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

## Full text

(1)

ISSN 2319-8133 (Online)

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

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2

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2

### 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.

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G

G

1

2

n

i

j

i

j

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14, 31

14

) ( 2

2

) (

1

G G

G E uv G

G V

u

) (

1

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) ( 2

) (

1

G G

G E uv G

G G E

uv

1

10

3

) (

G

G V

u

2

2

) (

G G

G E uv

33,34

) ( ) , (

G

G V v

u

27

2

) ( ) , (

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

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3

k

3

3

k

k

Figure 1.

1

2

3

d

1

2

3

d

2

3

1

2

3

d

1

2

3

d

Figure 2.

k

1

= 1

=

d i

i i

d

i

i

i

i

i

ik=1

i

(4)

5, 13, 22.

18

19

s

1

s

i

28, 29

28

1

2

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.

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### 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.

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1

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1

=

1 i

k

i

1

=

2 i

k

i

1

= 1

=

3 i

k

i i d

i

1

= 1

=

4 i

k

i i d

i

Ed(T)

Ed(T)

i

Ed(T)

i

Ed(T)

i

Ed(T)

1

Ed(T)

2

Ed(T)

3

Ed(T)

4

1

3

2

2

2 ) ( 1

=

1

Ed T

n

i

## 

2

1 ) ( 2

1 ) ( 2

)

(

k

i

i T Ed k

i

i T Ed T

Ed

2 2 ) ( 2

1 ) ( 2

1 )

(

Ed T

Ed T

d

i

i T

Ed

2 4 ) ( 2

3 )

(

Ed T

Ed T

(7)

2

2

2

2

2

2

2

2

1

3

2

2

1

2

3

4

ED(T)

ED(T)

i

ED(T)

i

ED(T)

i

ED(T)

1

ED(T)

2

ED(T)

3

ED(T)

4

1

3

2

2

2 ) ( 1

=

1

EDT

n

i

2

1 ) ( 2

1 ) ( 2

)

(

k

i

i T ED k

i

i T ED T

ED

2 2 ) ( 2

1 ) ( 2

1 )

(

EDT

EDT

d

i

i T

ED

2 4 ) ( 2

3 )

(

EDT

EDT

2

2

2

2

2

2

2

2

1

3

2

2

### 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

(8)

3

2

2

2

2

4

3

2

3 ) ( 1

=

Ed T

n

i

3

1 ) ( 3

1 ) ( 3

)

(

k

i

i T Ed k

i

i T Ed T

Ed

3 2 ) ( 3

1 ) ( 3

1 )

(

Ed T

Ed T

d

i

i T

Ed

3 4 ) ( 3

3 )

(

Ed T

Ed T

3

3

3

3

3

3

3

3

3

2

2

2

2

4

3

2

1

2

3

4

3

2

2

2

4

3

2

3 ) ( 1

=

EDT

n

i

3

1 ) ( 3

1 ) ( 3

)

(

k

i

i T ED k

i

i T ED T

ED

(9)

3 2 ) ( 3

1 ) ( 3

1 )

(

EDT

EDT

d

i

i T

ED

3 4 ) ( 3

3 )

(

EDT

EDT

3

3

3

3

3

3

3

3

3

2

2

2

4

3

2

Ed(T)

i

Ed(T)

i

Ed(T)

i

4

1

= )

Ed(T i

i

1

= )

Ed(T i

d

i

i

1

= )

Ed(T i

k

i

i

Ed(T)

i

j

Ed(T)

i

j

Ed(T)

i

j

1

= )

Ed(T i

d

i

i

4

1

= )

Ed(T i

i

i

4

1

= )

Ed(T i

i

i

4

1

= )

Ed(T i

i

i

Ed(T)

i

j

2

(10)

Ed(T)

) , (

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

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

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

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

2

ED(T)

i

ED(T))

i

ED(T)

i

4

1

= )

ED(T i

i

1

= )

ED(T i

d

i

i

1

= )

ED(T i

k

i

i

ED(T)

i

j

ED(T)

i

j

ED(T)

i

j

1

= )

ED(T i

d

i

i

( , )=6

4

1

= )

ED(T i

i

i D

x

d

4

1

= )

ED(T i

i

i

(11)

4

1

= )

ED(T i

i

i

ED(T)

i

j

2

Ed(T)

) , (

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

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

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

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

2

ED(T)

i

ED(T))

i

ED(T)

i

4

1

= )

ED(T i

i

1

= )

ED(T i

d

i

i

1

= )

ED(T i

k

i

i

ED(T)

i

j

ED(T)

i

j

(12)

ED(T)

i

j

1

= )

ED(T i

d

i

i

4

1

= )

ED(T i

i

i

4

1

= )

ED(T i

i

i

4

1

= )

ED(T i

i

i

ED(T)

i

j

2

ED(T)

) , (

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

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

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

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

2

(13)

4

2

2

2

2

Ed(T) Ed(T) 2

) ( ) , (

G V v u

Ed(T) 2

1

= )

Ed(T 1

=

i d

i i d

i

Ed(T)

1

= 2 )

Ed(T 1

= )

Ed(T 1

=

i k

i i

k

i i k

i

Ed(T)

2

1

= )

Ed(T 1

= 2 )

Ed(T 1

=

i i k

i i i k

i i

k

i

Ed(T)

2

1

= 1

= )

Ed(T 1

= 1

=

i i d

i k

i i i d

i k

i

Ed(T)

2

1

= 1

= )

Ed(T 1

= 1

=

i i d

i k

i i i d

i k

i

Ed(T)

<

2 )

Ed(T

<

) Ed(T

<

j i j

i j

i j

i j i j

i

Ed(T)

2

<

) Ed(T

<

2 )

Ed(T

<

j i j

i j i j

i j i j

i

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 4

1

= )

Ed(T i

i i k

i i

i i

i

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 1

=

i i i k

i i

i i k

i

4

1

= ) Ed(T 1

= 2 4

1

= ) Ed(T 1

=

i i i d

i i

i i k

i

Ed(T)

2

<

) Ed(T

<

2 4

1

= ) Ed(T 1

=

j i j

i j i j

i i i i d

i

4

2

2

2

2

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