Abstract- A novel graph invariant: degree distance index was introduced by Andrey A. Dobrynin and Amide A. Kochetova [3] and the same time by Gutman as true Schultz index [4]. In this paper we establish explicit formulate to calculate the degree distance of some planar graphs.

1. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 54-57
ISSN: 2278-2397
54
Degree Distance of Some Planar Graphs
S. Ramakrishnan1
J. Baskar Babujee2
1
Department of Mathematics, Sri Sai Ram Engineering College, Chennai, India
2
Department of Mathematics, Anna University, Chennai, India
E-mail: sai_shristi@yahoo.co.in
Abstract - A novel graph invariant: degree distance index was
introduced by Andrey A. Dobrynin and Amide A. Kochetova
[3] and at the same time by Gutman as true Schultz index [4].
In this paper we establish explicit formulae to calculate the
degree distance of some planar graphs.
Index terms— Degree distance, planar graph, tree graph,
regular graph, star graph.
I. INTRODUCTION
Topological indices are numbers associated with molecular
graphs for the purpose of allowing quantitative structure-
activity/property/toxicity relationships. Wiener index is the
best known topological index introduced by Harry Wiener to
study the boiling points of alkane molecules. By now there
exist a lot of different types of such indices (based either on the
vertex adjacency relationship or on the topological distances)
which capture different aspects of the molecular graphs
associated with the molecules under study. For a graph
the degree distance of is defined as
where is
the degree of the vertex in and is the shortest
distance between and Klein et al. [5] showed that if is
a tree on vertices then where
is the Wiener index of the graph
Thus the study of the degree distance for trees is equivalent to
the study of the Wiener index. But the degree distance of a
graph is a more sensitive invariant than the Wiener index.
In this paper we compute the degree distance of some planar
and tree graphs. In Section II, we introduce multi-star graph
[2] and calculate its degree distance, in Section III,
we calculate the degree distance of planar graphs
and and in Section IV, We introduce the
graph operation and compute the degree distance of
II. MULTI-STAR GRAPH
In this section we give the construction of multi-star graph and
calculate its degree distance.
2.1 Construction
Starting from the star graph with vertices
introduce an edge to each of the pendant vertices to
get the resulting graph with vertices
again introduce an edge to each of
the pendant vertices to get the graph .
Repeating this procedure times the resulting graph
with,
vertices
and
edges is obtained as shown in Fig. 1.
Fig. 1. Multi-star graph .
Result 2.2: Degree distance of the Multi-star graph is
Proof: Let the vertices of the multi-star graph
be
 
,
G V E
 G
     
 
   
 
,
deg deg ,
G G G
u v V G
DD G u v d u v

 
  
degG u
u G  
,
G
d u v
u .
v G
n      
4 1
DD G W G n n
  
   
 
{ , }
,
G
u v V G
W G d u v

  .
G
times
1, , ,...,
m
n n n
K
 
5
n
Pl n 
 
2, 3
m n
P C m n
  
ê
   
1 1 2 2
ˆ .
m n m n
P C e P C
 
times
1, , ,...,
m
n n n
K
1,n
K  
0 1 2
, , ,..., n
v v v v
1 2
, ,..., n
v v v
1, ,
n n
K
 
 
0 1 2
1
, ,..., , ,..., ,
n n
n
v v v v v

  2
1 ,..., ,
n
n
v v
 1, , ,
n n n
K
 
1
m
times
1, , ,...,
m
n n n
K
 
1
mn
     
 
0 1 2 2 3
1 2 1 1 1
{ , , ,..., , ..., , ,..., ,..., ,..., }
n n n mn
n n m n
v v v v v v v v v v
   
mn
times
1, , ,...,
m
n n n
K
times
1, , ,...,
m
n n n
K
times
2 2
1, , ,..., 6 4 3 1
3
m
n n n
nm
DD K m n m mn
 
 
     
 
 
 
 
1
mn
times
1, , ,...,
m
n n n
K    
 
0 1 2 2
1 1 1
{ , , ,..., , ,..., ,..., ,..., }
n n mn
n m n
v v v v v v v v
  

2. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 54-57
ISSN: 2278-2397
55
III. PLANAR GRAPHS
In this section we compute the degree distance of planar graph
with maximal edges [1], (Fig. 2) and structured
web graph (Fig. 3).
Result 3.1: Degree distance of graph is
Fig. 2. graph.
Proof: Let the vertices of the graph be
. Then the degree distance of is
. Note that
.
Result 3.2: Degree distance of structured web 3-regular graph
is
Fig. 3. graph.
Proof: Case (i) when is odd.
   
   
   
   
 
     
 
   
 
   
 
 
times
1, , ,...,
Then,
deg deg ,
= 2 1 2 ... 1 1
4 2 1 2 2 ... 2 1 1 2 ... 1
4 1 2 1 1 2 1 2 ... 2 1 2
4 1 2 2 1 2 2 2 ... 2 2 3
... 4 1 2
m
n n n
i j i j
i j
DD K
v v d v v
n n m n n m
m n n
n n m
n n m
n n m

 
 
 
 
 
 
       
 
 
            
 
           
           
   

 
 
       
 
 
   
   
   
   
 
 
2 2
2 1
3 1 2 1 1 2 2 2 ... 2 1 1
4 1 2 ... 2 4 1 2 ... 3 ... 4 1
3 1 2 ... 2 1 2 2 1 ... 1
= 6 4 3 1
3
n n m m m
n m n m n
n m m m n
nm
m n m mn
 
 
  
 

 
 
 
             
 
 
           
 
   
           
   
  
 
5
n
Pl n 
 
2, 3
m n
P C m n
  
 
5
n
Pl n 
  2
10 48 68
n
DD Pl n n
  
n
u
 
5
n
Pl n 
n  
5
n
Pl n 
 
1 2
, ,..., n
u u u n
Pl
     
   
deg deg ,
n i j i j
i j
DD Pl u u d u u

 

       
1 2 3
deg deg 1;deg deg 3;
G G G G n
u u n u u
    
   
   
4 1
1
deg ... deg 4; , 1, 2,..., ;
G G j
n
u u d u u j n

    
   
2 1
, 1, 1,..., , 2; , 1, 3,..., 1;
G j G j j
d u u j n j d u u j n

     
   
, 2; 3,4,...,( 2); 2
i j
d u u i n i j n
     
 
            
   
 
   
 
4 5
5 6
1 2 3 4 4 2 1 2 3 4 4
+ 3 1 2 2 ... 2 4 1 2 2 ... 2 2 3
+ 4 1 2 2 ... 2 4 1 2 2 ... 2 2 3
+ 4 1 2 2 ..
n
n n
n n
DD Pl
n n n n n n
 
 
              
   
 
   
 
   
         
 
   
   
 
 
   
 
   
         
 
   
   
 
  
   
 
               
6 7
2
. 2 4 1 2 2 ... 2 2 3
+ 4 1 2 2 4 1 2 2 3 4 1 2 4 1 2 3 4 1 3 1
10 48 68
n n
n n
 
 
   
 
   
      
 
   
   
 
              
   
  
 
2, 3
m n
P C m n
  
 
   
 
2 2 2 2
3 2 2 2
3
1 1 when n is odd
4
3
1 when n is even
4
m n
m n n mn m
DD P C
n m mn m

  


  
  


5
m
P C

n
     
   
deg deg ,
m n i j i j
i j
DD P C u u d u u

  


3. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 54-57
ISSN: 2278-2397
56
Case (ii) when is even.
IV. NEW GRAPH OPERATION
In this section we introduce new graph operation and
establish the degree distance of .
Definition 4.1: is a connected graph obtained from
and by introducing an edge between an arbitrary vertex
of and an arbitrary vertex of .
If has vertices and edges and has
vertices and edges then will have
vertices and edges. If and
an interesting graph structure
(example shown in Fig. 4) is obtained by using the new graph
operation defined above and we prove the following result.
Result 4.2: Degree distance of is
Where,
Fig. 4. Graph .
Proof: Let an edge be introduced between the arbitrary vertex
of and the arbitrary vertex of , then the degree
distance of is given by
 
 
 
 
  
 
 
 
 
2
2
2
1 1
6 2 1 2 ...
2 2
1
1 1
12 1
2 2 8
1
2 1 1
12 2
2 2 8
1
1 2 1
... 12 1
2 2 8
6 2 ..
m n
DD P C
n
mn
n
m m n
n m
n
m m n
n m
n
n
n
n n
 
 
 
 

 
     
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
  
 
 
 
 
 
 
 
 

 
  
 
 
  
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
    
   
   
   
2 2 2 2
. 1 6 2 ... 2 ... 6
3
1 1
4
m n n n m n n
m n n mn m
 
        
 
   
n
 
   
   
   
 
   
2 2
1
2
deg deg ,
1
= 6 2 1 2 ... 1
2 2 2
6 2 1 1 1 1
2 2 2 2
6 2 1 1 2 2
2 2 2 2
m n
i j i j
i j
m m
m m
DD P C
u u d u u
n n
mn
n n n n
n C m m C
n n n n
n C m m



 
 
 
 
 
 
      
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    
       
 
 
    
    
 
 
    
      
    
    

 
     
 
1
2
2 2 2
2 2
3 2 2 2
+...+ 6 2 1 1 1 1 1
2 2 2 2
3
= 1
4
C
n n n n
n C C mn m
n m mn m

 
 
 
 
 
 
 
 
    
      
 
 
    
    
 
 
 
ê
ê
   
1 1 2 2
ˆ
m n m n
P C e P C
 
1 2
ˆ
G eG 1
G
2
G e
1
G 2
G
 
1 1 1
,
G p q 1
p 1
q  
2 2 2
,
G p q
2
p 2
q 1 2
ˆ
G eG  
1 2
p p

 
1 2 1
q q
  1 1
1 m n
G P C
 
2 2
2 m n
G P C
  1 2
ˆ
G G eG

   
1 1 2 2
ˆ
m n m n
G P C e P C
  
         
   
1
2
1 2 2 1
1 1 1 2 1 2
+ 6 2 ,
6 2 , 6
G
G
DD G DD G DD G n D u u
n D v v n n n n
  
    
   
 
 
 
 
1
1 1
1 1 1
2
1 1
1 2 1
1 1 2
1 1
1 2 1
+ when is even
4
, ,
1
+ when is odd
4
m
G G
u u V G m
n m
n C n
D u u d u u
n m
n C n
 



  





   
1 2
3 6
ˆ
m m
P C e P C
 
   
 
 
 
 
2
2 2
1 2 2
2
2 2
2 2 2
1 1 2
2 2
2 2 2
+ when is even
4
, ,
1
+ when is odd
4
m
G G
v v V G m
n m
n C n
D v v d v v
n m
n C n
 



  





1
u 1
G 1
v 2
G
   
1 1 2 2
1 2
ˆ ˆ
m n m n
G G eG P C e P C
   
 
   
   
   
   
   
   
   
 
   
 
   
     
1 1 1
2 2 2
1 1 2 2
1 2
1 2
1 1 1 2
1 1 2 2
1 2 1 2
1 1
1 1 2 1
1 1
1 1
deg deg ,
deg deg ,
+ deg deg ,
, ,
deg deg , 1 ,
G i G j G i j
i j
G i G j G i j
i j
m n m n
G i G j G i j
i j
G G
u u V G v v V G
m n m n
G i G j G i G j
i j
DD G
u u d u u
v v d v v
u v d u v
DD G d u u DD G d v v
u v d u u d v v


 
   
 
 
 

   
 
   
 


 
 

   
 
   
 
 
 
 
 
   
 
 
 
 
         
   
1 2
1 1 1 2
1
1 1
1
1 1
2 2
1 2 1 2
1
2
1 1 2 1
2 1 2 1
1 1 2
1 1 1
1 2 2 1
1 1 1
, ,
+6 , 6
, 1 2 1
6 , ,
+ 6 2 ,
6 2 ,
G G
u u V G v v V G
G
u u V G
G
u u V G
G G
v v V G v v V G
G
G
DD G d u u DD G d v v
n d u u n n
d u u n n
n d v v d v v
DD G DD G DD G n D u u
n D v v n
   
 
 
   
   

     
 
   
   

 


 
2 1 2
6
n n n


4. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 54-57
ISSN: 2278-2397
57
To find when is even,
Similarly,
To find when is odd,
Similarly,
Using Result 3.2,
V. CONCLUSIONS
In this paper we have computed the degree distance index of
some planar graphs. Nevertheless, there are still many classes
of interesting and chemically relevant graphs not covered by
our approach. So it would be interesting to find explicit
formulae for computing the degree distance and other
topological indices for various classes of chemical graphs.
REFERENCES
[1] J. Baskar Babujee, “Planar graphs with maximum edges-antimagic
property”, The Mathematics Education, vol. XXXVII, No. 4, pp. 194-
198, 2003.
[2] J. Baskar Babujee, D. Prathap, “Wiener number for some class of Planar
graphs”, National Symposium on Mathematical Methods and
Applications organized by IIT, Madras, 2005.
[3] Andrey A. Dobrynin, Amide A. Kochetova, “Degree Distance of a
Graph: A Degree analogue of the Wiener Index”, J. Chem. Inf. Comput.
Sci., vol. 34, pp. 1082 -1086, 1994.
[4] I. Gutman, “Selected properties of the Schultz molecular topological
index”, J. Chem. Inf. Comput. Sci., vol. 34, pp. 1087 –1089, 1994.
[5] D. J. Klein, , and ,
“Molecular topological index: A relation with the Wiener index”, J.
Chem. Inf. Comput. Sci., vol. 32, pp. 304-305, 1992.
 
1 1,
G
D u u 1
n
 
 
 
   
   
 
 
1
1
1 1
1
1
1 1
1 1
1 1
,
,
2 1 2 ... 1
2 2
+ 2 1 1 2 1 ... 1 1 1 1
2 2
+ 2 1 2 2 2 ... 1 2 2 2+...
2 2
+ 2 1 1
G
G
u u V G
D u u
d u u
n n
n n
n n
m
 

 
 
 
     
 
 
 
 
 
 
 
 
 
   
         
 
 
 
   
   
 
 
 
 
 
 
 
   
         
 
 
 
   
   
 
 
 
 
 

   
 
 
1 1
1 1
1
2
1 1 1 1 1
... 1 1 1
2 2
1
1
2 4
n n
m m
m
n m m m n
 
 
 
   
       
 
 
 
   
   
 
 
 
 
 

 
   
 
 
2 2
1 2
1 1
2
2 2 2 2 2
2
, ,
1
when is even
2 4
G G
v v V G
D v v d v v
n m m m n
n
 


 

 
1 1,
G
D u u 1
n
   
 
1 1
1 1
1 1
, ,
G G
u u V G
D u u d u u
 
 
 
   
   
 
     
1 1
1
1
1
1
1 1
,
1
2 1 2 ...
2
1
+ 2 1 1 2 1 ... 1 1
2
1
+ 2 1 2 2 2 ... 2 2
2
1
+...+ 2 1 1 ... 1 1
2
G
D u u
n
n
n
n
m m m
 
 

 
   
 
 
 
 
 
 
 
 
 

 
      
 
 
 
 
 
 
 
 
 
 
 
 

 
      
 
 
 
 
 
 
 
 
 
 
 
 

 
       
 
 
 
 
 
 
 
 
 

   
2
1 1
1 1 1
1
1
2 4
m n
n m m 


   
 
   
2 2
1 2
1 1
2
2 2
2 2 2
2
, ,
1
1
when is odd
2 4
G G
v v V G
D v v d v v
m n
n m m
n
 



 

   
   
 
1 1
2 2 2 2
1 1 1 1 1 1 1
1
3 2 2 2
1 1 1 1 1 1
3
1 1 when is odd
4
3
1 when is even
4
m n
m n n mn m n
DD G DD P C
n m mn m n

  


   
  


   
   
 
2 2
2 2 2 2
2 2 2 2 2 2 2
2
3 2 2 2
2 2 2 2 2 2
3
1 1 when is odd
4
3
1 when is even
4
m n
m n n m n m n
DD G DD P C
n m m n m n

  


   
  


Z. Mihalic D. Plavsic N. Trinajstic