On Third Leap Zagreb Index of Some Generalized Graph Structures

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International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-11, September 2019

3170

Published By:

Blue Eyes Intelligence Engineering & Sciences Publication

Retrieval NumberK25100981119/2019©BEIESP DOI:10.35940/ijitee.K2510.0981119

Abstract— Let G be a connected graph with n vertices. The 2-degree of a vertex v in G is the number of vertices which are at distance two from v in G. The third leap Zagreb index of G is the sum of product of degree and 2-degree of all vertices in G. In this paper, we determine the exact values for the third leap Zagreb index of some generalized graph structures

Keywords— Thorn graphs, generalized graph structures, leap Zagreb indices.

I. INTRODUCTION

Topological indices of graphs play a vital role in Mathematical Chemistry to study the structural properties of some complicated chemical compounds. In general, topological indices are widely classified into two types: degree based indices and distance based indices. Thorn tree is a well known graph structure in Chemical graph Theory. In 2012 K.M.Kathirasen and C.Parameswaran [1] introduced the idea of Generalized Thorn graphs as a generalization of thorn trees. In 2017, Naji et al. [5] introduced a new topological invariant called leap Zagreb indices and studied their properties. Also they discussed the first leap Zagreb index of some graph operations [4]. Basavanagoud and Praveen [2] computed first leap Zagreb index of some nano structures. Shao et al. [6] found some interesting results on leap Zagreb indices of trees and unicyclic graphs. Shiladhar et al. [7] computed leap Zagreb indices of some windmill type graphs.

Venkatakrishnan et al. [8] found eccentric connectivity index of certain generalized thorn graphs. In this sequel, we are interested in computing exact values for the third leap Zagreb index of some generalized thorn graphs. We recall their definitions in the following:

II. BASIC DEFINITIONS

Definition 1

The generalized thorn graphs

' ' ,

,

,

,

,

C K A C K

P

G

G

G

G

G

G

and

G

A' are constructed

using the following graph operations:

Revised Manuscript Received on September2, 2019.

S.Swathi, Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, Tamil Nadu, India.

C.Natarajan, Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, Tamil Nadu, India.

(Email: natarajan_c@maths.sastra.edu)

K.Balasubramanian, Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, Tamil Nadu, India.

M.Swathi Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, Tamil Nadu, India.

P

G

: Attach

t

icopies of a path of order

r

2

to each vertex

v

iof

G

by identifying

v

ias the initial vertex of such paths.

C

G

: Attach

t

i copies of a cycle of order

r

3

to each vertex

v

iby identifying

v

ias one of the vertices in

C

r.

K

G

: Attach

t

i copies of a complete graph

K

rof order

4

r

to every vertex

v

iof

G

by identifying

v

ias a vertex of

K

r.

A

G

: Attach

t

i copies of a complete bipartite graph

s r

K

, to every vertex

v

iof

G

by identifying

v

ias a vertex

of

K

r,s. ' C

G

: Attach

t

i copies of a cycle of order

r

3

to each vertex

v

iof

G

by an edge.

' K

G

: Attach

t

i copies of a complete graph

K

rof order

4

r

to each vertex

v

iof

G

by an edge.

' A

G

: Attach

t

i copies of a complete bipartite graph

K

r,s to each vertex

v

iof

G

by an edge.

Recently Naji et al. [5] introduced a new set of topological invariants called Leap Zagreb Indices in 2017 and they are defined as follows:

Definition 2

The first leap Zagreb index of G is denoted by

and defined as

) (

2 2 1

(

)

(

:

)

G V v

G

v

d

G

LM

.

Definition 3

The second leap Zagreb index of G is defined as

) (

2 2

2

(

)

(

:

)

(

:

)

G E uv

G

v

d

G

u

d

G

LM

.

Definition 4

The third leap Zagreb index of a graph G is defined as

) (

2 3

(

)

deg(

:

)

(

:

)

G V v

G

v

d

G

v

G

LM

.

On Third Leap Zagreb Index of Some

Generalized Graph Structures

(2)

III. THE THIRD LEAP ZAGREB INDEX OF GENERALIZED THORN GRAPHS & RESULTS In this section, we compute the exact values of the third leap Zagreb index of generalized thorn graphs , , ,

, , and .

Let

.

2.1 Generalized Thorn Graph

Observation 1

Let be the vertices of . The degree and

2-degree of any vertex in is given as follows:

(1) (2 )

(3)

(4) ; 3 ≤ j ≤ r−1 (5)

(6) (7)

(8)

Theorem 5

 

 

n

i

n

i i i

n

i

n

i i i i

i

n

i

i i i

P

t

r

t

S

t

t

G

v

d

S

t

G

v

G

LM

G

LM

1 1

2

1 1

2

1 3

3

)

13

4

(

3

)

/

(

]

3

)[

/

deg(

)

(

)

(

Proof.

By observation 1 we have

 

 

 

 

 

 

 

 

n

i i n

i

i i n

i i n

i i i

n

i i n

i

i i n

i

i i

n

i

i i n

i

i i

i i

n

i i n

i i n

i i

n

i i n

i

i i i

i i

i i

n

i i n

i i n

i i

n

i i G

V v

P i P i

G V v

P P

P

t

t

G

v

t

r

S

t

t

t

G

v

d

S

G

v

t

G

v

G

v

d

G

v

t

G

v

t

t

r

t

t

S

t

G

v

d

t

G

v

t

G

v

t

t

r

t

t

G

v

d

G

v

G

v

d

G

v

G

LM

P i

P

1 2

1 1

1

1 2

1 2 1

1 1

2

1 1

1

1 1

2 1 1

1

1 )

(

2 ) (

2 3

2

)

:

deg(

2

)

13

4

(

)

:

(

)

:

deg(

)

:

deg(

)

:

(

)

:

deg(

]

)

:

[deg(

2

)

4

(

4

2

]

)

:

(

][

)

:

[deg(

]

)

:

[deg(

2

)

4

(

4

2

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

(

Thus the result follows.

Corollary 6

If t

i=t for all i=1,2,…,n in GP, then

nt

r

nt

S

t

G

v

d

t

S

G

v

mt

G

LM

G

LM

n

i i n

i

i

n

i

i i P

)

13

4

(

3

)

:

(

)

:

deg(

6

)

(

)

(

2

1 1

2

1 3

3

 

Example 7

(1) If t copies of (r≥2) are attached to each vertex of

, then

.

(2) In particular if we attach t copies of P

2 to every vertex of by identifying one of the vertices in as a

vertex in then the resulting graph is a Thorn tree and

.

Figure 1: Thorn tree

(3) If t copies of (r≥2) are attached to each vertex of

, then

2.2 Generalized Thorn Graph

C

G

Observation 2:

Let be the vertices in a cycle Then the degree and 2-degree of any vertex in

is as follows:

(1) (2)

(3) (4)

(5)

(6)

Theorem8

 

 

 

  

 

n

i i n

i i i n

i i n

i

i i

n

i

i i i C

t S t t r t G v d

S t G v G

LM G LM

1 2

1 1 1

2

1 3

3

4 4

) 2 ( 4 ) : ( 2

] 2 )[ : deg( 2 ) ( ) (

(3)

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-11, September 2019

3172

Published By:

Blue Eyes Intelligence Engineering & Sciences Publication

Retrieval NumberK25100981119/2019©BEIESP DOI:10.35940/ijitee.K2510.0981119

Proof:

By Observation 2 we have

) (

2

3

(

)

deg(

:

)

(

:

)

C

G V v

C C

C

v

G

d

v

G

G

LM

=

 

 

)) ( ) : ( ( \ ) (

2 )

( \ ) : ( u

2 )

(

2

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

k

G V G v N G V u

C j C j G V G v N

C j C j G

V v

C i C i

C i C j

C i C i

G

u

d

G

u

G

u

d

G

u

G

v

d

G

v

 

n

i

n

i i i

i i

n

i

i i i

i i

t

r

t

G

v

t

S

t

G

v

d

t

G

v

1 1

1

2

)

3

(

4

]

2

)

:

[deg(

2

]

2

2

)

:

(

][

2

)

:

[deg(

 

 

n

i i n

i i i n

i i n

i

i i

n

i

i i i

t

S

t

r

t

t

G

v

d

S

t

G

v

G

LM

1 2

1 1

1 2

1 3

4

4

)

2

(

4

)

:

(

2

]

2

)[

:

deg(

2

)

(

Corollary 9

If

t

i

t

for all

i

1

,

2

,...,

n

in

G

C, then

2

1 1

2

1 3

3

4

4

)

2

(

4

)

:

(

2

)

:

deg(

2

8

)

(

)

(

nt

S

t

nt

r

G

v

d

t

S

G

v

mt

G

LM

G

LM

n

i i n

i

i

n

i

i i C

 

Example 10

(1) If t copies of

C

r.,

r

5

are attached to each vertex

of

G

P

n, then

)

14

4

(

2

))

3

(

3

(

4

)

5

2

(

2

)

(

3

t

t

r

t

nt

n

G

LM

C

(2) If t copies of

C

r.,

r

5

are attached to each vertex

of

G

C

n, then

2 3

(

G

)

4

n

12

nt

4

rnt

12

nt

LM

C

.

2.3 Generalized Thorn graph

G

K

Observation 3

Let

u

1

,

u

2

,

...,

u

rbe the vertices in a clique

K

r. Then

the degree and 2-degree of any vertex

G

Kare as follows:

(1)

deg(

v

i

:

G

K

)

deg(

v

i

:

G

)

(

r

1

)

t

i where

)

(

G

V

v

i

(2)

d

2

(

v

i

:

G

K

)

d

2

(

v

i

:

G

)

(

r

1

)

S

i where

)

(

G

V

v

i

(3)

deg(

u

j

:

G

K

)

r

1

where

u

j

V

(

G

K

)

\

V

(

G

)

(4)

d

2

(

u

j

:

G

K

)

deg(

v

i

:

G

)

(

t

i

1

)(

r

1

)

where

)

(

\

)

(

G

V

G

V

u

j

K

Theorem 11

n

i

i i i

K

LM

G

r

v

G

t

S

G

LM

1 3

3

(

)

(

)

(

1

)

deg(

:

)[

]

 

n

i

n

i

i i i i

i

G

t

r

t

S

t

v

d

r

1 1

2 2

2

(

:

)

(

1

)

[

]

)

1

(

n

i i

t

r

1 2

)

1

(

.

Proof:

By Observation 3 we have

) (

2

3

(

)

deg(

:

)

(

:

)

K

G V v

K K

K

v

G

d

v

G

G

LM

 

) ( \ ) (

2 )

(

2

)

:

(

)

:

deg(

)

:

(

)

:

deg(

G V G V u

K j K j G

V v

K i K i

K j i

G

u

d

G

u

G

v

d

G

v

 

n

i

i i

i n

i

i i

i i

t

r

G

v

t

r

S

r

G

v

d

t

r

G

v

1 1

2

)]

1

)(

1

(

)

:

[deg(

)

1

(

]

)

1

(

)

:

(

][

)

1

(

)

:

[deg(

 

n

i

i i n

i i i i

i n

i

i i i

t

t

S

t

r

t

G

v

d

r

S

t

G

v

r

G

LM

1

2

1 2 2

1 3

]

[

)

1

(

)

:

(

)

1

(

]

)[

:

deg(

)

1

(

)

(

Corollary 12

If

t

i

t

for all

i

1

,

2

,...,

n

in

G

K, then

 

n

i i

n

i

i n

i

i i K

nt

r

nt

r

S

t

r

G

v

d

t

r

S

G

v

r

mt

r

G

LM

G

LM

1

2 2

2 2

1 2 1

3 3

)

1

(

)

1

(

)

1

(

)

:

(

)

1

(

)

:

deg(

)

1

(

)

1

(

2

)

(

)

(

Example 13

(1) If

t

copies of

K

r

(

r

4

)

are attached to each

vertex

of

G

P

n, then

nt

r

r

t

n

rt

n

t

r

n

t

rt

n

G

LM

K

)

1

10

(

)

2

3

(

)

4

6

(

)

2

3

(

12

12

)

5

2

(

2

)

(

2 2 2

2 2 3

(4)

(2) If

t

copies of

K

r

(

r

4

)

are attached to each

vertex of

G

C

n, then

).

2

(

3

)

3

(

3

)

10

(

4

)

(

2 3

r

nrt

t

nt

r

rnt

n

G

LM

K

2.4 Generalized Thorn graph

G

A

Observation 4:

Let

(

X

,

Y

)

be a partition of the vertex set of

t

i-copy of

s r

K

, and let

V

(

X

)

{

x

1

,

x

2

,...,

x

r

}

and

}.

,...,

,

{

)

(

Y

y

1

y

2

y

s

V

Then the degree and 2-degree of

any vertex in

G

Ais as follows:

(1)

deg(

v

i

:

G

A

)

deg(

v

i

:

G

)

st

i

(2)

d

2

(

v

i

:

G

A

)

d

2

(

v

i

:

G

)

(

r

1

)

t

i

sS

i

(3)

deg(

x

j

:

G

A

)

s

,

2

j

r

(4)

d

2

(

x

j

:

G

A

)

r

1

,

2

j

r

(5)

deg(

y

k

:

G

A

)

r

,

1

k

s

(6)

s

k

G

v

t

s

s

G

y

d

k A i i

1

),

:

deg(

)

1

(

)

1

(

)

:

(

2

Theorem 14

.

)

2

3

(

)

:

(

]

)

1

2

)(

:

[deg(

)

(

)

(

1 1

2

1 1

2 2

1 3

3

 

 

n

i i n

i i i

n

i

n

i i i

i i

n

i

i i

A

t

r

S

t

s

s

rs

t

G

v

d

t

s

sS

t

r

G

v

G

LM

G

LM

Proof:

By Observation 4 we have

 

   

n

i

i i

i n

i n

i

i i i

i i

G V G V y

A k A k G V G V x

A j A j G

V v

A i A i

G V v

A A

A

G

v

t

s

s

t

r

t

r

s

sS

t

r

G

v

d

st

G

v

G

y

d

G

y

G

x

d

G

x

G

v

d

G

v

G

v

d

G

v

G

LM

i A k

A j

i

A

1 1

2 1

2 )

( \ ) (

2 )

( \ ) (

2 )

(

2 ) (

2 3

)]

:

deg(

)

1

(

1

[

)

1

(

]

)

1

(

)

:

(

][

)

:

[deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

(

Thus the result follows.

Corollary 15

If

t

i

t

for all

i

1

,

2

,...,

n

in

G

A, then

.

2

)

2

3

(

)

:

(

)

:

deg(

)

1

2

(

2

)

(

)

(

2 2 2

1 2 1

3 3

rnt

t

ms

nt

s

rs

G

v

d

st

S

G

v

s

mt

r

G

LM

G

LM

n

i

i n

i

i i A

 

2.5 Generalized Thorn Graph

' C

G

Observation 5

Let

u

1

,

u

2

,...,

u

r be the vertices in the

t

i-copy of

C

r. Then degree and 2-degree of any vertex in

G

C' are as follows:

(1)

deg(

v

i

:

G

C'

)

deg(

v

i

:

G

)

t

i (2)

d

(

v

i

:

G

C

)

d

2

(

v

i

:

G

)

2

t

i

S

i

' 2

(3)

deg(

u

j

:

G

C

)

d

(

u

j

:

G

C

)

2

,

4

j

r

' 2

'

(4)

deg(

u

j

:

G

C'

)

2

,

for

j

2

,

3

(5)

d

2

(

u

j

:

G

C'

)

3

,

for

j

2

,

3

(6)

deg(

u

1

:

G

C'

)

3

,

for

u

1

N

(

v

i

:

G

C'

)

\

V

(

G

)

(7)

)

(

\

)

:

(

)

:

deg(

1

)

:

(

' 1

' 1 2

G

V

G

v

N

u

for

G

v

t

G

u

d

C i

i i

C

Theorem 16

.

)

7

4

(

5

)

:

(

]

5

)[

:

deg(

)

(

)

(

1 1

1 1

2 2

1 3

' 3

 

 

n

i i i n

i i n

i

n

i i i

i

n

i

i i i

C

S

t

t

r

t

t

G

v

d

S

t

G

v

G

LM

G

LM

Proof:

By Observation 5 we have

) (

' 2 ' '

3

'

)

:

(

)

:

deg(

)

(

C G V v

C C

C

v

G

d

v

G

G

LM

 

) ( \ ) : (

' 1 2 ' 1 )

(

' 2

'

' 1

)

:

(

)

:

deg(

)

:

(

)

:

deg(

G V G v N u

C C

G V v

C i C i

C i i

G

u

d

G

u

G

v

d

G

v

)

:

(

)

:

deg(

)

:

(

)

:

deg(

' 2

'

' 2 2 ' 2

C r C r

C C

G

u

d

G

u

G

u

d

G

u

1

3

' 2

'

)

:

(

)

:

deg(

r

j

C j C

j

G

d

u

G

u

]

2

)

:

(

[

]

)

:

[deg(

2

1

i i i

n

i

i

i

G

t

d

v

G

t

S

v

(5)

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-11, September 2019

3174

Published By:

Blue Eyes Intelligence Engineering & Sciences Publication

Retrieval NumberK25100981119/2019©BEIESP DOI:10.35940/ijitee.K2510.0981119

 

n

i

n

i i i

i

i

t

v

G

r

t

t

1 1

)]

2

(

4

12

[

)]

:

deg(

1

[

3

Thus the result follows.

Corollary 17

If

t

i

t

for

all

i

1

,

2

,...,

n

in

G

C' , then

 

n

i

i n

i

i i C

G

v

d

t

S

G

v

nt

r

mt

nt

mt

G

LM

G

LM

1 2 1

2 2

3 '

3

).

:

(

)

:

deg(

)

7

4

(

2

5

10

)

(

)

(

2.6 Generalized Thorn Graph

' K

G

Observation 6

Let

u

1

,

u

2

,...,

u

rbe vertices in a clique

K

r. Then the

degree and 2-degree of any vertex in

G

K' are as follows:

(1)

deg(

v

i

:

G

K'

)

deg(

v

i

:

G

)

t

i

(2)

d

(

v

i

:

G

K

)

d

2

(

v

i

:

G

)

(

r

1

)

t

i

S

i

' 2

(3)

deg(

u

j

:

G

K

)

r

1

,

2

j

r

'

(4)

d

(

u

j

:

G

K

)

1

,

2

j

r

' 2

(5)

)

(

\

)

:

(

,

)

:

deg(

'

1 '

1

G

r

where

u

N

v

G

V

G

u

K

i K

(6)

)

(

\

)

:

(

)

1

(

)

:

deg(

)

:

(

' 1

' 1 2

G

V

G

v

N

u

where

t

G

v

G

u

d

K i

i i

K

Theorem 18

.

)

1

2

(

)

:

(

]

)

1

2

)[(

:

deg(

)

(

)

(

1

1

1 1

2 2

1 3

' 3

 

n

i i i

n

i i n

i

n

i i i

i i

n

i

i i

K

S

t

t

t

r

t

G

v

d

S

t

r

G

v

G

LM

G

LM

Proof:

) (

' 2 ' '

3

'

)

:

(

)

:

deg(

)

(

K G V v

K K

K

v

G

d

v

G

G

LM

  

) ( \ ) : (

' 1 2 ' 1 )

(

' 2

' )

(

' 2 '

' 1

'

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

G V G v N u

K K

G V u

K j K j G V v

K K

K i K j i

G

u

d

G

u

G

u

d

G

u

G

v

d

G

v

 

n

i

n

i

i i

i i

n

i

i i i

i i

t

G

v

t

r

t

r

S

t

r

G

v

d

t

G

v

1 1

1

2

]

1

)

:

[deg(

)

1

(

]

)

1

(

)

:

(

][

)

:

[deg(

.

)

1

2

(

)

:

(

)

:

deg(

)

:

deg(

)

1

2

(

)

(

1

1 1

2

1 2 1

1 3

 

 

n

i i i n

i

n

i i i

n

i

i i n

i

i i

n

i

i i

S

t

t

t

r

t

G

v

d

S

G

v

t

G

v

r

G

LM

Corollary 19

If

t

i

t

for

all

i

1

,

2

,...,

n

in

G

K' , then

 

n

i i

n

i

i n

i

i i K

S

t

nt

nt

r

G

v

d

t

S

G

v

mt

r

G

LM

G

LM

1

1

2 2

1

3 '

3

.

)

1

2

(

)

:

(

)

:

deg(

)

1

2

(

2

)

(

)

(

2.7 Generalized Thorn Graph

' A

G

Observation 7

Let

(

X

,

Y

)

be a bipartition of the vertex set of

t

i-copy

of

K

r,sand let

V

(

X

)

{

x

1

,

x

2

,...,

x

r

}

and

}.

...,

,

{

)

(

Y

y

1

y

2,

y

s

V

Then the degree and 2-degree of

any vertex in

G

A' are as follows: (1)

deg(

v

i

:

G

A'

)

deg(

v

i

:

G

)

t

i (2)

d

2

(

v

i

:

G

A'

)

d

2

(

v

i

:

G

)

st

i

S

i

(3)

r

j

G

V

G

v

N

G

V

x

where

s

G

x

A i A

j

A j

2

)},

(

)

:

(

{

\

)

(

,

)

:

deg(

' '

'

(4)

r

j

G

V

G

v

N

G

V

x

where

r

G

x

d

A i A

j

A j

2

)},

(

)

:

(

{

\

)

(

,

1

)

:

(

' '

' 2

(5)

)

(

\

)

:

(

,

1

)

:

deg(

x

1

G

A'

s

where

x

1

N

v

i

G

'A

V

G

(6)

d

2

(

x

1

:

G

A'

)

(

r

1

)

deg(

v

i

:

G

)

(

t

i

1

)

where

x

1

N

(

v

i

:

G

A'

)

\

V

(

G

)

(7)

deg(

y

k

:

G

'A

)

r

,

1

k

s

(8)

d

2

(

y

k

:

G

A'

)

s

,

1

k

s

Theorem 20

(

)

)

(

3

'

3

G

LM

G

(6)

.

]

2

)

1

(

3

[

)

1

(

)

:

(

]

)

1

(

)[

:

deg(

1 1

2

1 1

2 1

 

 

n

i i n

i i n

i

n

i i i i i n

i

i i i

t

r

s

t

s

s

S

t

t

G

v

d

S

t

s

s

G

v

Proof:

By virtue of Observation 7, we have

) (

' 2 ' '

3

'

)

:

(

)

:

deg(

)

(

A G V v

A A

A

v

G

d

v

G

G

LM

 

 

)} ( ) : ( \{ ) (

' 2

' )

(

' 2

' )

(

' 2

'

' ' '

)

:

(

)

:

deg(

)

:

(

)

:

deg(

)

:

(

)

:

deg(

G V G v N G V x

A j A j G

V y

A k A k G V v

A i A i

A i A j

A k i

G

x

d

G

x

G

y

d

G

y

G

v

d

G

v

) ( \ ) : (

' 1 2 ' 1

' 1

)

:

(

)

:

deg(

G V G v N x

A A

A i

G

x

d

G

x

 

n

i

i i

i

n

i i n

i i n

i

i i i

i i

t

G

v

r

t

s

t

r

s

t

rs

S

st

G

v

d

t

G

v

1

1 1

1

2

)]

1

(

)

:

deg(

)

1

[(

)

1

(

)

1

(

]

)

:

(

][

)

:

[deg(

Thus the result follows.

Corollary 21

If

t

i

t

,

for

all

i

1

,

2

,...,

n

in

G

A' , then

.

2

)

1

(

3

)

1

(

)

:

(

)

:

deg(

)

1

(

2

)

(

)

(

2

1

1 1

2 3

' 3

nt

nt

r

s

nt

s

s

S

t

G

v

d

t

S

G

v

mt

s

s

G

LM

G

LM

n

i i n

i

n

i

i i

i A

 

IV. CONCLUSION

The exact values of the third leap Zagreb index of some generalize thorn graphs have been determined. Results on the second leap Zagreb index of these graph structures will be reported in near future.

REFERENCES

1. B. Basavanagoud, E. Chitra, “On leap Zagreb indices of some nano structures”, Malaya Journal of Matematik, vol.6, no.4, pp.816-822, 2018.

2. B. Basavanagoud, J. Praveen, “Computing first leap Zagreb index of some nano structures”, International Journal of Mathematics and its Applications, vol.6, no.2B, pp.141-150, 2018.

3. K.M. Kathiresan, C. Parameswaran, “Certain generalized thorn graphs and their Wiener indices”, Journal of Applied Mathematics and iNformatics, vol.30, no.5-6, pp.793-807, 2012.

4. A.M. Naji, N.D. Soner, “The first leap Zagreb index of some graph operations”, International Journal of Applied Graph Theory, vol.2, no.1, 2018.

5. A.M. Naji, N.D. Soner, I. Gutman, “On leap Zagreb indices of graphs”, Communications in combinatorics and optimization, vol.2, no.2, pp.99-117, 2017.

6. Z. Shao, I. Gutman, Z. Li, S. Wang, P. Wu, “Leap Zagreb indices of trees and unicyclic graphs”, Communications in combinatorics and optimization, vol.3, no.2, pp.179-194, 2018.

7. P. Shiladhar, A.M. Naji, N.D. Soner, “Computational of leap Zagreb indices of some windmill graphs”, International Journal of Mathematics and its Applications, vol.6, no.2-B, pp.183-191, 2018. 8. Y.B. Venkatakrishnan, S. Balachandran, K.Kannan, “On the eccentric

Figure

Figure 1: Thorn tree

Figure 1:

Thorn tree p.2

References