Study on Some Topological Indices of Semientire and Entire Dominating Graphs of Kragujevac Trees

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Study on Some Topological Indices of Semientire and Entire Dominating Graphs of Kragujevac Trees

B. Basavanagoud

¹

and Sujata Timmanaikar

²

1

Department of Mathematics, Karnatak University, Dharwad, INDIA.

2

Department of Mathematics,

Government Engineering College, Haveri, INDIA.

email: b.basavanagoud@gmail.com,sujata123rk@gmail.com (Received on: January 6, 2018)

ABSTRACT

A topological index of a graph is a single unique number characteristic of the graph and is mathematically invariant under graph automorphism. In this paper we compute the some of the topological indices like first Zagreb index, first zagreb coindex, forgotten index, Wiener and hyper-Wiener indices of semientire and entire dominating graphs of Kragujevac tree.

AMS Subject Classification: 05C05, 05C07, 05C69.

Keywords: degree; Zagreb indices; Zagreb coindices; forgotten index; distance;

Wiener index; hyper-Wiener index ; d-transformation graphs.

1. INTRODUCTION

The graphs in this paper are simple and undirected. The terminology not defined here can be found in

¹⁷

. Let G = (V, E) be such a graph. The number of vertices of G we denote by n and the number of edges we denote by m, thus ∣V (G)∣ = n and ∣E(G)∣ = m. By the open neighborhood of a vertex v of G we mean the set N

G

(v) = {u ∈ V (G) : uv ∈ E(G)}. The degree of a vertex v, denoted by d

G

(v), is the cardinality of its open neighborhood. Let S be a finite set and let F = {S

1

, S

2

, ..., S

n

} be a partition of S. Then the intersection graph Ω(F ) of F is the graph whose vertices are the subsets in F and in which two vertices S

i

and S

j

are adjacent if and only if S

i

∩ S

j

∕= Φ.

A graph invariant is any function on a graph that does not depend on a labeling of its

vertices. Such quantities are also called topological indices. Hundreds of different invariants

have been employed to date (with unequal success)in QSAR/QSPR studies. Among more useful of them appear two that are known under various names, but mostly as Zagreb indices.

Due to their chemical relevance they have been subject to numerous papers in chemical literature

^{14, 31}

. There are two invariants called the first Zagreb index and second Z agreb index

¹⁴

defined as

( ) = ( ) ( ) = ( ) ( ),

) ( 2

2

) (

1

G d u and M G d u d v

M

_{G G}

G E uv G

G V

u

 





respectively.

In fact, one can rewrite the first Zagreb index as ( ) = [ ( ) ( )].

) (

1

G d u d v

M

_{G G}

G E uv

 



In view of above equations the first and second Zagreb coindices were recently introduced and defined as

), ( ) (

= ) ( )]

( ) ( [

= ) (

) ( 2

) (

1

G d u d v and M G d u d v

M

_{G G}

G E uv G

G G E

uv

 





 respectively

¹

.

Also, another topological index namely forgotten index is

¹⁰

defined as the sum of cubes of vertex degrees and is given by

³

) (

) (

= )

( G d u

F

_G

G V

u





In fact, one can rewrite the forgotten index as

( ) = [ ( )

²

( )

²

].

) (

v d u d G

F

_{G G}

G E uv

 



Usage of topological indices in biology and chemistry began in 1947 when H.

Wiener

^33,34

introduced Wiener index which is denoted by W and is given by ( ) = ( , )

) ( ) , (

v u d G

W

_G

G V v

u





The hyper-Wiener index of acyclic graphs was introduced by Milan Randic in 1993. Then Klein et al.

²⁷

, generalized Randic’s definition for all connected graphs, as a generalization of the Wiener index. It is defined as

[ ( , ) ( , ) ] 2

= 1 )

(

²

) ( ) , (

v u d v u d G

WW

_{G G}

G V v u

 



Readers interested in more information on computing topological indices of graph operations can be referred to

2, 3, 4, 6, 7, 8, 9, 11, 12, 15, 16, 20, 21, 23, 24, 25, 26, 30, 32, 35

.

2. KRAGUJEVAC TREES

A connected acyclic graph is called a tree. A rooted tree is a tree in which one

particular vertex is distinguished; this vertex is referred to as the root. The vertex of degree

one is a pendant vertex. The vertex adjacent to pendant vertex is called support vertex. The

formal definition of a Kragujevac tree was introduced in

¹³

.

Definition 1.[13] Let P

₃

be the 3-vertex tree,rooted at one of its terminal vertices.For 2,3,...

=

k , construct the rooted tree B

_k

by identifying the roots of k copies of P

₃

. The vertex obtained by identifying the roots P

₃

-trees is the root of B

_k

.

Examples illustrating the structure of the rooted tree B

_k

depicted in Figure 1.

Figure 1.

Definition 2.[13] Let d  2 be an integer. Let 

₁

, 

₂

, 

₃

... 

_d

be rooted trees specified in Definition 1, i.e., 

₁

, 

₂

, 

₃

..., 

_d

 { B

₂

, B

₃

,...} . A Kragujevac tree T is a tree possessing a vertex of degree d, adjacent to the roots of 

₁

, 

₂

, 

₃

..., 

_d

. This vertex is said to be the central vertex of T, where d is the degree of T. The subgraphs 

₁

, 

₂

, 

₃

..., 

_d

are the branches of T.

Recall that some (or all) branches of T may be mutually isomorphic.

A typical Kragujevac tree of degree d=5 is depicted in Figure 2.

Figure 2.

Clearly, the branch B

_k

has 2k+1 vertices. Therefore, the vertex set and the edge set of the Kragujevac tree T are:

1) (2

= ) ( 1)

(2 1

= ) (

1

= 1

=





  

^{d i}

i i

d

i

k T

E and k

T V

We denote the vertices of Kragujevac trees as follows:

• pendant vertices by x

_i

• support vertices by w

_i

• vertices belongs to the set N ( w

_i

)  { x

_i

}

_i^k₌₁

by v

_i

• the central vertex by u

Recent information on Kragujevac trees can be found in

^{5, 13, 22.}

3. d-TRANSFORMATION GRAPHS

The theory of domination has emerged as one of the most studied area in graph theory and its allied branch in mathematics. The wide variety of domination parameters have been defined and studied their applications in various fields

¹⁸

. A subset D ⊆ V (G) is a dominating set of G if every vertex of V (G) ∖ D has a neighbor in D. The domination number of graph G, denoted by (G), is the minimum cardinality of a dominating set of G. The dominating set D is called a minimal dominating set if no proper subset of D is a dominating set. For a comprehensive survey of domination in graphs, see

¹⁹

.

In the mathematical literature several transformation graphs have been considered and constructed. Let G denote the set of simple undirected graphs. Various important results in graph theory have been obtained by considering some functions ℱ : G → G or ℱ

s

: G

1

× ... × G

s

→ G called transformations or operations (here each G

i

= G) and by establishing how these operations affect certain properties or parameters of graphs. The complement, the k-th power of graph, line graph and the total graph are well known examples of such transformations or operations. The same concept of transformation has been applied to dominating sets and defined a variety of dominating transformation(d-transformation) graphs by Prof. V.R. Kulli and his students

^{28, 29}

. In what follows we define the best known representatives of such graphs.

The semientire dominating graph Ed(G) of a graph G is the graph with the vertex set V ∪S where S is the set of all minimal dominating sets of G and with two vertices u, v in V ∪ S adjacent if u, v ∈ D where D is a minimal dominating set or u ∈ V and v = D is a minimal dominating set of G containing u.

²⁸

.

The entire dominating graph ED(G) of a graph G is the graph with the vertex set V ∪ S where S is the set of all minimal dominating sets of G and with two vertices u, v in V ∪ S adjacent if u = D

1

and v = D

2

are two nondisjoint minimal dominating sets in S or u, v ∈ D where D is minimal dominating set or u ∈ V and v = D is a minimal dominating set of G containing u.

²⁹

.

In Figure 3, a graph G and its semientire and entire dominating graphs are shown.

Figure 3 : A graph G and its semientire and entire dominating graphs.

In Figure 4, Kragujevae tree T and its semientire graph Ed(T) and entire graph ED(T) are given.

Figure 4 : The Kragujevac tree and its semientire and entire dominating graphs.

In this paper, expressions for the first Zagreb index , first Zagreb coindex,forgotten index, Wiener index and hyper Weiner index are obtained for semientire and entire dominating graphs of Kragujevac tree.

4. COMPUTING FIRST ZAGREB INDEX AND FIRST ZAGREB COINDEX FOR ED(T ) AND ED(T ) OF KRAGUJEVAC TREE T.

We begin with the following straightforward, previously known auxiliary result.

Lemma 1. [1] Let G be any nontrivial graph of order n and size m. Then

M

₁

( G )  M

₁

( G ) = 2 m ( n  1).

Lemma 2. The dominating sets of the Kragujevac tree T are:

} , {

=

1

=

1 i

k

i

w u

D 

} , {

=

1

=

2 i

k

i

x u

D 

} , {

=

1

= 1

=

3 i

k

i i d

i

w v

D  

}.

, {

=

1

= 1

=

4 i

k

i i d

i

x v

D  

Before going to prove our next result we need the following lemma.

Lemma 3. The degree of each vertex in the semientire dominating graph of Kragujevac tree T is:

1. d

Ed(T)

(u) = 2(k + 1) 2. d

Ed(T)

(w

i

) = (k + d + 2) 3. d

Ed(T)

(x

i

) = (k + d + 2) 4. d

Ed(T)

(v

i

) = (2k + d + 1) 5. d

Ed(T)

(D

1

) = (k + 1) 6. d

Ed(T)

(D

2

) = (k + 1) 7. d

Ed(T)

(D

3

) = (k + d) 8. d

Ed(T)

(D

4

) = (k + d)

Theorem 4. The first Zagreb index of the semientire dominating graph of Kragujevac tree T is

M

1

(Ed(T )) = 2 [k

³

+ 8k

²

+ 10k + 3 ]+ 2kd[ 4k + 3d + 8 ] + d [d

²

+ 4d + 1] .

Proof. By the definition of Ed(G) it is clear that V (Ed(T )) = V (T ) ∪ S(T ). By using the definition of first Zagreb index and Lemma 3, we have

2 ) ( 1

=

1

( Ed ( T )) = d ( u )

M

_{Ed T}

n

i



²

1 ) ( 2

1 ) ( 2

)

(

( )] [ ( ) ] [ ( ) ]

[

=  









^k

i

i T Ed k

i

i T Ed T

Ed

u d w d x

d

2 2 ) ( 2

1 ) ( 2

1 )

(

( ) ] [ ( )] [ ( )]

[ d v d

_{Ed T}

D d

_{Ed T}

D

d

i

i T

Ed

 

 



2 4 ) ( 2

3 )

(

( )] [ ( )]

[ d

_{Ed T}

D  d

_{Ed T}

D



= 2(k + 1)]

²

+ k(k + d + 2)

²

+ k(k + d + 2)

²

+ d(2k + d + 1)

²

+ (k +1)

²

+ (k + 1)

²

+ (k + d)

²

+ (k + d)

²

M

1

(Ed(T )) = 2[ k

³

+ 8k

²

+ 10k + 3 ]+ 2kd[ 4k + 3d + 8 ] + d [d

²

+ 4d + 1] . By the definition of entire dominating graph ED(G), V (ED(T )) = V ∪ S, where S= {D

1

, D

2

, D

3

, D

4

}. Hence the following lemma gives the information about the degree of each vertex of ED(T ) of Kragujevac tree T.

Lemma 5. The degree of each vertex entire dominating graph of Kragujevac tree T is:

1. d

ED(T)

(u) = 2(k + 1) 2. d

ED(T)

(w

i

) = (k + d + 2) 3. d

ED(T)

(x

i

) = (k + d + 2) 4. d

ED(T)

(v

i

) = (2k + d + 1) 5. d

ED(T)

(D

1

) = (k + 3) 6. d

ED(T)

(D

2

) = (k + 3) 7. d

ED(T)

(D

3

) = (k + d + 2) 8. d

ED(T)

(D

4

) = (k + d + 2).

Theorem 6. The first Zagreb index of the entire dominating graph of Kragujevac tree T is M

1

(ED(T )) = 2 [k

³

+ 8k

²

+ 18k + 15] + 2kd[ 4k + 3d + 8] + d[ d

²

+ 4d + 9] .

Proof. By the definition of ED(G) it is clear that V (ED(T )) = V (T ) ∪ S(T ). By using the definition of first Zagreb index and Lemma 5, we have

2 ) ( 1

=

1

( ED ( T )) = d ( u )

M

_EDT

n

i



²

1 ) ( 2

1 ) ( 2

)

(

( )] [ ( ) ] [ ( ) ]

[

=  









^k

i

i T ED k

i

i T ED T

ED

u d w d x

d

2 2 ) ( 2

1 ) ( 2

1 )

(

( ) ] [ ( )] [ ( )]

[ d v d

_EDT

D d

_EDT

D

d

i

i T

ED

 

 



2 4 ) ( 2

3 )

(

( )] [ ( )]

[ d

_EDT

D  d

_EDT

D



= [2(k + 1)]

²

+ k(k + d + 2)

²

+ k(k + d + 2)

²

+ d(2k + d + 1)

²

+ (k + 3)

²

+ (k + 3)

²

+ (k + d+2)

²

+ (k + d+2)

²

M

1

(ED(T )) = 2 [k

³

+ 8k

²

+ 18k + 15] + 2kd[ 4k + 3d + 8] + d[ d

²

+ 4d + 9] . Applying Lemma 1, from the results of Theorems 4 and 6 we can deduce expressions for the first Zagreb coindex of the semientire dominating graph and entire dominating graphs of Kragujevac tree T. These are collected in the following.

Corollary 7. Let T be a Kragujevac tree of order n and size m. Then

. 1]}

+ 4d + [d d + ] 8 + 3d + 4k 2kd[

+ ] 3 + 10k + 8k + k 2[

{ 1)]

(2 2[

= ))

(Ed(T ^{2 3 2 2}

1

=

1



^{d i} 

i

k M

. 9]}

+ 4d + [d d + ] 8 + 3d + 4k 2kd[

+ ] 15 + 18k + 8k + k 2[

{ 1)]

(2 2[

= ))

(ED(T ^{2 3 2 2}

1

=

1



^{d i} 

i

k M

5. COMPUTING FORGOTTEN INDEX FOR ED(T) AND ED(T) OF KRAGUJEVAC TREE T.

Using the definition of forgotten index and Lemma 3 and 5 we can deduce the expression for the forgotten index of Ed(T ) and ED(T ) of Kragujevac tree T .

By the definition of semientire dominating graph Ed(G) and Lemma 3, we can deduce forgotten index for Ed(T ) for Kragujevac tree in the following theorem.

Theorem 8. The forgotten index of the semientire dominating graph of Kragujevac tree

T is

F (Ed(T )) = 2k[ k

³

+ 12k

²

+ 27k + 3kd + 3d

²

+ 23 ] + 2kd [7k

²

+ 4d

²

+ 18k + 12d + 9kd +15]+[d

⁴

+ 5d

³

+ 3d

²

+ d + 10] .

Proof. By the definition of Ed(G) it is clear that V (Ed(T )) = V (T ) ∪ S(T ). By using the definition of Forgotten index and Lemma 3, we have

3 ) ( 1

=

) (

= )) (

( Ed T d u

F

_{Ed T}

n

i



3

1 ) ( 3

1 ) ( 3

)

(

( )] [ ( ) ] [ ( ) ]

[

=  









^k

i

i T Ed k

i

i T Ed T

Ed

u d w d x

d

3 2 ) ( 3

1 ) ( 3

1 )

(

( ) ] [ ( )] [ ( )]

[ d v d

_{Ed T}

D d

_{Ed T}

D

d

i

i T

Ed

 

 



3 4 ) ( 3

3 )

(

( )] [ ( )]

[ d

_{Ed T}

D  d

_{Ed T}

D



= 2(k + 1)]

³

+ k(k + d + 2)

³

+ k(k + d + 2)

³

+ d(2k + d + 1)

³

+ (k + 1)

³

+ (k + 1)

³

+ (k + d)

³

+ (k + d)

³

F (Ed(T )) = 2k[ k

³

+ 12k

²

+ 27k + 3kd + 3d

²

+ 23 ] + 2kd [7k

²

+ 4d

²

+ 18k + 12d + 9kd + 15] +[d

⁴

+ 5d

³

+ 3d

²

+ d + 10] .

By the definition of entire dominating graph ED(G), V (ED(T )) = V ∪ S, where

S = {D

1

, D

2

, D

3

, D

4

}. By using the definition of forgotten index and Lemma 5, we deduce forgotten index for ED(T )) of Kragujevac tree T in next theorem.

Theorem 9. The forgotten index of the entire dominating graph of Kragujevac tree T is F (ED(T ) = 2k[ k

³

+ 12k

²

+ 39k + 59 ]+ 2kd[ 7k

²

+ 21k + 4d

²

+ 15d + 9kd + 27]

+[d

⁴

+ 5d

³

+ 15d

²

+ 25d + 78 ].

Proof. By the definition of forgotten index and Lemma 5, we have

3 ) ( 1

=

) (

= )) (

( ED T d u

F

_EDT

n

i



³

1 ) ( 3

1 ) ( 3

)

(

( )] [ ( ) ] [ ( ) ]

[

=  









^k

i

i T ED k

i

i T ED T

ED

u d w d x

d

3 2 ) ( 3

1 ) ( 3

1 )

(

( ) ] [ ( )] [ ( )]

[ d v d

_EDT

D d

_EDT

D

d

i

i T

ED

 

 



3 4 ) ( 3

3 )

(

( )] [ ( )]

[ d

_EDT

D  d

_EDT

D



=[2(k + 1)]

³

+ k(k + d + 2)

³

+ k(k + d + 2)

³

+ d(2k + d + 1)

³

+ (k + 3)

³

+ (k + 3)

³

+ (k + d + 2)

³

+ (k + d + 2)

³

F (ED(T ) = 2k[ k

³

+ 12k

²

+ 39k + 59 ]+ 2kd[ 7k

²

+ 21k + 4d

²

+ 15d + 9kd + 27]

+[d

⁴

+ 5d

³

+ 15d

²

+ 25d + 78 ].

6. COMPUTING WEINER INDEX FOR ED(T ) AND ED(T) OF KRAGUJEVAC TREE T.

Lemma 10. The distance between each vertex in the semientire dominating graph of Kragujevac tree T is:

1. d

Ed(T)

(u, v

i

) = 2 2. d

Ed(T)

(u, x

i

) = 1 3. d

Ed(T)

(u, w

i

) = 1

4. ( , ) = 6

4

1

= )

Ed(T i

i

D u

d 

5. ( , ) = 1

1

= )

Ed(T i

d

i

i

v

x

d 

6. ( , ) = 2

1

= )

Ed(T i

k

i

i

w

x

d 

7. d

Ed(T)

(x

i

, x

j

) = 5 8. d

Ed(T)

(w

i

,w

j

) = 5 9. d

Ed(T)

(v

i

, v

j

) = 1

10. ( , ) = 1

1

= )

Ed(T i

d

i

i

v

w

d 

11. ( , ) = 6

4

1

= )

Ed(T i

i

i

D

x

d 

12. ( , ) = 6

4

1

= )

Ed(T i

i

i

D

w

d 

13. ( , ) = 6

4

1

= )

Ed(T i

i

i

D

v

d 

14. d

Ed(T)

( D

_i

, D

_j

) = 12

Theorem 11. The Wiener index of the semientire dominating graph of Kragujevac tree T is

W [Ed(T )] = 2k

²

+ 2kd + 14k + 8d + 29.

Proof. By the definition of Ed(G) it is clear that V (Ed(T )) = V (T ) ∪ S(T ). By using the definition of Wiener index and Lemma 10, we have

) , (

= )) ( Ed

(

_Ed(T)

) , (

v u d T

W

V v

u





= ( , ) ( , ) ( , )

_Ed(T)

( , )

1

= )

Ed(T 1

= )

Ed(T 1

= )

Ed(T 1

=

i i k

i i k

i i k

i i d

i

w x d w

u d x

u d v

u

d   

 ^{  }

( , ) ( , )

_Ed(T)

( , )

<

) Ed(T 1

= 1

= )

Ed(T 1

= 1

=

j i j

i i i d

i k

i i i d

i k

i

x x d v

w d v

x

d   



 ^{ }



( , ) ( , ) ( , ) ( , )

4

1

= ) Ed(T 1

= 4

1

= ) Ed(T )

Ed(T

<

) Ed(T

<

i i i k

i i i j

i j

i j i j

i

D x d D

u d v v d w

w

d    

 ^{  }



( , ) ( , )

_Ed(T)

( , )

<

4

1

= ) Ed(T 1

= 4

1

= ) Ed(T 1

=

j i j

i i i i d

i i i i k

i

D D d

D v d D

w

d    

 ^{ }



W [Ed(T )] = 2k

²

+ 2kd + 14k + 8d + 29.

Thus, the result follows.

Lemma 12. The distance between each vertex in the entire dominating graph of Kragu-jevac tree T is:

1. d

ED(T)

(u, v

i

) = 2 2. d

ED(T))

(u, x

i

) = 1 3. d

ED(T)

(u, w

i

) = 1

4. ( , ) = 6

4

1

= )

ED(T i

i

D u

d 

5. ( , ) = 1

1

= )

ED(T i

d

i

i

v

x

d 

6. ( , ) = 2

1

= )

ED(T i

k

i

i

w

x

d 

7. d

ED(T)

(x

i

, x

j

) = 5 8. d

ED(T)

(w

i

,w

j

) = 5 9. d

ED(T)

(v

i

, v

j

) = 1

10. ( , ) = 1

1

= )

ED(T i

d

i

i

v

w

d 

11.

_{( , )=6}

4

1

= )

ED(T i

i

i D

x

d



12. ( , ) = 6

4

1

= )

ED(T i

i

i

D

w

d 

13. ( , ) = 6

4

1

= )

ED(T i

i

i

D

v

d 

14. d

ED(T)

( D

_i

, D

_j

) = 8

Theorem 11. The Wiener index of the semientire dominating graph of Kragujevac tree T is W [Ed(T )] = 2k

²

+ 2kd + 14k + 8d + 29.

Proof: By the definition of Ed(G) it is clear that V (Ed(T )) = V (T ) ∪ S(T ). By using the definition of Wiener index and Lemma 10, we have

) , (

= )) ( Ed

(

_Ed(T)

) , (

v u d T

W

V v

u





= ( , ) ( , ) ( , )

_Ed(T)

( , )

1

= )

Ed(T 1

= )

Ed(T 1

= )

Ed(T 1

=

i i k

i i k

i i k

i i d

i

w x d w

u d x

u d v

u

d   

 ^{  }

( , ) ( , )

_Ed(T)

( , )

<

) Ed(T 1

= 1

= )

Ed(T 1

= 1

=

j i j

i i i d

i k

i i i d

i k

i

x x d v

w d v

x

d   



 ^{ }



( , ) ( , ) ( , ) ( , )

4

1

= ) Ed(T 1

= 4

1

= ) Ed(T )

Ed(T

<

) Ed(T

<

i i i k

i i i j

i j

i j i j

i

D x d D

u d v v d w

w

d    

 ^{  }



( , ) ( , )

_Ed(T)

( , )

<

4

1

= ) Ed(T 1

= 4

1

= ) Ed(T 1

=

j i j

i i i i d

i i i i k

i

D D d

D v d D

w

d    

 ^{ }



W [Ed(T )] = 2k

²

+ 2kd + 14k + 8d + 29.

Thus, the result follows.

Lemma 12. The distance between each vertex in the entire dominating graph of Kragujevac tree T is:

1. d

ED(T)

(u, v

i

) = 2 2. d

ED(T))

(u, x

i

) = 1 3. d

ED(T)

(u, w

i

) = 1

4. ( , ) = 6

4

1

= )

ED(T i

i

D u

d 

5. ( , ) = 1

1

= )

ED(T i

d

i

i

v

x

d 

6. ( , ) = 2

1

= )

ED(T i

k

i

i

w

x

d 

7. d

ED(T)

(x

i

, x

j

) = 5

8. d

ED(T)

(w

i

,w

j

) = 5

9. d

ED(T)

(v

i

, v

j

) = 1

10. ( , ) = 1

1

= )

ED(T i

d

i

i

v

w

d 

11. ( , ) = 6

4

1

= )

ED(T i

i

i

D

x

d 

12. ( , ) = 6

4

1

= )

ED(T i

i

i

D

w

d 

13. ( , ) = 6

4

1

= )

ED(T i

i

i

D

v

d 

14. d

ED(T)

( D

_i

, D

_j

) = 8

Theorem 13. The Wiener index of the entire dominating graph of Kragujevac tree T is W [ED(T )] = 2k

²

+ 2kd + 14k + 8d + 25.

Proof: By the definition of ED(G) it is clear that V (ED(T )) = V (T ) ∪ S(T ). By using the definition of Wiener index and Lemma 12, we have

) , (

= )) ( ED

(

_ED(T)

) , (

v u d T

W

V v

u





= ( , ) ( , ) ( , )

_ED(T)

( , )

1

= )

ED(T 1

= )

ED(T 1

= )

ED(T 1

=

i i k

i i k

i i k

i i d

i

w x d w

u d x

u d v

u

d   

 ^{  }

( , ) ( , )

_ED(T)

( , )

<

) ED(T 1

= 1

= )

ED(T 1

= 1

=

j i j

i i i d

i k

i i i d

i k

i

x x d v

w d

v x

d   



 ^{ }



( , ) ( , ) ( , ) ( , )

4

1

= ) ED(T 1

= 4

1

= ) ED(T )

ED(T

<

) ED(T

<

i i i k

i i i j

i j

i j i j

i

D x d D

u d v v d w

w

d    

 ^{  }



( , ) ( , )

_ED(T)

( , )

<

4

1

= ) ED(T 1

= 4

1

= ) ED(T 1

=

j i j

i i i i d

i i i i k

i

D D d

D v d D

w

d    

 ^{ }



W [ED(T )] = 2k

²

+ 2kd + 14k + 8d + 25.

Thus, the result follows.

7. COMPUTING HYPER-WIENER INDEX FOR ED(T) AND ED(T) OF KRAGUJEVAC TREE T.

Using the definition of hyper-Wiener index and Lemma 10 and 12 we can deduce the

expression for the hyper-Wiener index of Ed(T ) and ED(T ) of Kragujevac tree T .

Theorem 14. The hyper-Wiener index of the semientire dominating graph of Kragujevac tree T is

].

260) 8

14 40

2 2

76 2 [(4

= 1 )) (

( Ed T k

⁴

 k

²

 k

²

d

²

 kd  d

²

 k  d  WW

Proof. By the definition of Ed (G ) it is clear that V ( Ed ( T )) = V ( T )  S ( T ). By using the definition of hyper-Wiener index and Lemma 10, we have

[ ( , ) ( , ) ]

2

= 1 )) (

(

_{Ed(T) Ed(T)} ²

) ( ) , (

v u d v u d T

Ed WW

G V v u

 



[[ ( , ) ( , ) ]

2

= 1

_Ed(T) ²

1

= )

Ed(T 1

=

i d

i i d

i

v u d v

u

d 

 ^

[ ( , ) ( , ) ] [

_Ed(T)

( , )

1

= 2 )

Ed(T 1

= )

Ed(T 1

=

i k

i i

k

i i k

i

w u d x

u d x

u

d  

 ^{ }



( , ) ] [ ( , )

_Ed(T)

( , )

²

]

1

= )

Ed(T 1

= 2 )

Ed(T 1

=

i i k

i i i k

i i

k

i

w x d w

x d w

u

d  

 ^{ }



[ ( , )

_Ed(T)

( , )

²

]

1

= 1

= )

Ed(T 1

= 1

=

i i d

i k

i i i d

i k

i

v x d v

x

d  



 ^



[ ( , )

_Ed(T)

( , )

²

]

1

= 1

= )

Ed(T 1

= 1

=

i i d

i k

i i i d

i k

i

v w d v

w

d  



 ^



[ ( , ) ( , ) ] [

_Ed(T)

( , )

<

2 )

Ed(T

<

) Ed(T

<

j i j

i j

i j

i j i j

i

w w d

x x d x

x

d  

 ^{ }



( , ) ] [ ( , )

_Ed(T)

( , )

²

]

<

) Ed(T

<

2 )

Ed(T

<

j i j

i j i j

i j i j

i

v v d v

v d w

w

d  

 ^{ }



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

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 4

1

= )

Ed(T i

i i k

i i

i i

i

D x d D

u d D u

d  ^  ^  



( , ) ] [ ( , )

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 1

=

i i i k

i i

i i k

i

D w d D

x

d   

 ^



( , ) ] [ ( , )

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 1

=

i i i d

i i

i i k

i

D v d D

w

d   

 ^



( , ) ] [ ( , )

_Ed(T)

( , )

²

]]

<

) Ed(T

<

2 4

1

= ) Ed(T 1

=

j i j

i j i j

i i i i d

i

D D d

D D d

D v

d   

 ^{ }



] 260) 8

14 40

2 2

76 2 [(4

= 1 )) (

( Ed T k

⁴

 k

²

 k

²

d

²

 kd  d

²

 k  d  WW

Thus, the result follows.