The Generalized Wiener Polarity Index of some Graph Operations

(1)

The Generalized Wiener Polarity Index of Some Graph

Operations

YANG WU,FUYI WEI•••• AND ZHEN JIA

Department of Mathematics, South China Agricultural University, Guangzhou, P. R. China, 510642

(ReceivedOctober 13, 2013; Accepted November 16, 2013)

A

BSTRACT

Let G be a simple connected graph. The generalized polarity Wiener index of

G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.

Keywords: Wiener index, Cartesian product, tensor product.

1. I

NTRODUCTION

Let G be a simple connected graph, with n vertices and m edges. The set

1

( ) { _n}

V G = v,...,v denotes the vertex set of G while E G( )={e₁,...,e_n} denotes it’s edge set. When we mention v_i =v_j it means the two vertices v_i and v_j are identical. The distance

( )

G i j

d v v, between two vertices v_i and v_j in G is the length of a shortest path between the two vertices. The diameter of G is the greatest distance between two vertices of G, denoted by D G( ). The Wiener index W G( ) and the Wiener polarity index W G_P( ) are, respectively, defined as [1]

•_{Research supported by National Natural Science Foundation of China (No.11201156).}

(2)

1 ( )

( ) ( )

i j

G i j i j n

v v V G

W G d v v

≤ < ≤ , ∈

=

∑

,

( ) |{( )| ( ) 3 ( )}|

P G

W G = u v d, u v, = , , ∈u v V G

by H. Wiener in the mid−20th century. Later in 1988, H. Hosoya [2] introduced the Hosoya(or wiener) polynomial

( )

1 1

( ) ( ) ( )

D G

k k

H G x d G k x d G k x

≥ =

, =

∑

, =

∑

,

where (d G k, ) denotes the number of unordered pairs of vertices whose distance is equal to

k. Actually, H G x( , ) provided a generalized way of studying the wiener index and the wiener polarity index for (d G, =3) W G_p( ) and

1

( )

( ) x

dH G x

W G dx =

, _{| =} _.

Motivated by the definition of W G_P( ), Ilić [3] recently defined (d G k, ) to be the generalized Wiener polarity index W G_k( ) for its significance. As a matter of fact, Gutman [4] showed that (d G k, ) is closely related with the first and the second Zagreb index M₁

and M₂. For example [4],

3 2

1 ) 2 ,

(G = M₁−m−

d

n



and

d(G,3) = M2−M1 + m− 4n

where m is the number of edges, while 3

n

 and 4

n

 denotes the number of triangles and

the number of quadrangles respectively. However, the relation between (d G k, ) and Zagreb indices has not yet been studied. The Cartesian product G H× of two graphs G and H is defined as:

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

( ) {( )( )| ( ) ] [ ( )}

E G H× = u v, u v u u, ∈E G and v =v or u =u and v v ∈E H

and the tensor product G⊗H of graphs G and H is defined as

1 1 2 2 1 2 1 2

( ) {( )( )| ( ) ( )}

E G⊗H = u v, u v u u, ∈E G and v v ∈E H .

More details about graph operations and generalized wiener polarity index can be found in articles [7, 8, 9].

(3)

2. T

HE

G

ENERALIZED

W

IENER

I

NDEX OF THE

C

ARTESIAN

P

RODUCT

G

××××

H

Before we turn to our main theorem of this section, there is a lemma to introduce

Lemma 2.1. [5] Let G₁ and G₂ be two simple connected graphs, then

1 2 1 2 1 2 2 1

( ) 2 ( ) ( ) | | ( ) | | ( ).

H G +G x, = H G x H G x, , +G H G x, +G H G x,

Since W G_k( )=0 when k >D G( ), we only consider k≤D G( ) in this paper. Now we are ready to present our main theorem of the section

Theorem 2.1. Let G₁ and G₂ be two simple connected graphs, then

1

1 2 1 2 2 1 1 2

1

( ) ( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

k

k k k i k i

i

W G G W G V G W G V G W G W G

−

− =

× = + +

∑

Proof. According to Lemma 2.1 and direct computation, we have

1 2

1

1 2 1 2 2 1

1

1 2 2 1 1 2

1

( ) ( )

2 ( ) ( ) | | ( ) | | ( )

( ) | ( ) | ( ) | ( ) | 2 ( ) ( ) k

k

i

k

k k i k i

i

W G H d G G k

d G i d G k i G d G k G d G k

W G V G W G V G W G W G

−

=

−

− =

× = + ,

= , , − + , + ,

= + +

∑

Thus the Theorem holds.  As a direct application to Theorem 2.1, it follows that

Corollary 2.1. Let G be a simple connected graph, then

1

( ) 2 ( ) | ( ) | 2 ( ) ( ) k

k k i k i

i

W G G W G V G W G W G

−

− =

× = +

∑

The graphs U =P_n×C_m and O=C_n×C_m are usually called C₄-nanotube and C₄ -nanotorus, where P_n denotes a path whose length is n−1 and C_m a cycle with m vertices.

4

C −nanotube and C₄−nanotorus are studied widely in materialogy and quantum chemistry for their extraordinary thermal conductivity and mechanical and electrical properties. By applying Theorem 2.1 we obtain:

(4)

2

( ) 2 min{ 1 }

2

( ) 2 min{ }

2 2

k

n W U kmn mk k n

n m W O kmn k

 

= − , < _{− ,  }

      = , < _{   },    

Proof. By Theorem 2.1 we have

1

2

( ) ( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

( ) 2 ( )

( ) ( 1)(2 )

2

k

k k n m k m n i n k i m

i k

i

W U W P V C W C V P W P W C

n k m mn n i m

n k m mn k n k m kmn mk

−

− =

−

=

= + +

= − + + −

= − + + − −

= −

∑

and

1

( ) ( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

2 2

2 2( 1)

2

k

k k n m k m n i n k i m

i k

i

W O W C V C W C V C W C W C

mn mn

mn k mn kmn

−

− =

−

=

= + +

= +

= + −

=

∑

The result follows.

Though W_k

(

P_n×C_m

)

(

min{_n− ,1_{ }n₂ }≤ ≤_k max{_n− ,1_{ }₂n }

)

    and W C_k

(

_n×C_m

)

(

min{ 2 2 } max{ 2 2 }

)

n , m ≤ ≤_k n , m

       

        are not considered in Corollary 2.2, we can discussed them in a similar way by applying Theorem 2.1.

3. T

HE

G

ENERALIZED

W

IENER

I

NDEX OF THE

T

ENSOR

P

RODUCT

G

⊗

H

In this section, we compute the generalized Wiener polarity index of the tensor product of graphs. A prepare work shall be introduced at first.

Lemma 3.1.[6] Let G₁ and G₂ be two simple connected graphs, and u=(u v₁, ₁),

2 2 1 2

( ) ( ) ( )

v= u v, ∈V G ×V G , then

1 2( ) max{ 1( 1 2) 2( 1 2)}

G G G G

d _⊗ u v, = d u u, ,d v v, .

Now we are ready for

(5)

1

1 2 1 2 2 1 1 2

1

2 1

1

( ) ( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

2 ( ) ( )

k

k k k k i

i k

k i

i

W G G W G V G W G V G W G W G

W G W G

−

=

⊗ = + +

+

∑

Proof. Let u=(u v1, 1), v=(u v2, 2)∈V G( 1)×V G( 2). According to Lemma 3.1, we have

1 2 1 2

( ) ( ) 0

( ) 0 ( )

( ) if and only if ( ) ( )

( ) ( )

d u u k d v v

d u u d v v k

d u v k d u u k d v v k

d v v k d u u k

, = , , =



 _, _{= ,} _, ₌



, = _ , = , , <

 _, _{= ,} _, _<



, = , , =



By summing up the numbers of five types of pairs of vertices above, we have

1 2 1 2 1 2 2

1 2 1 2 1

1

1 2 1 2

1

1 2 1 2

1

1 2 2 1 1 2

1

( ) | {( ) | ( ) ( )} |

|{( ) | ( ) ( )} |

{( ) | ( ) ( ) } |

( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

|

k

i k

i

k

k k k i

i

W G G u v d u u k v v V G u v d v v k u u V G

u v d u u k d v v i

u v d v v k d u u i

W G V G W G V G W G W G

−

=

−

=

× = , , = , = ∈

+ , , = , = ∈

+ , , = , , =

= + +

∑

2 1

1

2 ( ) ( )

k

k i

i

W G W G

=

+

∑

Therefore, the theorem is established.  A direct deduction of Theorem 3.1 is

Corollary 3.1. Let G be a simple connected graph, then

1

2

1

( ) 2 ( ) | ( )| 4 ( ) ( ) 2 ( ) k

k k k i k

i

W G G W G V G W G W G W G

−

=

⊗ = +

∑

+

And as an application of Theorem 3.1 we have:

(6)

2

( ) 4 3 min{ 1 }

2

( ) 4 min{ }

2 2

k

n W U kmn mk k n

n m W O kmn k

 

= − , < _{− ,  }

      = , < _{   },    

Proof. By Theorem 3.1 we have

( )

(

)

(

)

(

)

1 1 1 1 1 1 2

( ) ( ) | ( ) | ( ) | ( ) | 2 ( ) ( )

2

( ) 2 2

( ) 2 ( 1) (2 1)

4 3

k

k k n m k m n k n i m

i k

k m i n

i

k k

i i

W U W P V C W C V P W P W C

W C W P

n k m mn n k m m n i

n k m mn m n k k mk n k kmn mk − = = − = = = + + + = − + + − + − = − + + − − + − − = −

∑

and kmn n m m n mn C W C W C W C W C V C W C V C W O W k i k i k i n i m k k i m i n k n m k m n k k 4 2 2 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 1 1 1 1 1 1 = + + = + + + =

∑

= − = = − =

The result follows.  Acknowledgement. The author would like to thank the National Natural Science Foundation of China (No.11201156) for supporting this research.

R

EFERENCES

1. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1) (1947), 17–20.

2. H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math., 19 (1−3) (1988), 239–257.

3. A. Ilić, M. Ilić, Generalizations of Wiener polarity index and terminal Wiener index, Graphs Combin., 29 (5) (2013), 1403–1416.

(7)

polynomials, Distance in Molecular Graphs

−

Theory, Mathematical Chemistry Monograph 12 (2012), University of Kragujevac, 49–70.

5. D. Stevanović, Hosoya polynomial of composite graphs, Discrete Math., 235 (1−3) (2001), 237–244.

6. D. Stevannović, Distance regularity of compositions of graphs, Appl. Math. Lett., 17 (3)(2004), 337–343.

7. H. Khodashenas, M. J. Nadjafi−Arani, Distance−based topological indices of tensor product of graphs, Iranian J. Math. Chem., 3 (1) (2012), 45–53.

8. M. Faghani, A. R. Ashrafi, O. Ori, Remarks on The Wiener polarity index of some graph operations, J. Appl. Math. Informatics, 30 (3−4) (2012), 353–364.