Some Results on Generalized Degree Distance

(1)

Some Results on Generalized Degree Distance

Asma Hamzeh, Ali Iranmanesh*, Samaneh Hossein-Zadeh

Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran Email: *_{[email protected]}

Received January 14, 2013; revised February 15, 2013; accepted May 14, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In [1], Hamzeh, Iranmanesh Hossein-Zadeh and M. V. Diudea recently introduced the generalized degree distance of graphs. In this paper, we present explicit formulas for this new graph invariant of the Cartesian product, composition, join, disjunction and symmetric difference of graphs and introduce generalized and modified generalized degree distance polynomials of graphs, such that their first derivatives at x = 1 are respectively, equal to the generalized degree distance and

the modified generalized degree distance. These polynomials are related to Wiener-type invariant polynomial of graphs. Keywords: Generalized Degree Distance; Cartesian Product; Join; Symmetric Difference; Composition; Disjunction

1. Introduction

A graph invariant is any function on a graph that does not depend on a labeling of its vertices. Topological indices and graph invariants based on the distances between the vertices of a graph are widely used in theoretical chemistry to establish relations between the structures and the properties of molecules. Topological indices provide correlations with physical, chemical and thermodynamic parameters of chemical compounds [2]. In this paper, we only consider simple and connected graphs. Let G be a

graph on n vertices and edges. We denote the ver-

tex and the edge set of G by and , respec-

tively. As usual, the distance between the vertices and of G, denoted by

m

 

V G E G

 

u v dG

 

u v, (d u

 

,v for short),

is defined as the length of a minimum path connecting them. We let dG

 

v be the degree of a vertex in G.

The eccentricity, denoted by , is defined as the maximum distance from vertex to any other vertex. The diameter of a graph G, denoted by

v

v



v



 

diam G , is the

maximum eccentricity over all vertices in a graph G.

The Cartesian product of graphs G and H is a

graph such that

G H





   

V GH V G V H , and any two

vertices

 

a b, and

 

u v, are adjacent in G H if

and only if either a and b is adjacent to , or

and is adjacent to , see [3] for details. Let 1 and 2 be two graphs with disjoint vertex sets and 2 and edge sets and 2. The join G1 G2

u u

1 E



E

v bv

G V

a

G V₁



is the graph union 1 2 together with all the edges joining and . The composition

G  2 V

G

1

V G G₁

 

₂ is the

graph with vertex set V1V2 and is adja-

cent to



1 1

,

u u v





2, 2



v u v

u

whenever ( 1 is adjacent with 2) or (u1 2

u u

 and is adjacent to 2), [3, p. 185]. The disjunction of graphs G and

1 v GH

v

H is the graph

with vertex set V G

   

and



is adjacent

to

V H

 u v1, 1





u v2, 2



whenever u u1 2E G

 

or 1 2

 

. The symmetric difference G of two graphs

and

v v E H

G H



H is the graph with vertex set V G

   

V



H and



 





 

2

v

H



1 1

not both

u v E G

1 2 1

2 2

,u v,

E



or but

E GH 



  G

u V G

u

u v



2

.

The first Zagreb index was originally defined as

 

1

M G d







u [4]. The first Zagreb index can

be also expressed as a sum over edges of G,

 

G





.

 

uv E G

1 G

M G d







_ u d v _ We refer readers to [5]

for the proof of this fact and for more information on Zagreb index. The first Zagreb coindex of a graph G is

defined in [6] as:

 

_G

 

_G





.

 

E G

1

uv

M G d u d v







_  _

Let



,k



hat

d G

G t

be the number of pairs of vertices of a graph are at distance k,  be a real number,

and W

 

G d G k k



,



1

k

 







, that is called the Wiener- type invariant of G associated to , see [7,8

e

] for details. Additively weighted Harary ind x is defined in [9] as

(2)

 



 

   

1

,

A G

u v V G



.

G

H G d u v d u d







 v

Dobrynin and Kochetova in [10] and Gutman in [11] in



.

troduced a new graph invariant with the name degree distance that is defined as follows:

 

,



G

 

G

 

   u v, V G

D G 



d u v d u d



v



In [12], the modified degree distance was defined as follows:

 



   



   , ,

G G G

u v V G

d u v d u d v









.

The generalized degree distance, denoted by

S G

 

H_ G , is defined as follows in [1].

For every vertex x and real number , H_

 

x is defined by H

 

x D

   

x dG x



  , where

 

,

 

y V G

D x 



d x y . We then define 

 

   

 



 



   ,

, .

G

x V G x V G

G G

u v V G

H G H x D x d

d u v d u d v



 



 



 

 



x

If 0, then H

 

G 4 m. When 1, this new topological index



H_

 

G



is

ndex). T

equal to t egree dis-tance (or Schultz i here are many papers for studying this topological index, for example see [13-16]. Also if 1

he d

  , then H

 

G H_A

 

G . Therefore the study of w topolo portant and we try to obtain some new results related to this topological index. The modified generalized degree distance, denoted by

 

this ne gical index is im

H_ G , is defined in [1] as:

 

,



G



G

 

  

   u v, V G



.

H G d







d u v d u v

If 

1

 , then H_

 

G S

 

G .

ph polynomial

We construct gra s having the property such that their first derivatives at x1 are equal to the

generalized degree distance, the m fied generalized degree distance and Wiener-type invariant respectively. These polynomials are defined as follows:



,





 



H G x 



d u d v

odi

,

and

The Wiener index of the Cartesian product of graphs w

Gutman computed the Szeged index of the Cartesian

omplete gra on rtices. Th

     







   



 

   

, ,

d u v

G G

u v V G

d u v

G G

u v V G

x

H G x d u d v x

















 

   

, ,

, d u v.

u v V G

W_ G x x







as studied in [17,18]. In [19], Klavžar, Rajapakse and

product of graphs. In [9,20-24], exact formulae for the hyper-Wiener, the first Zagreb index, the second Zagreb index and Schultz polynomials of some graph operations were computed.

Throughout this paper, C P Kn, ,n n and Sn denote

the cycle, path, c ph and star

G is a

n ve

e complement of a graph graph H on the

same vertices such that two vertices of H are adjacent

if and only if they are not adjacent in G. The graph H

is usually denoted by G. Our other no ons are stan-

dard and taken mainly from [2,25,26].

In this paper we pres t explicit formulas for tati

en H

 

G

of graph operations containing the Cartesian co

product, fference mposition, join, disjunction and symmetric di

of graphs and introduce generalized and modified generalized degree distance polynomials of graphs, such that their first derivatives at x1 are respectively, equal to

the generalized degree distance and the modified gener-alized degree distance. e polynomials are related with Wiener-type invariant polynomial of graphs.

2. Main Results

Thes

tion is to compute the generalized ve graph operations. We start with a The aim of this sec

degree distance for fi

lemma which gives some information about the number of vertices and edges of operations on two arbitrary graphs. For a given graph Gi, the number of vertices

and edges will be denoted by ni and mi, respectively.

Lemma 2.1. [3,20] Let G and H be graphs. Then

we have: a)











 







 





 





 

 





 





 

2

2 2

,

, , ,

2 ,

V GH V G H V G H V G H

V G V H

E G H E G V H V G E H

E G H E G E H V G V H

E G H E G V H E H V G

E G H E G V H E H V G

E G E H

    

 

    

    

   

    

 

and





 

2 2

4 .

E G H E G V H E H V G

E G E H

    

 

G H

b) The graph is connected if and only if and

G

H are connected.

c) If

 

a c, an d

 

b d, are vertices of G H , th en

  







 

a b, d



c d,



.

tion and symmetric difference of graphs are associative

, , ,

G G

d __H a c b d d _H

(3)

an

f)

g)

h)

i)

d all of them are commutative except the composition of graphs.

e)

 

,

d u v

 



 



0,

1, or or .

2,otherwise

G

u v

uv E G uv E H u V G v V H

  

_    

 

.

H



 



   



 

, , 0, &

, , , .

1, & 2, &

G

G H

d a c a c

a c b d

d a b c d

a c bd E H

 



 



  _ _



 _ _



   





 

0, &

, , , 1, or .

2,otherwise

G H

a c b d

d  a b c d ac E G bd E H

 

 

  

 

   





 

, , , 0, &

1, or but not both.

2,otherwise

G H

d a b c d

a c b d

ac E G bd E H



 

 

_  

 

 



,



 

.

G H G H

d _ a b d a d b

j) dG H_{ }



 

a b,



V H d

   

G a dH



b



.

k)

 

,

. ,

G G H

H

d a V H a VG

d a

d a V G a V H



  

  

 



l)

 





   

 

   

,

.

G H G H

G H

d a b V H d a V G d

d a d b

  



b

m)

 





   

 

   

,

2 .

G H G H

G H

d a b V H d a V G d

d a d b

  



b

In Theorem 2.2, we give a formula for the generalized degree distance of the join of two graphs.

Theorem 2.2. Let G1 and G2 be two graphs. Then









 









1 2

H_ G G



1 2



,

u v V G G , the distance dG₁G₂

 

u,v is either 1

ula for

or 2. In the form H_



G₁G₂



, we partition the

set of pairs of vertices of G1G2 into three subsets 0, 1

A A and A₂. In A₀ we pairs of vertices u

acent i

collect all an

are adj

d v such that u is in G1 and v is in G2. Hence, they 1 G2

n G  . The set is the set of

hic

, 1, 2

i A i

h are in



pairs of vertices u and v w Gi. Also we parti-

tion the sum in the formula of H G1G2



into three sums Si so that Si is over Ai for i0,1,2. For S0

ain

 

we obt





   





1 2

0 1 2

1 2 1 2 2 2 1 2

G u V G v V G

S d u d n

n n n n n m

 

  

 

and

1 2,

G v

n m



n



 





 

   

 





 





 

1 1 1 2

1

1 2

1

1 2

1

S  ₂

,

2

1

1 1 2 1 1 1 2 1

2 ,

2

2 2

2 2 2 .

G G G G

u v V G

G G

uv E G

G G

uv E G

d u d v n d u v

d u d v n

M G n m M G n m



 

 







 

  

  

   



Similarly,

 

1

2 1 2 2 1 2 2 1 2 2 1

S M G  n m  M G   n m₂.

Therefore









 









1 2

0 1

S  S S2

1 2 1 2 2 1 1 2 1 1 1

1

1 2 2 1 1 2

4 4

2 .

H G G

n n n n n m n m M G M G

M G n m n m



  

 

     

  

□

Corollary 2.3. Let G be a connected graph with n ver-

tices and m edges. Then

2

1 1 2 M G











 

1 1

1 4

2 2

H K G n n m M G

M G m



 

    

  .

The exact formulas H_

 

G for the fan graph K1 + Pn

and for the wheel graph W_nK₁C_n are given in the

following Corollary. Corollary 2.4.



1 n



2 9 10 2



2 5



9

H_ K P n  n   n n



,

and



1 n



2



3 .



H_ K C n n

the

9n 3 2n

  

Remark 2.5. In above theorem, if 1, then we obtain D G



1G2



1 2 1 2 2 1 1 2 1 1 1 2

1

1 1 1 2 2 1 1 2

4 4

2 2 .

n n n n n m n m M G M G

M G M G n m n m

  

     

   

Proof. It is obvious from definition that for any

, which gives first vatives formula T 2] at

deri heorem 3 in [2 x1.

In the ge

the next theorem we obtain the exact formula for neralized degree distance of the composition of two graphs.

(4)



u1, , un1



and



v1, , vn2



, respectively. T

 





 

2

1 1 2 1 1 2 2 2 1

3

1 1 2

4 2

2

G G m m n n M G m

n M G H

 





  



2 2

2 1 4 2 2 1

H m n

n _ G n m W_ G

 

Proof. Suppose are

two set of vertices hen

by Lemma 2.1 and de

of G1 and finition of 2

G

H_, we have:

 





_{ }

 



_{ }

 

_{ }

 



    

 











 



 

 



 





 



 











 



 



1 2 1 2 dG G1 2 v 1 2

,

1

u v G

1 2 1 2

1 2

1 2 2

1 2

1 1 2 2

1 2

, ,

2 1 , 1

2 , 1,

, , , 2

, , ,

i k j l

G G G G

V G

i k j l G i G k G j G l

G G u v u v n n

p k p l G p G k G l

G G p k l

i k j l G i G j G k G l

G G i j i j

u

d u v u v nd u d v nd u d v

d u v u v n d u d v d v

d u v u v n d u d u d v d v



 

 

 

   

  

   

 

 





 

,

H G G 



d u v d

 

2 1

1 2

1 2 2

2

, 1

3

2 2 1 2 2 1

1 , 1,

2 3

2 1 2 1 1 2 2 2 1 1 1 2 2 1 2 2 1

2 2 4

4 2 2 4 .

k l

n n

k l

n n

G p G k G l

p k l u v E G

n d u d v d v n H G n m W G

m m n n M G m n m n M G n H G n m W G



 



  



    

     

 



So the proof of theorem is now completed. □

By composing paths and cycles with various small graphs we can obtain classes of polymer-like graphs.

ow we give the formula of the

and

 



n 2



9 8



 

n

 



H C K  n H C W Cn .

Remark 2.8. In Theorem 2.6, if  1, then we obtain

 

H

N  index for the fence

gr



1 G2



rem 5 in

D G

Theo

, which gives first tives formula [22] at

deriva 1

x .

aph P Kn

 

2 and the closed fence n

 

2 Corollary 2.7.



C K .





n 2



9 8 8



 

n

 

n



,

H_ P K  n  H_ P W_ P _erNow we prove the theorem that characterizes the gen- _nctio

et and be two graphs. Then

alized degree distance of the disju n of two graphs. Theorem 2.9. L G1 G2









 





 

   





 





 

   





3 3

2 4 2 2 1 1 1 1 1 2

H ₁ ₂ _{1 2} ₁ ₂ ₁ ₁ ₁ ₁ ₂

2

1 1 1 1 1 1 2

2

1 1 1 2 1 1 2 2 2 1

8 4

2 2

2

2 2 2 2 1

2 2 2

G G n n m m n n M G

G n m n m M G

M G M G n m m n m m



 

   

    

 

  

Proof. According to definition of , we have the following relations:

m  n  n m M G M G M G

n m n m M



  

1 2

G G

 





     

 

     

 

1 2 1 2 1 2 1 2

1 2

2 2 1 1 1 2 1 2

1 2

1 2 1 2 1

,

1 2

,

3

1 1 2 4 1 2 1 2 2 1 1 1 2 ,

G G G G G G G G

u v V G uv E G

G G G G G G G G

x y V G uv E G

S n d x n d u d x d u n d y n d v d y d v

u x d u d y d v

n M G n m m n n m M G

 

     



  



 







 





n d d v n d y d x d





   

 





     

 

1 2 1 2 1 2 1 2

1 2

2 2 1 2 1

,

3

2 1 1 4 1 2 1 2 2 2 2 1 1 ,

G G G G G G G G

xy E G u v V G

n M G n m m n n m M G

 

     

  



 

   





   

 

   

1 2 1 2 1 2 1 2

1 1

3 2 1 2 1

1 1 1 2 2 2 1 1 1 1 1 2

2 2 ,

G G G G G G G G

xy E G uv E G

S n d x n d u d x d u n d y n d v d y

n m M G n m M G M G M G

 

     

  

 

d v

(5)

 





   

 



 





 







   

 



 





 







   





 





 

   

1 2 1 2 1 2 1 2

1 2

2 2 1 1

1 2

1 1 2 2

1 2

4 2 1 2 1

, ,

1 2

2 1

2 2

2 2 2 2 1 1 1 1 1 1 1 2 1 1 1 2

2

2 2

2 2 2 2 2

G G G G G G G G

xy E G x y uv E G u v

G G G G

xy E G u V G

G G G G

x V G uv E G

n d u n d u d x d y

n d x n d x d u d v

n m n m M G n m n m M G M G M G



   

 

 

     

   



      



 





2

1 1 2 2 2 1

2  n m m n m m .

  

 

So we have:









 





 

   





 





 

   





1 2

3 3

1 2 4 3 1 2 1 2 1 1 1 1 2 2 2 2 1 1 1 1 1 2

2 2 2

2 2 2 1 1 1 1 1 1 1 2 1 1 1 2 1 1 2 2 2 1

8 4 4

2 2 2 2 2 .

H G G

S S S S n n m m n n m M G n n m M G M G M G

n m M G n m n m M G M G M G n m m n m m



  



         



     _  

This completes the proof. □

Now we prove the theorem that characterizes the generalized degree distance of the symmetric difference of o graphs.

Theorem 2.10. Let G1 and G2 be two graphs. Then 2

2 2n m

 

tw





 





 

  









 





 

   





3 3

1 2 1 2 1 1 1 1 2 2 2 2 1 1 1 1 1 2

2

1 1 2 2 2 1

8 4

n n m m n n m M G n n m M G M G M G

n m m n m m

    



Proof. We consider four sums as follows:

1 2 8 8

H G G 

2 2

2 2 2 2 1 1 1 1 1 1 1 2

2 2n m n 4m M G 2n m n 4m M G 

 _      _

1 1 1 2

2M G M G 2

 

1, , 4

S  S

 





     

 

1 2 1 2 1 2 1 2

1 2

1 2 1 2 1

, 3

1 1 2 1 2 1 2 1 1 1 2

2 2

4 4 ,

G G G G G G G G

x y V G uv E G

n M G n m m n n m M G

 

     

  



similarly to S1

 





   

 

   

1 2 1 2 1 2 1 2

1 2

3

2 2 1 1 1 2 1 2 2 2 1 1

3 2 1 2 1

1 1 1 2 2 2 1 1 1 1 1 2

4 4 ,

2 2

2 2 2 ,

G G G G G G G G

xy E G uv E G

S n M G n m m n n m M G

n m M G n m M G M G M G

 

  

     

  



and

 





   

 



 





 







   

 



 





 







   





 





 

1 2 1 2 1 2 1 2

1 2

2 2 1 1

1 2

1 1 2 2

1 2

4 2 1 2 1

, ,

1 2

2 1

2 2

2 2 2 2 1 1 1 1 1 1 1 2 1 1

2 2 2

2 2 4 2 4 2

G G G G G G G G

xy E G x y uv E G u v

G G G G

xy E G u V G

G G G G

x V G uv E G

n d u n d u d x d y

n d x n d x d u d v

n m n m M G n m n m M G M G



   

 

 

     

   

      



 

 





1 2

2

1 1 2 2 2 1

2 .

M G

n m m n m m

 

 

 

 

By the definition of G1G2, we have:









 





 

   





 





 

   





3 3

1 2 1 2 4 3 1 2 1 2 1 1 1 1 2 2 2 2 1 1 1 1 1 2

2 2

2 2 2 2 1 1 1 1 1 1 1 2 1 1 1 2

2 1 1

H

2 2 2 1

2

2 8 8 8 4

2 2 4 2 4 2

.

G G S S S S n n m m n n m M G n n m M G M G M G

n m n m M G n m n m M G M G M G

n m m



          

 

 _       _

So the proof is now completed. □

n m m



(6)

In the next theorem we find the generalized degree distance of the Cartesian product of two graphs. Theorem 2.11. Let G1 and G2 be two graphs. Then





 



 



   







2

1 2 1 2 1 1 2 1 1 1 2 1 1 2

1 1 1 2

2 1 2 2 1

4 2

1

H G G n H G m n W G H G W G W G H G

G H G

G n                 _{ }     

 



2  H G W

 ₂ ₁ ₂

 

₂ ₂

 

₁ ₂

 

₂



2 ₁

   

₁ ₂



 

2 ₄ _.

n H m W G

 

2

Proof. Suppose and are two set of vertices of and , respectively. Then by Lemma 2.1 and definition of

1

G W G H G H G W G W

     _   

   _ 

    



u1, , un1





v1, , vn2



G1 G2 H_, we have:





 



 



   















 



  _ 











 















1 2 1 2 1 2

1 2

1 2 1 2 1 2

1 2 1

1 2

,

, 1 , 1

0 ,

1 _, _, _,

2 , , 2 1 , , 2

G G G G G G

u v V G G

G G i k j l G i G k G j G l

u

G i j G k l G i G k G j G l

k l i j

r r

r G i j G k l G

r

H G G d u v d u d v

d u v u v d u d v d u d v

d u u d v v d u d v d u d v

d u u d v v d u

                              _ _  



 





 



 



 



 





   





 

2 1

2 1 2

, 1 , 1

2

1 2 1 1 2 1 1 1 2 1 1 2

2 1 2 2 2 1 2 2

2

1 1 2 1 1 1 2 2 1 2 2 1

4 2

1 2

2

2 4 .

1

n n

i G k G j G l

k l i j

d v d u d v

n H G m n W G H G W G W G H G

H G W G W G H G

2 1

, ,

1

ivk u vj l

n n _



 

H G W G W G H G n H G m n W G

                              _{ }       _{ }        _ _      

 



So the proof is now completed. □

As an application of the above theorem, we list explicit formulae for the generalized degree distance of and These graphs are known

lar grid, nanotube, and the ctive

,

n m n m

P P PC

as the rectangu anotorus, respe

n m

C C .

the ly. 4

C C4

n

Lemma 2.12. Define

1

i i



. By [1,23], we have:

 





 

k r,  



k r





 











 





 



1

1, 1, 1 ,

n

W P n n n

n n



1, is even

2 2 2, _,

1_, _, _{is odd} 2

2 3, 4 3,

4 3, 1 2 1 6 2 ,

4 1, , is even

2 2

1

4 , , is odd 2 n n n n n n n n n

H P n n n

n n n

n n n n H C n n n    W C                                   _ _               _ _{  }    _ _{  }              .   

Corollary 2.13. By Theorem 2.9 and Lemma 2.12 we have:





 

   









 



  

 





 

   





2 2 2 2

1 1 2 2 1 1 4 4 2 , 4 1 4 1 2 , n m

m m n

i n i m i n i m

i

n m

m m n

n

i n i m i n i m

i

H C C

n H C n W C m H C m W C

H C W C H C W C

i

H P P

n H P n n W P m H P

m m W P

H P W P H P W P

i                                   _{ }              _{ }   



n and           _ _        





 



  

 

   





2 2 2 1 4 1 4 2 . n m m m n n

i n i m i n i m

H P C

n H C n n W C

m H P m W P

H P W C H P W C

                    



_{ }  1

i  i

(7)

mula Theorem 1 in [22] at x1.

relation between ial and W

n betwee i aphs.

is a graph

 

2

d u

W_

 



have



 

,

d u v

d u



 

 

 



u



Now we obtain the the generalized degree distance polynom iener-type invariant polynomial and the relatio n the modified generalized degree distance polynom al and Wiener-type invariant polynomial for gr

Theorem 2.15. If with vertices an edges, then

oof. By definition, we



,

G n d

m







 



 





,

, 2

4 2 , .

d u v

G G

u v V G

H G x d v x

m n G x



 



 



Pr









 



 

   

 





 





 



 

, ,

,

2 2

2

d u v

G G

u v V G

d u v

G G

u v V G

d u v

G G

u v V G

u

H G x d v x

d v x

v x





 

 





  



 

,

2

v V G

d u v

G G

x

d u d 

 

 



 





4 2 2 , .

u v V G

m n W_ G x

 

  

Theorem 2.16. If G is a graph with n vertices and m edges, then



,



H G x



 



 









,

1

1 1

4 ,

d u v

G G

u v V G

d d v x

.

M G m H G x W G x



 

 

 

   



n



have







 





 





 





,

1 1

4 ,

1 1

4 ,

d u

v x

d v

v

v x

d v

H

G x



 

 

Proof. By definition, we



,



H_ G x

  

 

   

 







 





 

   

     

 







 

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