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 KP
G
G
G
G
G
G
andG
A' are constructedusing 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
: Attacht
icopies of a path of orderr
2
to each vertexv
iofG
by identifyingv
ias the initial vertex of such paths.C
G
: Attacht
i copies of a cycle of orderr
3
to each vertexv
iby identifyingv
ias one of the vertices inC
r.K
G
: Attacht
i copies of a complete graphK
rof order4
r
to every vertexv
iofG
by identifyingv
ias a vertex ofK
r.A
G
: Attacht
i copies of a complete bipartite graphs r
K
, to every vertexv
iofG
by identifyingv
ias a vertexof
K
r,s. ' CG
: Attacht
i copies of a cycle of orderr
3
to each vertexv
iofG
by an edge.' K
G
: Attacht
i copies of a complete graphK
rof order4
r
to each vertexv
iofG
by an edge.' A
G
: Attacht
i copies of a complete bipartite graphK
r,s to each vertexv
iofG
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
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
.
[image:2.595.45.551.55.790.2]
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 ) ( ) (
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(
:
)
(
:
)
CG 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 alli
1
,
2
,...,
n
inG
C, then2
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 vertexof
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 vertexof
G
C
n, then2 3
(
G
)
4
n
12
nt
4
rnt
12
nt
LM
C
.2.3 Generalized Thorn graph
G
KObservation 3
Let
u
1,
u
2,
...,
u
rbe the vertices in a cliqueK
r. Thenthe 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
whereu
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
KTheorem 11
ni
i i i
K
LM
G
r
v
G
t
S
G
LM
1 3
3
(
)
(
)
(
1
)
deg(
:
)[
]
ni
n
i
i i i i
i
G
t
r
t
S
t
v
d
r
1 1
2 2
2
(
:
)
(
1
)
[
]
)
1
(
ni i
t
r
1 2
)
1
(
.Proof:
By Observation 3 we have
) (
2
3
(
)
deg(
:
)
(
:
)
KG 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 alli
1
,
2
,...,
n
inG
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 ofK
r(
r
4
)
are attached to eachvertex
of
G
P
n, thennt
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
(2) If
t
copies ofK
r(
r
4
)
are attached to eachvertex of
G
C
n, then).
2
(
3
)
3
(
3
)
10
(
4
)
(
2 3
r
nrt
t
nt
r
rnt
n
G
LM
K2.4 Generalized Thorn graph
G
AObservation 4:
Let
(
X
,
Y
)
be a partition of the vertex set oft
i-copy ofs r
K
, and letV
(
X
)
{
x
1,
x
2,...,
x
r}
and}.
,...,
,
{
)
(
Y
y
1y
2y
sV
Then the degree and 2-degree ofany 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 alli
1
,
2
,...,
n
inG
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 thet
i-copy ofC
r. Then degree and 2-degree of any vertex inG
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
13
' 2
'
)
:
(
)
:
deg(
r
j
C j C
j
G
d
u
G
u
]
2
)
:
(
[
]
)
:
[deg(
21
i i i
n
i
i
i
G
t
d
v
G
t
S
v
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
ni
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
inG
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 cliqueK
r. Then thedegree 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
inG
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 oft
i-copyof
K
r,sand letV
(
X
)
{
x
1,
x
2,...,
x
r}
and}.
...,
,
{
)
(
Y
y
1y
2,y
sV
Then the degree and 2-degree ofany 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
1G
A'
s
where
x
1
N
v
iG
'AV
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
.
]
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
inG
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
