Computing Some Topological Indices of Tensor Product of Graphs

Computing

 

Some

 

Topological

 

Indices

 

of

 

Tensor

 

Product

 

of

 

Graphs

 

 

ZAHRA YARAHMADI•

Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad University, Khorramabad, I. R. Iran

(Received June 10, 2011)

A

BSTRACT

A topological index of a molecular graph G is a numeric quantity related to G which is invariant under symmetry properties of G. In this paper we obtain the Randić, geometric-arithmetic, first and second Zagreb indices , first and second Zagreb coindices of tensor product of two graphs and then the Harary, Schultz and modified Schultz indices of tensor product of a graph G with complete graph of order n are obtained.

Keywords: Topological index, tensor product.

1.

I

NTRODUCTION

A topological index of a molecular graph G is a numeric quantity related to G which is

invariant under symmetry properties of G. The first and second Zagreb indices were

originally defined as M₁(G)=∑_a∈V(G)δ_G2a and M₂(G)=∑_ab∈E(G)δ_Gaδ_Gb,

respectively. The first Zagreb index can be also expressed as a sum over edges of G, ]

[ )

( _{( )}

1 G a b

M =

∑

_{ab∈E G} δ_G + δ_G , see [1, 2]. The first and second Zagreb coindices are

defined as M₁(G)=∑_ab∉E(G)[δ_Ga+δ_Gb] and M₂(G)=∑_ab∉E(G)δ_Gaδ_Gb, see [3]. In

1975, the chemist Milan Randić proposed a topological index based on the degrees of the end vertices of an edge in studying the properties of alkane [4]. The Randić index of a graph G is defined as R(G)=∑_ab∈E(G)(1/ δ_Ga δ_Gb .) The geometric−arithmetic index

(GA) was conceived, GA(G)=∑_ab∈E(G)( δ_Gaδ_Gb/1₂(δ_Gaδ_Gb)). Other topological indices

that will be used in this paper are the Schultz and modified Schultz indices and they are defined as follows:

W+(G)=

∑

{a,b}⊆V(G)(δGa+δGb)dG(a,b),

) , ( )

( _{{ , } ( )}

* G a bd a b

W _G

G V b

a G G

∑

⊆

= δ δ ,

respectively, see [5, 6] for details. The Harary index H(G) is defined as

) /

1 ( )

(G _{{ , } ( )} a b

H =∑ _{a b ⊆V G} δ_G δ_G [7]. For any two simple graphs G and H, the tensor

product G⊗H of G and H has vertex set V(G⊗H)=V(G)×V(H) and edge set

| ) , )( , ( { )

(G H a b c d

E ⊗ = ac∈E(G) and bd∈E(H)}. It is easy to prove that

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

|E G⊗H = E G E H [8]. In [9], the vertex PI index was proposed and the Wiener and vertex PI indices of this graph operation were computed in [10]. In this paper we study on some topological indices of tensor product of graph. At the beginning the Randić, GA, first and second Zagreb indices and first and second Zagreb coindices are computed. For obtaining Zagreb coindices of tensor product of graphs, we need another graph operations that recall them in the next stage.

The disjunction G∨H of two graphs G and H is the graph with vertex set

) ( )

(G V H

V × in which (a,b) is adjacent with (c,d) whenever a is adjacent with c in G or b

is adjacent with d in H.

The symmetric difference G⊕H of two graphs G and H is the graph with vertex

set V(G)×V(H) in which (a,b) is adjacent with (c,d) whenever a is adjacent with c in G

or b is adjacent with d in H, but not both.

For computing topological indices which related to distance in graphs, we use the useful and simple definitions and result in [11] for distance of vertices in tensor product of graphs.

Definition 1.1. Let G be a graph. We define dG′(x,y) for x,y∈V(G) as follows:

i. If dG(x,y) is odd then dG′(x,y) is defined as the length of the shortest even walk joining x and y in G, and if there is no shortest even walk then d_G′ (x,y)=+∞.

ii. If d_G(x,y) is even then d_G′(x,y) is defined as the length of the shortest odd walk

joining x and y in G, and if there is no shortest odd walk then dG′(x,y)=+∞.

iii. If dG(x,y)=+∞, thendG′ (x,y)=+∞.

Definition 1.2. Let G and H be two graphs and (a,b),(c,d)∈V(G⊗H). The relation Ron

the vertices of G⊗His defined as follows:

(a,b) R (c,d) if and only if d_G(a,c),d_H(b,d)<+∞ and d_G(a,c)+d_H(b,d) is even.

i. If (a,b) R (c,d) , then d_G⊗H((a,b),(c,d))=Max{d_G(a,c),d_H(b,d)}.

ii. If (a,b)R/ (c,d) then,

)}} , ( ), , ( { )}, , ( ), , ( { { ))

, ( ), ,

((a b c d Min Max d a c d b d Max d a c d b d

dG⊗H = G H′ G′ H .

We use the above definitions and Theorem for computing Schultz, modified Schultz and Harary indices of tensor product of complete graph Kn and a graph G.

2.

M

AIN

R

ESULTS

In this section, the Zagreb indices and coindices are computed for tensor product of graphs.

Theorem 2.1. Let G and H be graphs. The first and second Zagreb indices of tensor

product of G and H are given by:

), ( ) ( )

( _{1 1}

1 G H M G M H

M ⊗ =

M2(G⊗H)=2M2(G)M2(H).

Proof. By definition of Zagreb indices,

). ( ) (

) ( ) (

) ( ) (

) (

)) , ( ( )

(

1 1

) (

2

) (

2 ) ( ( )

2 2

2

) ( ) , (

2

) ( ) , ( 1

H M G M

b a

b a

b a

b a H

G M

H V b

H G

V a

G G V

a bV H

H G

H H G V b a

G H G V b a

H G

= = = = = ⊗

∑

∑

∑ ∑

∑

∑

∈ ∈

∈ ∈

⊗ ∈

⊗

∈ ⊗

δ δ

δ δ

δ δ δ

Also,

), ( ) ( 2

2

) (

) (

2

) , ( )

, ( )

(

2 2

) ( )

(

) ( ), (

) ( ) , )( , ( 2

H M G M

d b c

a

d c b a

d c b

a H

G M

H E

bd H H G

E

ac G G H E bd G E

ac G H G H H

G E d c b

a G H G H

=

∑ ∑

= ∑ =

∑ = ⊗

∈ ∈

∈ ∈

⊗

∈ ⊗ ⊗

δ δ δ

δ

δ δ δ δ

δ δ

which completes the proof. □

Theorem 2.2. Let G and H be graphs. The first and second Zagreb coindices of tensor product of G and H are computed as follows:

), (

) (

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

1 1

1 1

1 1

1

H G M H G M

G M G M H E H

M H M G E H

G M

∨ +

⊕ +

+ +

+ =

). ( ) ( )) ( ) ( )( ( 2 )) ( ) ( ( ) ( 2 ) ( 2 2 2 2 1 2 2 1 2 H G M H G M G M G M H M H M H M G M H G M ∨ + ⊕ + + + = ⊗

Proof. By definition

By similar method the second Zagreb coindex are obtained. □

Theorem 2.3. Let G and H be graphs. The Randić index of tensor product of G and H is

computed as follows:

). ( ) ( 2 )

(G H R G R H

R ⊗ =

Proof. By definition

). ( ) ( 2 1 1 2 1 1 2 ) , ( ) , ( 1 ) ( ) ( ( ) ) ( ( ) ) ( ) , )( , ( H R G R d b c a d b c a d c b a H G R G E

ac bd E H G G H H

G E

ac bd E H G G H H

H G E d c b

a G H G H

= = = = ⊗

∑ ∑

∑ ∑

∑

∈ ∈ ∈ ∈ ⊗ ∈ ⊗ ⊗ δ δ δ δ δ δ δ δ δ δ □ ). ( ) ( )) ( ) ( ( | ) ( | 2 )) ( ) ( ( | ) ( | 2 ) ( ) ( ) ( ) ( ) ( ) ( ] [ ] [ ] [ ] [ ] [ )] , ( ) , ( [ ) ( 1 1 1 1 1 1 1 1 ) ( ) ( ) ( ) ( ) ( ), ( ) ( ), ( ) ( ), ( , ) ( ) , )( , ( ) ( ) , )( , ( ) ( ) , )( , ( 1 H G M H G M G M G M H E H M H M G E H G M H G M c a b c a b d b a d b a d c b a d c b a d c b a d c b a d c b a d c b a H G M G E

ac H G G

G E

ac H G G

H E

bd G H H

H E

bd G H H

H E bd G E

ac G H G H

H E bd G E

ac G H G H

H E bd G E

ac G H G H

d b H G E d c b

a G H G H

H G E d c b

a G H G H

H G E d c b

a G H G H

Theorem 2.4. Let G and H be graphs and G be k-regular. The GA index of tensor product

of G and H is computed as follows:

). ( ) ( 2 )

(G H E G GA H

GA ⊗ =

Proof. By definition

). ( ) ( 2 ) (

) (

)) , ( )

, ( (

) , ( ) , ( )

(

) ( ) , )( ,

( 21

) ( ) , )( ,

( 21

) ( ) , )( ,

( 21

H GA G E d

b k

d b k

d c b a

d c b a

d c b

a

d c b

a H

G GA

H G E d c b

a H H

H H H

G E d c b

a G H G H

H G H G H G E d c b

a G H G H

H G H

G

= +

=

+ =

+ =

⊗

∑

∑

∑

⊗ ∈

⊗ ∈

⊗

∈ ⊗ ⊗

⊗ ⊗

δ δ

δ δ

δ δ δ δ

δ δ δ δ

δ δ

δ δ

□

Suppose G is a graph. Define the set TG ⊆E(G) as follows:

} |

) (

{ab E G abiscontained in atriangle

TG = ∈ .

Theorem 2.5. Let G be a graph and Kn be a complete graph of order n. The Harary index of tensor product of K_n and G is computed as follows:

. )

( )

( 2 ) ( )

( 1₆

3 2 2

1 2

G

n V G nE G nT

n G

H n G K

H _⎟⎟ − +

⎠ ⎞ ⎜⎜ ⎝ ⎛ + =

⊗

Proof. By definition of Harary index,

. )) , ( ), , ((

1 )

(

) ( )} , )( ,

{(

∑

⊆ ⊗

= ⊗

G K V d c b a n

n d a b c d

G K H

For each (a,b),(c,d)∈V(K_n⊗G) exactly one of the following cases hold:

{

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

}

1 a b c d a c b d a b R c d

A = ≠ ≠ ,

{

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

}

2 ab c d a c b d a b R c d

A = ≠ ≠ / ,

A₃=

{

{(a,b),(c,d)}|a≠c,b=d

}

,

A₄ =

{

{(a,b),(c,d)}|a=c,b≠d,(a,b)R(c,d)

}

,

Therefore,

∑

∑

∑

∈ ⊗

∈ ⊗

∈ ⊗

+ +

= ⊗

5 3

1

)} , )( , {(

)} , )( , {(

)} , )( , {(

)) , ( ), , ((

1

)) , ( ), , ((

1

)) , ( ), , ((

1 )

(

A d c b

a K G

A d c b

a K G

A d c b

a K G

n

d c b a d

d c b a d

d c b a d

G K H

n n n

We evaluate each sums separately. It is obvious, if {a,c}⊆V(K_n), then 1

) , (a c =

d

n

K and d′K_n(a,c)=2. By using notation of Definitions 1.1 and 1.2, one can see that, if (a,b) R/ (c,d)and a≠c,b≠d then,

), , ( )}} , ( ), , ( { )}, , ( ), , ( {

{Max d a c d b d Max d a c d b d d b d

Min _{K G K G G}

n

n ′ ′ =

) , ( )} , ( ), , (

{d a c d b d d b d

Max K_n G = G .

Hence

By attention to the set A3, we have:

∑

∑

∈⊆

∈ ⊗ ⎟⎟⎠

⎞ ⎜⎜ ⎝ ⎛ =

=

) (( ) } ,

{ 2

1

)} , )( ,

{( 2

) ( 2

1 ))

, ( ), , ((

1

3

G V b

K V c a A

d c b

a Kn G n

n G V d

c b a

d .

For computing the 4−th summation, we know that,

{

( , )( , )| ( ),{ , } ( ) 2| ( , )

}

4 a b a d a V K b d V G and d b d

A = ∈ n ⊆ G .

Hence,

) ( 2 2

) , (

1

)) , ( ), , ((

1

)) , ( ), , ((

1

)) , ( ), , ((

1 ))

, ( ), , ((

1

) ( } ,

{ { , } ( )

)} , )( , {(

)} , )( , {(

)} , )( , {( )}

, )( , {(

2 2 1 2

1

G H n

d b d

d c b a d

d c b a d

d c b a d

d c b a d

n

n n n n

K V c

a b d V G G A

d c b

a K G A

d c b

a K G A

d c b

a K G A

A d c b

a K G

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

∑ ∑

=

∑ +

∑ +

∑ =

∑

⊆ ⊆

∈ ⊗

∈ ⊗

∈ ⊗

∪

Now we can compute the 5−th summation,

∑

∈ ₅ ⊗

)} , )( ,

{( (( , ),( , ))

1

A d c b

a dKn G ab c d

∑

/ ⊆

∈ ⊗

=

) , ( | 2, } ( ) { ( )

)) , ( ), , ((

1

d b d

G V d b

K V

a K G

G

n d n a b c d

If dG(b,d)is odd then by Theorem 1.3,

)} , ( )}, , ( , 3 { {

)}} , ( ), , ( { )}, , ( ), , ( { { ))

, ( ), , ((

d b d d b d Max Min

d b d a a d Max d

b d a a d Max Min d

a b a d

G G

G K

G K

G

K_{n n n}

′ =

′ ′

=

⊗

By attention to different cases for d_G(b,d) and d_G′(b,d), we can see:

⎪ ⎩ ⎪ ⎨ ⎧ = ′

3 2

) , ( )}

, ( )}, , ( , 3 { {

d b d d b d d b d Max Min

G

G

G

4 ) , ( & 1 ) , (

2 ) , ( & ) , (

3 ) , (

≥ ′

=

= ′

≥

d b d d

b d

d b d d b d

d b d

G G

G G

G

Hence, the following sets are defined:

} )

, ( 3

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

5 ab a d a V K d b d and d a b isodd

A′ = ∈ _{n G} ≥ _G ,

A₅′′={{(a,b),(a,d)}|a∈V(K_n), d_G(b,d)=1and d_G′(a,b)=2},

A₅′′′={{(a,b),(a,d)}|a∈V(G),d_G(b,d)=1and d′_G(a,b)≥4}.

Thus,

. ) , (

1

)) , ( ), , ((

1

)) , ( ), , ((

1

) , ( | 2, } ( ) { ( )

) , ( | 2, } ( ) { ( )

)} , )( ,

{( 4

∑

∑

∑

⊆ ∈

⊆

∈ ⊗

∈ ⊗

= =

d b d

G V d b

K V

a G

d b d

G V d b

K V

a K G

A d c b

a K G

G n G

n n n

d b d

d c b a d

|). | | ) ( (| | | | ) ( | ) , ( 1 3 1 2 1 ) , ( 1 )) , ( ), , (( 1 )) , ( ), , (( 1 )) , ( ), , (( 1 )) , ( ), , (( 1 )) , ( ), , (( 1 3 1 2 1 ) , ( | 2 )} , ( ), , {( )} , ( ), , {( )} , ( ), , {( )} , )( , {( )} , )( , {( )} , )( , {( )} , )( , {( )} , )( , {( 5 5 5 5 5 5 5 5 G G d b d G A d a b a A d a b a A d a b a G A d c b

a K G A

d c b

a K G A

d c b

a K G A

d c b

a K G A

d c b

a K G

T G E n T n G E d b d n d b d d c b a d d c b a d d c b a d d c b a d d c b a d G n n n n n − + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∑ = ∑ + ∑ + ∑ = ∑ + ∑ + ∑ + ∑ = ∑ / ′′′ ′ ∈ ′′ ∈ ′ ∈ ∈ ⊗ ∈ ⊗ ∈ ⊗ ∈ ⊗ ∈ ⊗

By above calculations,

( ) ( ) .

2 ) ( )

( ₆1

3 2 2 1 2 G

n V G nE G nT

n G H n G K

H _⎟⎟ − +

⎠ ⎞ ⎜⎜ ⎝ ⎛ + = ⊗

Theorem 2.6. Let G be a graph and K_n be a complete graph of order n. The Schultz and

modified Schultz indices of tensor product of K_n and G are given by:

] ) ( 2 ) ( 2 ) ( ) 1 ( 8 ) ( 2 [ 2 )

( _{+ 1}

∑

_{= ∉}

+ ⎟⎟ + − + + + ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⊗ G T bd

e G G

n nW G n n E G M G b d

n G K

W δ δ ,

]. ) ( ) 1 2 ( ) ( ) 1 ( 4 ) ( [ ) 1 ( )

( 2 2 _{* 1 2}

* ⊗ = − + − + − +

∑

_{= ∉}

G

T bd

e G G

n G n n W G n n M G n M G n d b d

K

W δ δ .

Proof. We just prove the Schultz index of Kn⊗G, modified Schultz index is obtained

similarly. By using the proof of Theorem 2.5 and definition of Schultz index, we have:

∑

⊗ ⊆ ⊗ ⊗ ⊗ + ⊗ = + ) ( )} , ( ), , {( )) , ( ), , (( )] , ( ) , ( [ ) ( G K V d c b a G K G K G K n n n n

n ab c d d a b c d

G K

W δ δ

∑ ∑

− ∈ ⊗ ⊗ ⊗

+ = 5

1{( , ),( , )}

)) , ( ), , (( )] , ( ) , ( [

i ab cd A

G K G K G K i n n

∑ ∑

− ∈ ⊗

+ −

= 5

1{( , ),( , )}

)) , ( ), , (( ]

[ )

1 (

i ab cd A

G K G G

i

n a b c d

d d b

n δ δ

( ) 2 ) 1 ( 8 ) ( 2 ) 1 (

2 n n W G n n_⎟⎟E G

⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟

⎠ ⎞ ⎜⎜ ⎝ ⎛ −

= ₊

] )[

, ( )

1 (

) , ( |

2, } ( ) {

d b d b d n

n _{G G}

d b d

G V d b

G

G

δ δ + −

+

∑

⊆

] )[

, ( )

1 (

) , ( |

2, } ( ) {

d b d b d n

n _{G G}

d b d

G V d b

G

G

δ δ + −

+

∑

/ ⊆

] )[

, ( )

1 (

) (( ) } , {

d b d b d n

n _{G G}

G E bd e

G V d b

G δ +δ

−

−

∑

∈ = ⊆

∑

∈ = ⊆

+ −

+

G

T bd e

G V d b

G

Gb d

n n

) ( } , {

] [

) 1 (

2 δ δ

.] [

) 1 ( 3

) ( } , {

∑

∉ = ⊆

+ −

+

G

T bd

ebd V G

G

Gb d

n

n δ δ

By above calculations, we conclude that:

]. ) (

2 ) ( 2 ) ( ) 1 ( 8 ) ( 2 [ 2 )

( _{+ 1}

∑

_{= ∉}

+ ⎟⎟ + − + + +

⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⊗

G

T bd

e G G

n nW G n E G M G b d

n G K

W δ δ

R

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