Degree Distance of Some Planar Graphs

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 54-57 ISSN: 2278-2397

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.

2.1 Construction Starting from the star graph K1,n with vertices v0 , v1 , v2 ,..., vn  introduce an edge to each of the pendant vertices v1 , v2 ,..., vn to get

the

resulting

v , v ,..., v , v

Index terms— Degree distance, planar graph, tree graph, regular graph, star graph.

0

1

n

n 1 ,..., v2 n

graph

K1,n,n with

vertices

, again introduce an edge to each of

the pendant vertices v n 1 ,..., v2n , to get the graph K1,n,n,n .

I. INTRODUCTION

Repeating this procedure

  degG u   degG  v dG u, v  where

u ,vV G 

the resulting graph

K1,n,n,...,n with,

Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structureactivity/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 G  V , E  the degree distance of G is defined as DD  G  

 m  1 times

m times

 mn  1 vertices {v0 , v1 , v2 ,..., vn , v n1 ..., v2n , v 2n1 ,..., v3n ,..., v m1n1 ,..., vmn } and mn edges is obtained as shown in Fig. 1.

degG  u  is

the degree of the vertex u in G and dG  u, v  is the shortest distance between u and v. Klein et al. [5] showed that if G is a tree on n vertices then DD  G   4W G   n  n  1 where W G  



{u ,v}V  G 

dG  u, v  is the Wiener index of the graph G.

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 K1, n, n,..., n [2] and calculate its degree distance, in Section III,

Fig. 1. Multi-star graph K1, n, n,..., n . m times

m times

we calculate the degree distance of planar graphs Pln  n  5

Result 2.2: Degree distance of the Multi-star graph K1,n,n,...,n is

and Pm  Cn  m  2, n  3 and in Section IV, We introduce the graph operation





m times

 DD  K1, n, n,..., n  m times 

ê and compute the degree distance of



Pm1  Cn1 eˆ Pm2  Cn2 .

 nm   6m 2 n  4m 2  3mn  1   3  

Proof: Let the  mn  1 vertices of the multi-star graph

II. MULTI-STAR GRAPH K1,n,n,...,n m times

K1,n,n,...,n be {v0 , v1 , v2 ,..., vn , v n1 ,..., v2n ,..., v m1n1 ,..., vmn }

In this section we give the construction of multi-star graph and calculate its degree distance.

m times

54

Integrated Intelligent Research (IIR)

  Then, DD  K1, n, n,..., n     m times 



    d  ui , u j   2; i  3, 4,...,(n  2); i  2   j  n .

dG u2 , u j  1, j  1,..., n, j  2; dG u j , u j 1  1, j  3,..., n 1;

  

  deg  vi   deg v j d vi , v j i j

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 54-57 ISSN: 2278-2397



DD  Pln    n  n  1  2  3  4  n  4    n  2  n  1  2  3  4  n  4 

=  n  n  2  1  2  ...   m  1   n  n  1 m  

      + 3 1  2  2  ...  2   4 1  2  2  ...  2   2  3         n  4  n 5      

 4  2  1  2  2  ...  2   m  1    n  1   n  2   ...  1    4n  n  1  2  1  1  2  1  2  ...  2  1   m  2        4n  n  1  2  2  1  2  2  2  ...  2  2   m  3       ...  4n  n  1  2   m  2   1 

      +  4 1  2  2  ...  2   4 1  2  2  ...  2   2  3        n  5  n 6             +  4 1  2  2  ...  2   4 1  2  2  ...  2   2  3        n 6  n 7      

3n  n  1  2  1   m  1  2  2   m  2   ...  2   m  1  1    4n 1  2  ...   m  2    4n 1  2  ...   m  3   ...  4n 1  

+  4 1  2  2   4 1  2   2  3  4 1  2   4 1  2 3  4 1  3 1

3n   m  1   m  2   ...  2  1    2  2m    n  1  ...  1 nm = 6m 2 n  4m 2  3mn  1 3



 10n 2  48n  68



Result 3.2: Degree distance of structured web 3-regular graph Pm  Cn  m  2, n  3 is

III. PLANAR GRAPHS













3 2 2 2 2  4 m n n  1  mn m  1 when n is odd DD  Pm  Cn     3 n3m2  mn2 m2  1 when n is even  4

In this section we compute the degree distance of planar graph with maximal edges [1], Pln  n  5 (Fig. 2) and structured web graph Pm  Cn  m  2, n  3 (Fig. 3).

Result 3.1: Degree distance of Pln  n  5 graph is DD  Pln   10n2  48n  68

Fig. 3. Pm  C5 graph.

Proof: Case (i) when n is odd. i j

Fig. 2. Pln  n  5 graph.

Proof:

Let

the n vertices

of

the

Pln  n  5 graph be

u1 , u2 ,..., un  . Then the degree distance DD  Pln     deg  ui   deg  u j  d  ui , u j  .

of Note

Pln is that

i j

degG  u1   degG u2   n  1;degG u3   degG un   3;







  

DD  Pm  Cn    deg  ui   deg u j d ui , u j

un





degG  u4   ...  deg u n1  4; dG u1 , u j  1, j  2,..., n;

55



Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 54-57 ISSN: 2278-2397

DD  Pm  Cn   1     n 1   6  2 1  2  ...       mn   2 2       



DD  G   DD  G1   DD  G2  +  6n2  2  DG1  u1, u    6n1  2 DG2  v1, v   n1  n2  6n1n2



  n2  1  12n  m  m  1  n  1    m  1      2  2  8      2  n 1    m  2  m  1  n  1   m  2   12n    2 8  2       n2  1   ...  12n  1  2  n  1   1    2  2   8    









                

Where,  n12m1 +n1 m1C2 when n1 is even   4 DG1  u1, u    dG1  u1, u    2 u  u1V G1   n1  1 m1 +n1 m1C2 when n1 is odd  4 

 





 

 6  n  2n  ...   m  1 n   6  n  2n  ...   m  2  n   ...  6  n  









3  m2n n 2  1  mn 2 m2  1 4

Case (ii) when n is even.

DD  Pm  Cn 



  

  deg  ui   deg u j d ui , u j i j



1      n  n  =   6  2 1  2  ...    1      mn    2   2    2       n m n  n  n  m  6n  2   1 C2    1   m  1   m  1  C2    2  2  2     2     n   m 1  n  n  n  m 1 C2    1   m  2    m  2    C2   6n  2   1 2  2  2     2 

1











i j

i j

If





  degG  ui   degG  v j  dG  ui , v j  1

 DD  G1  



u  u1V  G1 

+6n2 

ˆ 2 G  G1eG

dG1  u , u1   DD  G2  



v  v1V  G2 

1



u  u1V  G1 



u  u1V  G1 



u  u1V  G1 

 6n1



2

1

dG1  u1, u   DD  G2  

2



v  v1V  G2 

dG1  u1 , u   6n2 n1

dG1  u1, u    n1  1  2   n2  1

v  v1V  G2 

dG2  v1, v  



v  v1V  G2 

d G2  v1, v 

DD  G   DD  G1   DD  G2   +  6n2  2  DG1  u1, u 



Result 4.2: Degree distance of G  Pm1  Cn1 eˆ Pm2  Cn2 is

  6n1  2  DG2  v1 , v   n1  n2  6n1n2

56

d G2  v, v1 

   degG  ui   degG  v j   dG  ui , u1   1  dG  v1, v j 

 DD  G1  

 p1  p2 

2

m1n1 m2 n2 i 1 j 1

(example shown in Fig. 4) is obtained by using the new graph operation defined above and we prove the following result.





  

i 1 j 1

G1  Pm1  Cn1 and

G2  Pm2  Cn2 an interesting graph structure



m1n1 m2 n2



 q1  q2  1 edges.

  



+

If G1  p1 , q1  has p1 vertices and q1 edges and G2  p2 , q2  has and



  deg G2  vi   deg G2 v j dG2 vi , v j

of G1 and an arbitrary vertex of G2 .

vertices

 



  deg G1  ui   deg G1 u j dG1 ui , u j

ê and

ˆ 2 is a connected graph obtained from G1 Definition 4.1: G1eG and G2 by introducing an edge e between an arbitrary vertex

ˆ 2 will have q2 edges then G1eG





establish the degree distance of Pm1  Cn1 eˆ Pm2  Cn2 .

p2 vertices and

.

ˆ 2  Pm1  Cn1 eˆ Pm2  Cn2 is given by distance of G  G1eG DD  G 





Proof: Let an edge be introduced between the arbitrary vertex u1 of G1 and the arbitrary vertex v1 of G2 , then the degree

ê

In this section we introduce new graph operation

 C6

 



IV. NEW GRAPH OPERATION

2

 n22m2 +n2 m2 C2 when n2 is even   4 DG2  v1, v    dG2  v1, v    2 v  v1V G2   n2  1 m2 +n2 m2 C2 when n2 is odd  4 

  n   n  n  n  +...+ 6n  2   1 2C2    1  1  1  2C2    mn m 2  1 2  2  2     2  3 = n3m2  mn 2 m2  1 4



 Pm  C3  eˆ  Pm

Fig. 4. Graph

d G2  v1 , v 

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 54-57 ISSN: 2278-2397

To find DG1  u1 , u  when n1 is even,

V. CONCLUSIONS

DG1  u1 , u 





u  u1 V  G1 

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.

d G1  u1 , u 

  n  n    2 1  2  ...   1  1   1  2  2       n    n  +  2  1  1   2  1  ...    1  1  1    1  1   1    2    2       n1    n1   +  2  1  2    2  2   ...     1  2      2    2+...       2   2    n   n   +  2  1   m  1   ...    1  1   m1  1     1   m1  1   2 2        

REFERENCES [1]

  m1  1



n1m1  m1  1 2

[2] mn2  1 1 4

[3]

Similarly, DG2  v1 , v   



v  v1 V  G2 

[4]

d G2  v1 , v 

n2 m2  m2  1 2

[5]

mn2  2 2 when n2 is even 4

To find DG1  u1 , u  when n1 is odd,



DG1  u1 , u  

u  u1 V  G1 

dG1  u1 , u 

DG1  u1 , u     n  1     2 1  2  ...   1    2         n  1    +  2  1  1   2  1  ...    1   1    1   2          n 1   +  2  1  2    2  2   ...    1   2    2  2          n 1   +...+  2  1   m  1   ...    1    m1  1      m1  1    2     

n1m1  m1  1 2



m1  n12  1 4

Similarly,



DG2  v1 , v  

v  v1 V  G2 

dG2  v1 , v 

n2 m2  m2  1



2



m2  n2 2  1 4

when n2 is odd

Using Result 3.2,

3 2 2 2 2  4 m1 n1  n1  1  m1n1  m1  1 when n1 is odd DD  G1   DD Pm1  Cn1    3 n 3m 2  m n 2  m 2  1 when n1 is even  4 1 1 1 1 1





3 2 2 2 2  4 m2 n2  n2  1  m2n2  m2  1 when n2 is odd DD  G2   DD Pm2  Cn2    3 n 3m 2  m n 2  m 2  1 when n2 is even 2 2 2  4 2 2





57

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