
( )2 2
2
n m m− +
= .                                                                      □ 
Theorem 2.3. The Geometric-arithmetic index (GA) of Dutch windmill graph is  
( )( ) ( )
2 2 4
1
m
n
m mn m n m
GA D
m
− + − +
=
+
.  
Proof. We know that ( ) ( ) ( )
2 u v
uv E G
u v
d d
GA G
d d∈
=
+∑  
( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )2,2 2 ,22,2 2 ,2
2 2
m
m u v u v
n muv E G uv E G
u v u v
d d d d
GA D E E
d d d d∈ ∈
= +
+ +∑ ∑   
( ) 2 2 2 2 2 22 2
2 2 2 2
mn m m
m
⋅ ⋅
= − +
+ +
 [From Table 1 and Figure 4] 
( )2 2 4
1
m mn m n m
m
− + − +
=
+
.                                                              □ 
Theorem 2.4. The Sum connectivity index ( )S G  of Dutch windmill graph is ( ) ( )2 2
2 1
n m mS G
m
−
= +
+
.  
Proof. We know that ( ) ( )
1
uv E G
u v
S G
d d∈
=
+
∑  
i.e., ( )( ) ( ) ( )( ) ( ) ( )( )2,2 2 ,22,2 2 ,2
1 1
m
m
n muv E G uv E G
u v u v
S D E E
d d d d∈ ∈
= +
+ +
∑ ∑   
( ) 1 12 2
2 2 2 2
n m m
m
= − +
+ +
 [From Table 1 and Figure 4] 
( )2 2
2 1
n m m
m
−
= +
+
.                                                                    □ 
Theorem 2.5. The fourth atom bond connectivity index of Dutch windmill graph is  
M. R. R. Kanna et al. 
 
 
78 
( )( )
( ) ( )
( )
4
4 6 1 2 3 if 4
1
2 1 3 if 3
2 11
n
m
m n m m n
m
ABC D
m m n
mm
  − + + + + ≥  +=   + + =  ++   
 
Proof. Any Dutch windmill graph ( )mnD  contains ( )1 1n m− +  vertices and mn edges. Let ud  denote the 
degree of the vertex u. We partition the edges of ( )mnD  into edges of the type ( )
*
,u vS S
E  where uv is an edge and 
uS  is the sum of the degrees of all neighbours of vertex u in G. In other words, ( )u vuv E GS d∈= ∑ , Similarly for 
vS . 
Case (1) If 4n ≥ : In ( )mnD  we get edges of the type ( )
*
4,4E , ( )
*
4,2 2mE +  and ( )
*
2 2,4m mE + . Edges of the type 
( )
*
4,4E , ( )
*
4,2 2mE +  and ( )
*
2 2,4m mE +  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 ( ) ( )4
2u v
uv E G
u v
S SABC G
S S∈
+ −
= ∑  
i.e., 
( )( ) ( ) ( )( ) ( ) ( )( )
( ) ( )( )
* *
4,4 4,2 2
*
2 2,4
* *
4 4,4 4,2 2
*
2 2,4
2 2
2
m
m m
m u v u v
n muv E G uv E G
u v u v
u v
m m uv E G
u v
S S S SABC D E E
S S S S
S SE
S S
+
+
+∈ ∈
+ ∈
+ − + −
= +
+ −
+
∑ ∑
∑
 
[From Table 2 and Figure 5] 
( ) ( ) ( )
4 4 2 4 2 2 2 2 2 4 24 2 2
4 4 4 2 2 4 2 2
m m mn m m m
m m m
+ − + + − + + −
= − + +
⋅ + +
 
( )4 6 2 3
4 1 1
n m mm m
m m
− +
= + +
+ +
  
 
 
Figure 5. ( )mnD .                                                         
 
Table 2. Edge partition based on degree sum of neighbors of end vertices of each edge.                                                         
Edges of the type Number of edges 
( )
*
4,4E  (n − 4)m 
( )
*
4,2 2mE +
 
2m 
( )
*
2 2,4m mE +  2m 
M. R. R. Kanna et al. 
 
 
79 
( ) ( )4 6 1 2 3 .
1
m n m m
m
 = − + + + + +
 
Case (2) If 3n = : In ( )3
mD  we get edges of the type ( )
*
2 2,2 2m mE + +  and ( )
*
2 2,4m mE + . The number of edges of 
these types are given in the Table 3. 
We know that ( ) ( )4
2u v
uv E G
u v
S SABC G
S S∈
+ −
= ∑  
i.e., 
( )( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( )
( )
( ) ( )
*
2 2,2 2
*
2 2,4
*
4 2 2,2 2
*
2 2,4
2
2
2 2 2 2 2 2 2 4 22
2 2
   
2
                   
2 2 2 4
2 2 1 3 2 1 3 .
2 1 1 2 11
          
m m
m m
m u v
n m m uv E G
u v
u v
m m uv E G
u v
S S
ABC D E
S S
S S
E
S S
m m m mm m
m m m m
m m mm m
m m mm
+ +
+
+ + ∈
+ ∈
+ −
=
+ −
+
+ + + − + + −
= +
+ + +
 + +
= + = + 
+ + ++   
∑
∑

  
Theorem 2.6. The fifth Geometric-arithmetic index ( 5GA ) of Dutch windmill graph is  
( )( )
( ) ( ) ( )
( )5
4 2 1 2 2 1
4 if 4
3 3 1
4 2 1
1 if 3
3 1
m
n
m m m m m
n m n
m
GA D
m m
m n
m
 + +
 − + + ≥
+
=   +  + = +  
 
Proof. We know that ( ) ( ) ( )5
2 u v
uv E G
u v
S S
GA G
S S∈
=
+∑  
Case (1) If 4n ≥ : 
( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( )( ) ( )
* *
4,4 4,2 2
*
2 2,4
* *
5 4,4 4,2 2
*
2 2,4
2 2
2
m
m m
m u v u v
n muv E G uv E G
u v u v
u v
m m uv E G
u v
S S S S
GA D E E
S S S S
S S
E
S S
+
+
+∈ ∈
+ ∈
= +
+ +
+
+
∑ ∑
∑
 [From Table  
2 and Figure 5] 
( ) ( ) ( )
2 4 2 2 2 2 2 42 4 44 2 2
4 4 4 2 2 2 2 4
m m m
n m m m
m m m
+ +⋅
= − + +
+ + + + +
 
( ) ( ) ( )
4 2 1 2 2 1
4 .
3 3 1
m m m m m
n m
m
+ +
= − + +
+
 
Case (2) If 3n = : 
( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )* *2 2,2 2 2 2,4
* *
5 2 2,2 2 2 2,4
2 2
m m m m
m u v u v
n m m m muv E G uv E G
u v u v
S S S S
GA D E E
S S S S+ + ++ + +∈ ∈
= +
+ +∑ ∑  
 
Table 3. Edge partition based on degree sum of neighbors of end vertices of each edge.                                           
Edges of the type Number of edges 
( )
*
2 2,2 2m mE + +  m 
( )
*
2 2,4m mE +  2m 
M. R. R. Kanna et al. 
 
 
80 
[From Table 3] 
( ) ( ) ( )2 2 2 2 2 2 2 2 4
2
2 2 2 2 2 2 4
m m m m
m m
m m m m
+ + +
= +
+ + + + +
 
( )4 2 1
1
3 1
m m
m
m
 +
 = +
+  
.                                 □ 
3. Conclusion 
The problem of finding the general formula for ABC index, 4ABC  index, Randic connectivity index, Sum 
connectivity index, GA index and 5GA  index of Dutch Windmill Graph is solved here analytically without 
using computers. 
Acknowledgements 
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. 
Conflict of Interests 
The authors declare that there are no conflicts of interests regarding the publication of this paper. 
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Computation of Topological Indices of Dutch Windmill Graph

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Computation of Topological Indices of Dutch Windmill Graph

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