Open Access Library Journal
2017, Volume 4, e2987
ISSN Online: 2333-9721
ISSN Print: 2333-9705
Sharp Upper Bounds for Multiplicative Degree
Distance of Graph Operations
R. Muruganandam1, R. S. Manikandan2, M. Aruvi3
Department of Mathematics, Government Arts College,Tiruchirappalli, India
Department of Mathematics, Bharathidasan University Constituent College, Lalgudi, Tiruchirappalli, India
3
Department of Mathematics, Anna University, Tiruchirappalli, India
1
2
How to cite this paper: Muruganandam, R.,
Manikandan, R.S. and Aruvi, M. (2017)
Sharp Upper Bounds for Multiplicative Degree Distance of Graph Operations. Open
Access Library Journal, 4: e2987.
https://doi.org/10.4236/oalib.1102987
Abstract
Received: August 18, 2016
Accepted: July 8, 2017
Published: July 11, 2017
Discrete Mathematics
Copyright © 2017 by authors and Open
Access Library Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Join, Disjunction, Composition, Symmetric Difference, Multiplicative Degree
Distance, Zagreb Indices and Coindices
Open Access
In this paper, multiplicative version of degree distance of a graph is defined
and tight upper bounds of the graph operations have been found.
Subject Areas
Keywords
1. Introduction
A topological index of a graph is a numerical quantity which is structural
invariant, i.e. it is fixed under graph automorphism. The simplest topological
indices are the number of vertices and edges of a graph. In this paper, we define
and study a new topological index called multiplicative degree distance. All
graphs considered are simple and connected graphs.
We denote the vertex and the edge set of a graph G by V ( G ) and E ( G ) ,
respectively. dG ( v ) denotes the degree of a vertex v in G. The number of
elements in the vertex set of a graph G is called the order of G and is denoted by
v ( G ) . The number of elements in the edge set of a graph G is called the size of
G and is denoted by e ( G ) . A graph with order n and size m edges is called a
( n, m ) -graph. For any u, v ∈V ( G ) , the distance between u and v in G, denoted
by dG ( u , v ) , is the length of a shortest ( u , v ) -path in G. The edge connective
the vertices u and v will be denoted by uv. The complement G of the graph G
is the graph with vertex set V ( G ) , in which two vertices in G are adjacent if
DOI: 10.4236/oalib.1102987
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R. Muruganandam et al.
and only if they are not adjacent in G.
The join of graphs G1 and G2 is denoted by G1 + G2 , and it is the graph
with vertex set V ( G1 ) ∪ V ( G2 ) and the edge set E ( G1 + G2 )= E ( G1 ) ∪ E ( G2 )
∪ {u1u2 | u1 ∈ V ( G1 ) , u2 ∈ V ( G2 )}. The composition of graphs G1 and G2 is
denoted by G1 [G2 ] , and it is the graph with vertex set V ( G1 ) × V ( G2 ) , and two
vertices u = ( u1 , u2 ) and v = ( v1 , v2 ) are adjacent if ( u1 is adjacent to v1 ) or
( u1 = v1 and u2 and v2 are adjacent). The disjunction of graphs G1 and G2
is denoted by G1 ∨ G2 , and it is the graph with vertex set V ( G1 ) × V ( G2 ) and
=
E ( G1 ∨ G2 )
{( u1 , u2 )( v1 , v2 ) | u1v1 ∈ E ( G1 ) or u2v2 ∈ E ( G2 )}.
The symmetric di-
fference of graphs G1 and G2 is denoted by G1 ⊕ G2 , and it is the graph
with vertex set V ( G1 ) × V ( G2 ) and edge set
=
E ( G1 ⊕ G2 ) {( u1 , u2 )( v1 , v2 )) | u1v1 ∈ E ( G1 ) or u2 v2 ∈ E ( G2 ) but not both}.
Let G be a connected graph. The Wiener index W ( G ) of a graph G is
defined as
=
W (G )
=
∑ dG ( u, v )
{u ,v}⊆V ( G )
1
∑ dG ( u, v ) .
2 u ,v∈V ( G )
Dobrynin and Kochetova [1] and Gutman [2] independently proposed a
vertex-degree-Weighted version of Wiener index called degree distance or
Schultz molecular topological index, which is defined for a connected graph G as
=
DD ( G )
∑
dG ( u , v )  dG=
( u ) + dG ( v )
{u ,v}⊆V ( G )
1
∑ dG ( u, v )  dG ( u ) + dG ( v ) .
2 u ,v∈V ( G )
The Zagreb indices have been introduced more than thirth years ago by
Gutman and Trianjestic [3]. The first Zagreb index M 1 ( G ) of a graph G is
defined as
M1 =
(G )
∑
uv∈E ( G )
v ) 
 dG ( u ) + dG (=
∑ dG2 ( v ) .
v∈V ( G )
The second Zagreb index M 2 ( G ) of a graph G is defined as
M 2 (G ) =
dG ( u ) dG ( v ) .
∑
uv∈E ( G )
The Zagreb indices are found to have applications in QSPR and QSAR studies
as well, see [4].
Note that contribution of nonadjacent vertex pair should be taken into account
when computing the Weighted Wiener Polynomials of certain Composite graphs,
see [5]. A.R. Ashrafi, T. Doslic, A. Hamzeha, [6] [7] defined the first Zagreb
coindex of a graph G is
=
M1 ( G )
∑
uv∉E ( G )
 dG ( u ) + dG ( v ) 
The second Zagreb coindex of a graph G is
M 2 (G ) =
∑
dG ( u ) dG ( v ) ,
uv∉E ( G )
respectively.
In [8], Hamzeh, Iranmanesh Hossein-Zadeh and M.V. Diudea recently
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introduced the generalized degree distance of graphs. Asma Hamzeh, Ali
Iranmanesh and Samaneh Hossein-Zadeh, Cartesian product, composition,
join, disjunction and symmetric difference of graphs and introduce generalized
and modified generalized degree distance Polynomials of graphs, such that their
first derivatives at x = 1 , see [9].
In this paper, we defne a new graph invariant named multiplicative version of
degree distance of a graph denoted by DD* ( G ) and defined by
 DD* ( G ) 
=
2
∏
dG ( u , v )  dG ( u ) + dG ( v )  .
u ,v∈V ( G ),u ≠v
Therefore the study of this new topological index is important and we have
obtained Sharp upper bounds for the graph operations such as join, disjunction,
composition, symmetric difference of graphs.
2. The Multiplicative Degree Distance of Graph Operations
Lemma 2.1. [10] [11], Let G1 and G2 be two simple connected graphs. The
number of vertices and edges of graph Gi is denoted by ni and ei respectively
for i = 1, 2 . Then we have
1, uv ∈ E ( G1 ) or uv ∈ E ( G2 ) or ( u ∈ V ( G1 ) and v ∈ V ( G2 ) )
1. dG1+G2 ( u , v ) = 
otherwise
2,
For a vertex u of G1 , dG1+=
dG1 ( u ) + n2 , and for a vertex v of G2 ,
G2 ( u )
dG1+=
dG2 ( v ) + n1.
G2 ( v )
u1 ≠ u2
dG1 ( u1 , u2 ) ,

2. dG1[G2 ] ( ( u1 , v1 ) , ( u2=
, v2 ) ) 1,
=
u1 u2 , v1v2 ∈ E ( G2 )
2,
otherwise

dG1[G=
( u, v ) n2 dG1 ( u ) + dG2 ( v ) .
2]
1, u1u2 ∈ E ( G1 ) or v1v2 ∈ E ( G2 )
3. dG1∨G2 ( ( u1 , v1 ) , ( u2 , v2 ) ) = 
otherwise
2,
dG1∨G2 ( ( u, v ) ) = n2 dG1 ( u ) + n1dG2 ( v ) − dG1 ( u ) dG2 ( v ) .
1, u1u2 ∈ E ( G1 ) or v1v2 ∈ E ( G2 ) but not both
4. dG1⊕G2 ( ( u1 , v1 ) , ( u2 , v2 ) ) = 
otherwise
2,
dG1⊕G2 ( ( u, v ) ) = n2 dG1 ( u ) + n1dG2 ( v ) − 2dG1 ( u ) dG2 ( v ) .
Lemma 2.2. (Arithmetic Geometric inequality)
Let x1 , x2 , , xn be non-negative numbers. Then
x1 + x2 +  + xn n
≥ x1 x2  xn
n
Remark 2.3. For a graph G, let A ( G ) = { ( x, y ) ∈ V ( G ) × V ( G ) | x and y
are adjacent in G } and let B ( G ) = { ( x, y ) ∈ V ( G ) × V ( G ) | x and y are not
adjacent in G }. For each x ∈ V ( G ) , ( x, x ) ∈ B ( G ) . Clearly,
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{( x, x ) | x ∈V ( G )}
C (G )
A ( G ) ∪ B ( G ) = V ( G ) × V ( G ) . Let=
and
D=
∅.
( G ) B ( G ) − C ( G ) . Clearly B=
(G ) C (G ) ∪ D (G ) , C (G ) ∩ D (G ) =
∑
The summation
( x , y )∈A( G )
we write the summation
∑
( x , y )∈B( G )
as
∑
xy∉G
∑
( x , y )∈C ( G )
∑
∑
as
( x , y )∈A( G )
xy∈G
∑
∑
xy∈E ( G )
∑
by
x , y∈G
runs over the edges of G. We
and similarly
∑
eqivalent to
( x , y )∈D( G )
eqivalent to
∑
x , y∈V ( G )
. Similarly, we write the summation
∑
. Also the summation
denote the summation
summation
runs over the ordered pairs of A ( G ) . For simplicity,
xy∉G , x ≠ y
∑
x , y∈V ( G )
by
∏
x , y∈G
. The
and similarly the summation
.
xy∉G , x =
y
Lemma 2.4. Let G be a graph. Then
∑ 1 = 2e ( G )
xy∈G
Proof:
=
∑ 1 2=
∑ 1 2e ( G )
xy∈E ( G )
xy∈G
Lemma 2.5.
∑ dG ( x ) = M 1 ( G )
xy∈G
Proof: Let x ∈ V ( G ) and t = dG ( x ) . Let y1 , y2 , , yt be the neighbours of
( x, yi ) ,1 ≤ i ≤ t , contributes dG ( x )
orderd pairs contribute dG2 ( x ) to the sum. Hence
x. Each orderd pair
=
∑ dG ( x )
to the sum. Thus these
=
∑ dG2 ( x ) M1 ( G )
x∈V ( G )
xy∈G
Lemma 2.6.
∑ dG ( x ) dG ( y ) = 2M 2 ( G )
xy∈G
Proof: Clearly,
=
∑ dG ( x ) dG ( y ) 2=
∑ dG ( x ) dG ( y ) 2M 2 ( G ) .
xy∈E ( G )
xy∈G
Lemma 2.7.
( )
=
∑ 1 2e G + v ( G )
xy∉G
Proof:
∑1 = ∑
xy∉G
1+
( x , y )∈D( G )
∑
( )
1 =2e G + v ( G )
( x , x )∈C ( G )
Lemma 2.8.
( )
( )
=
∑ dG ( x ) 2e G ( v ( G ) − 1) + 2e ( G ) − M1 G
xy∉G
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Proof.
=
∑ dG ( x )
∑
dG ( x ) +
( x , y )∈D ( G )
xy∉G
dG ( x )
∑
( x , x )∈C ( G )
∑ {v ( G ) − 1 − dG ( x )} + ∑
=
dG ( x )
( x , x )∈C ( G )
( x , y )∈D ( G )
=
( v ( G ) − 1)
∑
1−
( x , y )∈D ( G )
=
( v ( G ) − 1) 2e ( G ) −
=
( v ( G ) − 1) 2e ( G ) −
d G ( x ) + 2e ( G )
∑
( x , y )∈D ( G )
∑
dG2 ( x ) + 2e ( G )
( x , y )∈ A( G )
∑ dG2 ( x ) + 2e ( G )
xy∈G
= 2e ( G ) ( v ( G ) − 1) + 2e ( G ) − M 1 ( G ) by Lemma 2.5
Lemma 2.9.
) dG ( y )
∑ dG ( x=
xy∉G
2M 2 ( G ) + M1 ( G )
Proof:
=
∑ dG ( x ) dG ( y )
xy∉G
∑
dG ( x ) dG ( y ) +
∑
dG ( x ) dG ( y ) +
( x , y )∈D( G )
= 2
xy∉E ( G )
∑
dG ( x ) dG ( x )
( x , x )∈C ( G )
∑
dG2 ( x )
x∈V ( G )
= 2M 2 ( G ) + M1 ( G )
Lemma 2.10.
∑  dG ( x ) + dG ( y ) =
2 M 1 ( G ) + 4e ( G )
xy∉G
Proof:
( y )
∑  dG ( x ) + dG=
∑
( x , y )∈C ( G )
xy∉G
 dG ( x ) + dG ( y )  +
=∑ 2dG ( x ) + 2
x∈V ( G )
∑
xy∉E ( G )
∑
( x , y )∈D ( G )
 dG ( x ) + dG ( y ) 
 dG ( x ) + dG ( y ) 
= 4e ( G ) + 2 M 1 ( G )
3. The Multiplicative Degree Distance of Composition of Graph
Theorem 3.1. Let Gi , i = 1, 2 , be a
 DD* ( G1 [G2 ]) 
( ni , mi ) -graph. Then
2

1
 4 M 1 ( G2 )W ( G1 ) + 4n2 m2 DD ( G1 ) + 4n1 M 1 ( G2 )
≤
n
n
n
(
 1 2 1n2 − 1)
+ 8n22 m1 ( n2 − 1) + 2n1 M 1 ( G2 ) + 8m1n2 m2 + 4W ( G1 ) M 1 ( G2 )

+ 2n2 DD ( G1 )( 2m2 + n2 )  

n1n2 ( n1n2 −1)
Proof:
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 DD* ( G1 [G2 ]) 
∏
2
dG1 [G2 ] ( ( x, u ) , ( y, v ) )  dG1 [G2 ] ( x, u ) + dG1 [G2 ] ( y, v ) 


∏
x , y∈G1 u , v∈G2 , x ≠ u ( or ) y ≠ v
1

≤
∑
 n1n2 ( n1n2 − 1) x , y∈G1

∑ dG1[G2 ] ( ( x, u ) , ( y, v ) )  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v ) 
u , v∈G2 , x ≠ u ( or ) y ≠ v

n1n2 ( n1n2 −1)


1
dG1 [G2 ] ( ( x, u ) , ( y, v ) )  dG1 [G2 ] ( x, u ) + dG1 [G2 ] ( y, v )  
=
∑
∑


 n1n2 ( n1n2 − 1) x , y∈G1 u ,v∈G2


1


∑
 ∑ dG1 [G2 ] ( ( x, u ) , ( y, v ) ) dG1 [G2 ] ( x, u ) + dG1 [G2 ] ( y, v )
 n1n2 ( n1n2 − 1) x , y∈G1  uv∈G2
(
+
∑ d G [G ] ( ( x, u ) , ( y , v ) ) (
1
uv∉G2
2


1
=
 ∑
n
n
n
n
−
1
 1 2 ( 1 2 )  x , y∈G1 , x = y
+
+
+
)
 
dG1 [G2 ] ( x, u ) + dG1 [G2 ] ( y, v )  
 
)
n1n2 ( n1n2 −1)
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
uv∈G2
1
2
1
2
1
∑
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
x , y∈G1 , x ≠ y uv∈G2
1
x , y∈G1 , x = y uv∉G2
1
∑
x , y∈G1 , x ≠ y
n1n2 ( n1n2 −1)
2
1
1
2
2
1
2
1
2
2
2
 
∑ dG1[G2 ] ( ( x, u ) , ( y, v ) )  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v )  
uv∉G2
 
n1n2 ( n1n2 −1)


1
 DD G1 [G2 ] ≤ 
( S3 + S1 + S2 + S4 )
 n1n2 ( n1n2 − 1)

n1n2 ( n1n2 −1)
2
*
where S3 , S1 , S2 , S4 are terms of the above sums taken in order.
Next we calculate S1 , S2 , S3 and S4 separately.
=
S1
∑
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑
∑ dG ( x, y )  dG ( u ) + n2 dG ( x ) + dG ( v ) + n2 dG ( y )
x , y∈G1 , x ≠ y uv∈G2
=
x , y∈G1 , x ≠ y uv∈G2
= n2
1
2
1
1
2
x , y∈G1 , x ≠ y
= 4n2 m2 DD ( G1 ) + 4 M 1 ( G2 ) W ( G1 )
S2
=
2
2
∑ 1+ ∑
uv∈G2
1
dG1 ( x, y )
x , y∈G1 , x ≠ y
by Lemma 2.1
∑
uv∈G2
 dG2 ( u ) + dG2 ( v ) 
∑
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑

 ∑ dG1[G2 ] ( ( x, u ) , ( y, v ) )  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v ) 
y
uv∉G2 ,u= v
x , y∈G1 , x = y uv∉G2
=
1
1
dG1 ( x, y )  dG1 ( x) + dG1 ( y ) 
∑
2
x , y∈G1 , x=
1
2
1
2
1
2

dG1[G2 ] ( ( x, u ) , ( y, v ) )  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v )  


uv∉G2 ,u ≠v

= 0+ ∑
∑ dG1[G2 ] ( ( x, u ) , ( y, v ) )  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v )
x , y∈G , x = y uv∉G ,u ≠v
+
∑
1
=
∑
2
∑
dG1 ( x, y )  dG2 ( u ) + n2 dG1 ( x ) + dG2 ( v ) + n2 dG1 ( y )  by Lemma 2.1
x , y∈G1 , x = y uv∉G2 ,u ≠v
=2
∑
uv∉G2 ,u ≠v
 dG2 ( u ) + dG2 ( v ) 
∑
=
x , y∈G1 , x


1 + 2n2  ∑ 1 ∑  dG1 ( x ) + dG1 ( y ) 
y
x , y∈G1 , x y
 uv∉G2 ,u ≠v =
= 4n1M 1 ( G2 ) + 8n2 m1n2 ( n2 − 1)
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S3
=
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑
1
x , y∈G1 , x = y uv∈G2
2
1
2
1
2
1⋅ ∑
=
∑  dG1[G2 ] ( x, u ) + dG1[G2 ] ( y, v )
x , y∈G , x = y uv∈G
1
=
2
∑
∑
x , y∈G1 , x = y uv∈G2
 dG2 ( u ) + dG1 ( x ) n2 + dG2 ( v ) + n2 dG1 ( y )  by Lemma 2.1




=  ∑ 1 ∑  dG2 ( u ) + dG2 ( v )  + n2 ∑  dG1 ( x ) + dG1 ( y )   ∑ 1
x , y∈G1 ,=x y
 uv∈G2 
 x , y∈G1 ,=x y  uv∈G2
= 2n1M 1 ( G2 ) + 8n2 m1m2
S4
=
=
∑
∑ dG [G ] ( ( x, u ) , ( y, v ) )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑
∑ dG ( x, y )  dG [G ] ( x, u ) + dG [G ] ( y, v )
∑
∑ dG ( x, y )  dG ( u ) + dG ( x ) n2 + dG ( v ) + n2 dG ( y )
x , y∈G1 , x ≠ y uv∉G2
x , y∈G1 , x ≠ y uv∉G2
=
x , y∈G1 , x ≠ y uv∉G2
1
2
1
1
1
1
2
2
1
2
1
2
2
1
2
1
by Lemma 2.1


=  ∑ dG1 ( x, y )  ∑  dG2 ( u ) + dG2 ( v ) 
 x , y∈G1 , x ≠ y
 uv∉G2


+ n2 ∑ dG1 ( x, y )  dG1 ( x ) + dG1 ( y )   ∑ 1
x , y∈G1 , x ≠ y
 uv∉G2 
= 4W ( G1 ) M 1 ( G2 ) + 2n2 DD ( G1 )( 2m2 + n2 )

2
1
 DD* ( G1 [G2 ])  ≤ 
 4 M 1 ( G2 ) W ( G1 ) + 4n2 m2 DD ( G1 ) + 4n1M 1 ( G2 )
n
n
n
 1 2 ( 1n2 − 1)
+ 8n22 m1 ( n2 − 1) + 2n1M 1 ( G2 ) + 8m1n2 m2 + 4W ( G1 ) M 1 ( G2 )

+ 2n2 DD ( G1 )( 2m2 + n2 )  

n1n2 ( n1n2 −1)
Lemma 3.2.
DD* K n=
[ K r ]  2 ( nr − 1)
nr ( nr −1)
2
Proof: Clearly the graph K n [ K r ] is the complete graph K nr .
∴ DD* ( K n [ K r ]) =
DD* ( K nr ) =
 2 ( nr − 1) 
nr ( nr −1)
2
(1)
Remark 3.3. Let G1 = K n and G2 = K r . We get,
2 ( n − 1)
DD ( G1 ) =
n ( n − 1)
2
n ( n − 1)
n ( n − 1)
2
, W ( G1 ) =
n ( n − 1) , m1 =
=
2
2
r ( r − 1)
2
M 1 ( G2 ) =
r ( r − 1) , M 1 ( G2 ) =
0, m2 =
0, n1 =
n, n2 =
r , m2 =
2
∴ In Theorem 3.1, put G1 = K n and G2 = K r , we get
DD* K n [ K r ] ≤  2 ( nr − 1) 
nr ( nr −1)
2
(2)
From (1) and (2) our bound is tight
4. The Multiplicative Degree Distance of Join of Graph
Theorem 4.1. Let Gi , i = 1, 2 , be a
OALib Journal
( ni , mi ) -graph and let mi = e ( Gi ) . Then
7/18
R. Muruganandam et al.

2
1
 DD* ( G1 + G2 )  ≤ 
 2 M 1 ( G1 ) + 4n2 m1 + 4 M 1 ( G1 ) + 8n2 m1
+
n
n
(
)(
 1 2 n1 + n2 − 1)
+ 2m1n2 + 2m2 n1 + n1n2 ( n1 + n2 ) + 2 M 1 ( G2 )
( n1+ n2 )( n1+ n2 −1)

+ 4n1m2 + 4 M 1 ( G2 ) + 8n1m2  

Proof:
 DD* ( G1 + G2 ) 
=
2
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


∏
x , y∈V ( G1 + G2 ) , x ≠ y


1
≤
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y )  
∑


 ( n1 + n2 )( n1 + n2 − 1) x , y∈V ( G1 + G2 ), x ≠ y



1
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y )  

∑


 ( n1 + n2 )( n1 + n2 − 1) x , y∈V ( G1 + G2 )


1

∑
+
n
n
 ( 1 2 )( n1 + n2 − 1) x∈V ( G1 + G2 )
( n1 + n2 )( n1 + n2 −1)
( n1 + n2 )( n1 + n2 −1)

∑ d(G1 +G2 ) ( x, y )  d(G1 +G2 ) ( x ) + d(G1 +G2 ) ( y ) 
y∈V ( G1 + G2 )

( n1 + n2 )( n1 + n2 −1)


1

 ∑ d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 
∑
 ( n1 + n2 )( n1 + n2 − 1) x∈V ( G1 + G2 )  y∈V ( G1 )
+

∑ d(G1 +G2 ) ( x, y )  d(G1 +G2 ) ( x ) + d(G1 +G2 ) ( y ) 
y∈V ( G2 )


1

∑
 ( n1 + n2 )( n1 + n2 − 1) x∈V ( G1 + G2 )
+
∑
x∈V ( G1 + G2 )
+
+
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


y∈V ( G1 )
∑
∑
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


∑
∑
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


x∈V ( G1 ) y∈V ( G2 )
∑
x∈V ( G2 )
+2
OALib Journal
∑
( n1 + n2 )( n1 + n2 −1)
x∈V ( G2 ) y∈V ( G1 )

∑ d(G1 +G2 ) ( x, y )  d(G1 +G2 ) ( x ) + d(G1 +G2 ) ( y ) 
y∈V ( G2 )


1

∑
 ( n1 + n2 )( n1 + n2 − 1) x∈V ( G1 )
+
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


y∈V ( G1 )

∑ d(G1 +G2 ) ( x, y )  d(G1 +G2 ) ( x ) + d(G1 +G2 ) ( y ) 
y∈V ( G2 )


1

∑
 ( n1 + n2 )( n1 + n2 − 1) x∈V ( G1 )
+
∑
( n1 + n2 )( n1 + n2 −1)
∑
∑
∑
( n1 + n2 )( n1 + n2 −1)
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


y∈V ( G1 )
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


x∈V ( G1 ) y∈V ( G2 )
∑
x∈V ( G2 )

∑ d(G1 +G2 ) ( x, y )  d(G1 +G2 ) ( x ) + d(G1 +G2 ) ( y ) 
y∈V ( G2 )

( n1 + n2 )( n1 + n2 −1)
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R. Muruganandam et al.


1
 DD ( G1 + G2 )  ≤  ( n + n )( n + n − 1) ( J1 + 2 J 2 + J 3 ) 
 1 2 1 2

( n1+n2 )( n1+n2 −1)
2
*
where J1 , J 2 , J 3 are terms of the above sums taken in order.
Next we calculate J1 , J 2 and J 3 separately one by one. Now,
J1
=
=
∑
d( G1+G2 ) ( x, y )  d( G1+G2 ) ( x ) + d( G1+G2 ) ( y ) 


∑
x∈V ( G1 ) y∈V ( G1 )
d( G1+G2 ) ( x, y )  d( G1+G2 ) ( x ) + d( G1+G2 ) ( y ) 


∑
x , y∈V ( G1 )
∑ d(G +G ) ( x, y )  d(G +G ) ( x ) + d(G +G ) ( y )
=
1
xy∈G1
+
2
1
2
1
2
∑
d( G1+G2 ) ( x, y )  d( G1+G2 ) ( x ) + d( G1+G2 ) ( y ) 


∑
d( G1+G2 ) ( x, y )  d( G1+G2 ) ( x ) + d( G1+G2 ) ( y ) 


xy∉G1 , x ≠ y
+
xy∉G1 , x =
y
= 1⋅
∑
xy∈G1
 d G +G ( x ) + d G +G ( y ) 
( 1 2)
 ( 1 2)

∑
+ 2⋅
xy∉G1 , x ≠ y
∑
=
xy∈G1
 dG1 ( x ) + n2 + dG2 ( y ) + n2 
 dG1 ( x ) + n2 + dG2 ( y ) + n2  by Lemma 2.1
∑
+2
xy∉G1 , x ≠ y
∑
=
xy∈G1
d
x +d
y +0
 ( G1+G2 ) ( ) ( G1+G2 ) ( ) 
 dG1 ( x ) + dG2 ( y )  + 2n2
∑1
xy∈G1


+ 2  ∑  dG1 ( x ) + dG2 ( y )  + 2n2 ∑ 1
xy∉G1 , x ≠ y 
 xy∉G1 , x≠ y

= 2 M 1 ( G1 ) + 4n2 m1 + 4 M 1 ( G1 ) + 8n2 m1
=
J2
∑
∑
∑
∑
∑
∑
∑
dG1 ( x)
d( G1+G2 ) ( x, y )  d( G1+G2 ) ( x ) + d( G1+G2 ) ( y ) 


x∈V ( G1 ) y∈V ( G2 )
= 1
x∈V ( G1 ) y∈V ( G2 )
=
=
x∈V ( G1 ) y∈V ( G2 )
x∈V ( G1 )
 d G +G ( x ) + d G +G ( y ) 
( 1 2)
 ( 1 2)

 dG1 ( x ) + n2 + dG2 ( y ) + n1  by Lemma 2.1
∑
1+
y∈V ( G2 )
∑
∑
1
dG2 ( y ) + ( n1 + n2 )
x∈V ( G1 ) y∈V ( G2 )
∑
1
∑
1
x∈V ( G1 ) y∈V ( G2 )
= 2m1n2 + 2m2 n1 + ( n1 + n2 ) n1n2
J3
=
=
=
∑
∑
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


x∈V ( G2 ) y∈V ( G2 )
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


∑
x , y∈V ( G2 )
∑ d(G +G ) ( x, y )  d(G +G ) ( x ) + d(G +G ) ( y )
xy∈G2
+
1
2
1
2
1
2
∑
d( G1 + G2 ) ( x, y )  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


∑
d
xy∉G2 , x ≠ y
+
OALib Journal
( G1 + G2 )
xy∉G2 , x =
y
( x, y )  d(G +G ) ( x ) + d(G +G ) ( y )
1
2
1
2
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R. Muruganandam et al.
= 1 ∑  d( G1 + G2 ) ( x ) + d( G1 + G2 ) ( y ) 


xy∈G
2
∑
+2
xy∉G2 , x ≠ y
∑
=
xy∈G2
+2
 dG2 ( x ) + n1 + dG2 ( y ) + n1 
∑
xy∉G2 , x ≠ y
∑
=
xy∈G2
d G +G ( x ) + d G +G ( y ) + 0
( 1 2)
 ( 1 2)

 dG2 ( x ) + n1 + dG2 ( y ) + n1  by Lemma 2.1
 dG2 ( x ) + dG2 ( y )  + 2n1
∑1
xy∈G2


+ 2  ∑  dG2 ( x ) + dG2 ( y )  + 2n1 ∑ 1
xy∉G2 , x ≠ y 
 xy∉G2 , x ≠ y

= 2 M 1 ( G2 ) + 4n1m2 + 4 M 1 ( G2 ) + 8n1m2

2
1
*
 DD ( G1 + G2 )  ≤  ( n + n )( n + n − 1)  2 M 1 ( G1 ) + 4n2 m1 + 4 M 1 ( G1 ) + 8n2 m1
 1 2 1 2
+ 2m1n2 + 2m2 n1 + n1n2 ( n1 + n2 ) + 2 M 1 ( G2 )
( n1+ n2 )( n1+ n2 −1)

+ 4n1m2 + 4 M 1 ( G2 ) + 8n1m2  

Lemma 4.2.
DD* [ K n + K r ]
=  2 ( n + r − 1) 
( n + r )( n + r −1)
2
Proof: Clearly the graph K n + K r is the complete graph K n+r
DD* [ K n + K r ]
= DD* [ K n + r ]
(3)
=  2 ( n + r − 1) 
( n + r )( n + r −1)
2
Remark 4.3. Let G1 = K n and G2 = K r . We get,
M 1 ( G=
n ( n − 1) ,
1)
2
m1 =
n ( n − 1)
2
,
M 1 ( G=
r ( r − 1) ,
2)
2
M 1 ( G2 ) = 0,
=
m2
r ( r − 1)
2
∴ In Theorem 4.1, put
, M1 ( G
0,=
, n2 r ,=
=
n1 n=
m1 0,=
m2 0.
1)
G1 = K n , G2 = K r , we get
DD* [ K n + K r ] ≤  2 ( n + r − 1) 
( n + r )( n + r −1)
(4)
2
From (3) and (4) our bound is tight.
5. The Multiplicative Degree Distance of Disjunction of Graph
Theorem 5.1. Let Gi , i = 1, 2 , be a
OALib Journal
( ni , mi ) -graph and let mi = e ( Gi ) . Then
10/18
R. Muruganandam et al.
 DD* ( G1 ∨ G2 ) 
2

1
≤
2m2 n2 ( 2 M 1 ( G1 ) + 4m1 ) + 2n1 ( 2m1 + n1 ) M 1 ( G2 )
n
n
n
(
 1 2 1n2 − 1)
{
(
)
− 2 M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) + 2n2 M 1 ( G1 )( 2m2 + n2 )
(
+ 2m1n1 ( 2 M 1 ( G2 ) + 4m2 ) − 2 M 1 ( G1 ) 2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 )
)
+ 4n2 M 1 ( G1 ) m2 + 4n1m1 M 1 ( G2 ) − 2 M 1 ( G1 ) M 1 ( G2 )
+ 2n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + 2n2 ) + 2n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
(
)(
− 4 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 )

− 8m1n22 − 4n12 m2 + 16m1m2 

}
)
n1n2 ( n1n2 −1)
Proof:
 DD* ( G1 ∨ G2 ) 
∏
2
d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v ) 


∏
x , y∈G1 u , v∈G2 , x ≠ y ( or ) u ≠ v


 
1
≤
d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v )   
 ∑
∑


 n1n2 ( n1n2 − 1)  x , y∈G1 u ,v∈G2 , x ≠ y ( or )u ≠ v
 

 
1

=
 ∑ ∑ d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v )   
 n1n2 ( n1n2 − 1)  x , y∈G1 u ,v∈G2
 
n1n2 ( n1n2 −1)
n1n2 ( n1n2 −1)



1
 ∑  ∑ d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v ) 

−
n
n
n
n
1
 1 2 ( 1 2 )  x , y∈G1  uv∈G2
  
+ ∑ d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v )    


uv∉G2
  
n1n2 ( n1n2 −1)


1
=
 ∑ ∑ d( G1 ∨ G2 ) ( ( x, u ) , ( y, v ) )  d( G1 ∨ G2 ) ( x, u ) + d( G1 ∨ G2 ) ( y, v ) 
−
n
n
n
n
1
 1 2 ( 1 2 )  xy∈G1 uv∈G2
+
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
xy∉G1 uv∈G2
+
2
1
2
1
2
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
xy∈G1 uv∉G2
+
1
1
2
1
2
1
2
 
∑ ∑ d(G1 ∨G2 ) ( ( x, u ) , ( y, v ) )  d(G1 ∨G2 ) ( x, u ) + d(G1 ∨G2 ) ( y, v )  
xy∉G1 uv∉G2
 
n1n2 ( n1n2 −1)


2
1
*
 DD ( G1 ∨ G2 )  ≤  n n ( n n − 1) ( A3 + A1 + A2 + A4 ) 
 1 2 1 2

n1n2 ( n1n2 −1)
where A3 , A1 , A2 , A4 are terms of the above sums taken in order.
Next we calculate A1 , A2 , A3 and A4 separately one by one. Now,
OALib Journal
11/18
R. Muruganandam et al.
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
=
A1
1
xy∉G1 uv∈G2
=
2
1
2
1
2
∑ ∑ 1 ⋅  n2 dG ( x ) + n1dG ( u ) − dG ( x ) dG ( u ) + n2 dG ( y )
1
xy∉G1 uv∈G2
2
1
2
1
+ n1dG2 ( v ) − dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
∑ ∑
xy∉G1 uv∈G2
 dG1 ( x ) + dG1 ( y )  + n1
∑ ∑
xy∉G1 uv∈G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − ∑ ∑ dG ( y ) dG ( v )
−
2
1
xy∉G1 uv∈G2
xy∉G1 uv∈G2
1
2






=n2  ∑  dG1 ( x ) + dG1 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∉G1
  uv∈G2 
 xy∉G1   uv∈G2



 


−  ∑ dG1 ( x )   ∑ dG2 ( u )  −  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∉G1
  uv∈G2
  xy∉G1
  uv∈G2

= 2m2 n2 ( 2 M 1 ( G1 ) + 4m1 ) + 2n1 ( 2m1 + n1 ) M 1 ( G2 )
(
( ))
− 2 M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 G1
=
A2
=
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
xy∈G1 uv∉G2
∑ ∑
xy∈G1 uv∉G2
1
2
1
2
1
2
 n2 dG1 ( x ) + n1dG2 ( u ) − dG1 ( x ) dG2 ( u ) + n2 dG1 ( y )
+ n1dG2 ( v ) − dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2 ∑ ∑  dG1 ( x ) + dG2 ( y )  + n1 ∑ ∑  dG2 ( u ) + dG2 ( v ) 
xy∈G1 uv∉G2
−
xy∈G1 uv∉G2
∑ ∑ dG ( x ) dG ( u ) − ∑ ∑ dG ( y ) dG ( v )
1
xy∈G1 uv∉G2
2
1
xy∈G1 uv∉G2
2






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∈G1
  uv∉G2 
 xy∈G1   uv∉G2



 


−  ∑ dG1 ( x )   ∑ dG2 ( u )  −  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∈G1
  uv∉G2
  xy∈G1
  uv∉G2

= 2n2 M 1 ( G1 )( 2m2 + n2 ) + 2n1m1 ( 2 M 1 ( G2 ) + 4m2 )
− 2 M 1 ( G1 )  2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 ) 
=
A3
=
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
1
xy∈G1 uv∈G2
∑ ∑
xy∈G1 uv∈G2
2
1
2
1
2
 n2 dG1 ( x ) + n1dG2 ( u ) − dG1 ( x ) dG2 ( u ) + n2 dG1 ( y )
+ n1dG2 ( v ) − dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
−
∑ ∑
xy∈G1 uv∈G2
 dG1 ( x ) + dG2 ( y )  + n1
∑ ∑
xy∈G1 uv∈G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − ∑ ∑ dG ( y ) dG ( v )
xy∈G1 uv∈G2
1
2
xy∈G1 uv∈G2
1
2






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∈G1
  uv∈G2 
 xy∈G1   uv∈G2



 


−  ∑ dG1 ( x )   ∑ dG2 ( u )  −  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∈G1
  uv∈G2
  xy∈G1
  uv∈G2

= 4n2 m2 M 1 ( G1 ) + 4n1m1M 1 ( G2 ) − 2 M 1 ( G1 ) M 1 ( G2 )
OALib Journal
12/18
R. Muruganandam et al.
∑ ∑ d(G ∨G ) ( ( x, u ) , ( y, v ) )  d(G ∨G ) ( x, u ) + d(G ∨G ) ( y, v )
=
A4
xy∉G1 uv∉G2
= 2
∑ ∑
xy∉G1 uv∉G2
−2
∑
1
2
1
2
1
2
d
x, u ) + d( G1 ∨ G2 ) ( y, v ) 
 ( G1 ∨ G2 ) (

∑
xy∉G1 , x = y uv∉G2 , u = v
 d G ∨ G ( x, u ) + d G ∨ G ( y , v ) 
( 1 2)
 ( 1 2)

=
A4 2 A5 − 2 A6 , where A5 and A6 are terms of the above sums taken in
order.
Now,
=
A5
=
∑ ∑
 d G ∨ G ( x, u ) + d G ∨ G ( y , v ) 
( 1 2)
 ( 1 2)

∑ ∑
 n2 dG1 ( x ) + n1dG2 ( u ) − dG1 ( x ) dG2 ( u ) + n2 dG1 ( y )
xy∉G1 uv∉G2
xy∉G1 uv∉G2
+ n1dG2 ( v ) − dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
−
∑ ∑
xy∉G1 uv∉G2
 dG1 ( x ) + dG1 ( y )  + n1
∑ ∑
xy∉G1 uv∉G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − ∑ ∑ dG ( y ) dG ( v )
xy∉G1 uv∉G2
1
2
1
xy∉G1 uv∉G2
2






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∉G1
  uv∉G2 
 xy∉G1   uv∉G2



 


−  ∑ dG1 ( x )   ∑ dG2 ( u )  −  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∉G1
  uv∉G2
  xy∉G1
  uv∉G2

= n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + n2 ) + n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
)(
(
− 2 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 )
A6
=
∑
∑
 d G ∨ G ( x, u ) + d G ∨ G ( y , v ) 
( 1 2)
 ( 1 2)

∑
∑
 n2 dG1 ( x ) + n1dG2 ( u ) − dG1 ( x ) dG2 ( u ) + n2 dG1 ( y )
xy∉G1 , x =y uv∉G2 , u =v
=
xy∉G1 , u =v uv∉G2 , u =v
)
+ n1dG2 ( v ) − dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
−
∑
∑
∑
∑
xy∉G1 , x =y uv∉G2 , u =v
 dG1 ( x ) + dG1 ( y )  + n1
dG1 ( x ) dG2 ( u ) −
xy∉G1 , x =y uv∉G2 , u =v
∑
∑
∑
xy∉G1 , x =y uv∉G2 , u =v
∑
 dG2 ( u ) + dG2 ( v ) 
dG1 ( y ) dG2 ( v )
xy∉G1 , x =y uv∉G2 , u =v






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
,u v 
,u v
1,x y
1 , x y   uv∉G2=
 xy∉G=
  uv∉G2=
 xy∉G=



 


−  ∑ dG1 ( x )   ∑ dG2 ( u )  −  ∑ dG1 ( y )   ∑ dG2 ( v ) 
=
 xy∉G1 , x y
=
 uv∉G2 ,u v
=
 xy∉G1 , x y
=
 uv∉G2 ,u v

= 4m1n22 + 4m2 n12 − 8m1m2
OALib Journal
13/18
R. Muruganandam et al.
 DD* ( G1 ∨ G2 ) 

1
2m2 n2 ( 2 M 1 ( G1 ) + 4m1 ) + 2n1 ( 2m1 + n1 ) M 1 ( G2 )
≤
 n1n2 ( n1n2 − 1)
− 2 M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) + 2n2 M 1 ( G1 )( 2m2 + n2 )
2
{
(
(
)
+ 2m1n1 ( 2 M 1 ( G2 ) + 4m2 ) − 2 M 1 ( G1 ) 2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 )
)
+ 4n2 M 1 ( G1 ) m2 + 4n1m1 M 1 ( G2 ) − 2 M 1 ( G1 ) M 1 ( G2 )
+ 2n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + 2n2 ) + 2n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
(
)(
− 4 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 )

− 8m1n22 − 4n12 m2 + 16m1m2 

}
n1n2 ( n1n2 −1)
)
Lemma 5.2.
DD* [ K m ∨ K n ] =
( 2mn − 2 )
mn( mn −1)
2
Proof: Clearly the graph K m ∨ K n is the complete graph K mn .
DD* ( K m ∨ K n ) = DD* ( K mn ) =
( 2mn − 2 )
mn( mn −1)
2
(5)
Remark 5.3. Let G1 = K m and G2 = K n . We get
=
n1 m=
, n2 n=
, m1
m ( m − 1)
2
n ( n − 1)
=
, m2
2
, m1 0,=
=
m2 0
M 1 ( G1 ) =
M1 ( K m ) =
m ( m − 1) , M 1 ( G2 ) =
M1 ( K n ) =
n ( n − 1)
2
( )
2
( )
=
M 1 G1 M=
0, M
=
M
=
0, =
M 1 ( G1 ) M=
0
1 ( Km )
1 G2
1 ( Kn )
1 ( Km )
∴ In Theorem 5.1, put G1 = K m and G2 = K n , we get
DD* [ K m ∨ K n ] ≤ ( 2mn − 2 )
mn( mn −1)
2
(6)
From (5) and (6) our bound is tight.
6. The Multiplicative Degree Distance of Symmetric
difference of Graph
Theorem 6.1.
 DD* ( G1 ⊕ G2 ) 

1

≤
 2n2 m2 ( 2M 1 ( G1 ) + 4m1 ) + 2n1 M 1 ( G2 )( 2m1 + n1 )
n
n
n
(
 1 2 1n2 − 1)
2
(
)
− 4M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) + 2n2 M 1 ( G1 )( 2m2 + n2 )
(
+ 2n1m1 ( 2 M 1 ( G2 ) + 4m2 ) − 4 M 1 ( G1 ) 2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 )
+ 4n2 M 1 ( G1 ) m2 + 4n1m1 M 1 ( G2 ) − 4 M 1 ( G1 ) M 1 ( G2 )
)
+ 2  n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + n2 ) + n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
(
)(
)
− 4 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 ) 


− 2 4n22 m1 + 4n12 m2 − 16m1m2  

(
OALib Journal
)
n1n2 ( n1n2 −1)
14/18
R. Muruganandam et al.
Proof:
 DD* ( G1 ⊕ G2 ) 
∏
2
d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v ) 


∏
x , y∈G1 u ,v∈G2 , x ≠ y( or )u ≠v

 
1

≤
d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v )   
 ∑
∑


 
 n1n2 ( n1n2 − 1)  x , y∈G1 u ,v∈G2 , x≠ y( or )u ≠v


 
1
=
 ∑ ∑ d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v )   
 
 n1n2 ( n1n2 − 1)  x , y∈G1 u ,v∈G2
n1n2 ( n1n2 −1)
n1n2 ( n1n2 −1)



1
 ∑  ∑ d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v ) 

 n1n2 ( n1n2 − 1)  x , y∈G1  uv∈G2
  
+ ∑ d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v )    


uv∉G2
  
n1n2 ( n1n2 −1)

1

=
 ∑ ∑ d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v ) 
 n1n2 ( n1n2 − 1)  xy∈G1 uv∈G2
+
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
xy∉G1 uv∈G2
+
1
2
1
2
1
2
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
xy∈G1 uv∉G2
1
2
1
2
1
2
 
+ ∑ ∑ d( G1⊕G2 ) ( ( x, u ) , ( y, v ) )  d( G1⊕G2 ) ( x, u ) + d( G1⊕G2 ) ( y, v )   


xy∉G1 uv∉G2
 
n1n2 ( n1n2 −1)


1
 DD ( G1 ⊕ G2 )  ≤ 
( C3 + C1 + C2 + C4 )
 n1n2 ( n1n2 − 1)

n1n2 ( n1n2 −1)
2
*
where C3 , C1 , C2 , C4 are terms of the above sums taken in order.
Next we calculate C1 , C2 , C3 and C4 separately.
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
C1
=
xy∉G1 uv∈G2
=
1
2
1
2
1
2
∑ ∑ 1⋅  n2 dG ( x ) + n1dG ( u ) − 2dG ( x ) dG ( u ) + n2 dG ( y )
1
xy∉G1 uv∈G2
2
1
2
1
+ n1dG2 ( v ) − 2dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
∑ ∑
xy∉G1 uv∈G2
−2
 dG1 ( x ) + dG1 ( y )  + n1
∑ ∑
xy∉G1 uv∈G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − 2 ∑ ∑ dG ( y ) dG ( v )
xy∉G1 uv∈G2
1
2
xy∉G1 uv∈G2
1
2






=n2  ∑  dG1 ( x ) + dG1 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∉G1
  uv∈G2 
 xy∉G1   uv∈G2



 


− 2  ∑ dG1 ( x )   ∑ dG2 ( u )  − 2  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∉G1
  uv∈G2
  xy∉G1
  uv∈G2

= 2n2 m2 ( 2M 1 ( G1 ) + 4m1 ) + 2n1 M 1 ( G2 ) ( 2m1 + n1 )
(
− 4 M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 )
OALib Journal
)
15/18
R. Muruganandam et al.
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
C2
=
1
xy∈G1 uv∉G2
∑ ∑
=
xy∈G1 uv∉G2
2
1
2
1
2
 n2 dG1 ( x ) + n1dG2 ( u ) − 2dG1 ( x ) dG2 ( u )
+ n2 dG1 ( y ) + n1dG2 ( v ) − 2dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
∑ ∑
xy∈G1 uv∉G2
 dG1 ( x ) + dG2 ( y )  + n1
∑ ∑
xy∈G1 uv∉G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − 2 ∑ ∑ dG ( y ) dG ( v )
−2
xy∈G1 uv∉G2
2
1
xy∈G1 uv∉G2
1
2






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∈G1
  uv∉G2 
 xy∈G1   uv∉G2



 


− 2  ∑ dG1 ( x )   ∑ dG2 ( u )  − 2  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∈G1
  uv∉G2
  xy∈G1
  uv∉G2

= 2n2 M 1 ( G1 )( m2 + n2 ) + 2n1m1 ( 2 M 1 ( G2 ) + 4m2 )
(
− 4 M 1 ( G1 ) 2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 )
C3
=
=
)
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
1
xy∈G1 uv∈G2
∑ ∑
xy∈G1 uv∈G2
2
1
2
1
2
 n2 dG1 ( x ) + n1dG2 ( u ) − 2dG1 ( x ) dG2 ( u )
+ n2 dG1 ( y ) + n1dG2 ( v ) − 2dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2
∑ ∑
xy∈G1 uv∈G2
−2
 dG1 ( x ) + dG2 ( y )  + n1
∑ ∑
xy∈G1 uv∈G2
 dG2 ( u ) + dG2 ( v ) 
∑ ∑ dG ( x ) dG ( u ) − 2 ∑ ∑ dG ( y ) dG ( v )
xy∈G1 uv∈G2
1
2
xy∈G1 uv∈G2
1
2






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∈G1
  uv∈G2 
 xy∈G1   uv∈G2



 


− 2  ∑ dG1 ( x )   ∑ dG2 ( u )  − 2  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∈G1
  uv∈G2
  xy∈G1
  uv∈G2

= 4n2 M 1 ( G1 ) m2 + 4n1m1 M 1 ( G2 ) − 4 M 1 ( G1 ) M 1 ( G2 )
C4
=
∑ ∑ d(G ⊕G ) ( ( x, u ) , ( y, v ) )  d(G ⊕G ) ( x, u ) + d(G ⊕G ) ( y, v )
xy∉G1 uv∉G2
= 2
∑ ∑
xy∉G1 uv∉G2
−2
∑
1
2
1
2
1
2
d
x, u ) + d( G1 ⊕G2 ) ( y, v ) 
 ( G1 ⊕G2 ) (

∑
xy∉G1 , x = y uv∉G2 , u = v
d
x, u ) + d( G1 ⊕G2 ) ( y, v ) 
 ( G1 ⊕G2 ) (

=
C4 2C5 − 2C6 , where C5 and C6 denote the sums of the above terms in
order.
Now,
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R. Muruganandam et al.
=
C5
=
∑ ∑
 d G ⊕ G ( x, u ) + d G ⊕ G ( y , v ) 
( 1 2)
 ( 1 2)

∑ ∑
 n2 dG1 ( x ) + n1dG2 ( u ) − 2dG1 ( x ) dG2 ( u )
xy∉G1 uv∉G2
xy∉G1 uv∉G2
+ n2 dG1 ( y ) + n1dG2 ( v ) − 2dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2 ∑ ∑  dG1 ( x ) + dG2 ( y )  + n1 ∑ ∑  dG2 ( u ) + dG2 ( v ) 
xy∉G1 uv∉G2
∑ ∑ dG ( x)dG
−2
1
xy∉G1 uv∉G2
2
xy∉G1 uv∉G2
(u ) − 2
∑ ∑ dG ( y )dG
xy∉G1 uv∉G2
1
2
(v )






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
 xy∉G1
  uv∉G2 
 xy∉G1   uv∉G2



 


− 2  ∑ dG1 ( x )   ∑ dG2 ( u )  − 2  ∑ dG1 ( y )   ∑ dG2 ( v ) 
 xy∉G1
  uv∉G2
  xy∉G1
  uv∉G2

= n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + n2 ) + n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
(
)(
− 4( 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 )
C6
=
=
∑
∑
d
x, u ) + d( G1 ⊕G2 ) ( y, v ) 
 ( G1 ⊕G2 ) (

∑
∑
[n2 dG1 ( x ) + n1dG2 ( u ) − 2dG1 ( x ) dG2 ( u )
xy∉G1 , x =y uv∉G2 , u =v
)
xy∉G1 , x =y uv∉G2 , u =v
+ n2 dG1 ( y ) + n1dG2 ( v ) − 2dG1 ( y ) dG2 ( v )  by Lemma 2.1
= n2 ∑
∑  dG1 ( x ) + dG2 ( y ) + n1 ∑ ∑  dG2 ( u ) + dG2 ( v )
xy∉G1 , x =y uv∉G2 , u =v
−2
∑
∑ dG1 ,u =v ( x ) dG2 ( u ) − 2
xy∉G1 , x =y uv∉G2
xy∉G1 , x =y uv∉G2 ,u =v
∑
dG1 ( y ) dG2 ( v )
∑
xy∉G1 , x =y uv∉G2 , u =v






=n2  ∑  dG1 ( x ) + dG2 ( y )    ∑ 1 + n1  ∑ 1  ∑  dG2 ( u ) + dG2 ( v )  
,u v 
,u v
1,x y
1 , x y   uv∉G2=
 xy∉G=
  uv∉G2=
 xy∉G=



 


− 2  ∑ dG1 ( x )   ∑ dG2 ( u )  − 2  ∑ dG1 ( y )   ∑ dG2 ( v ) 
=
=
 xy∉G1 , x y
=
 uv∉G2 ,u v
 =
 xy∉G1 , x y
  uv∉G2 ,u v

= 4n22 m1 + 4n12 m2 − 16m1m2
 DD* ( G1 ⊕ G2 ) 
1


≤
 2n2 m2 ( 2M 1 ( G1 ) + 4m1 ) + 2n1 M 1 ( G2 )( 2m1 + n1 )
 n1n2 ( n1n2 − 1)
2
(
)
− 4M 1 ( G2 ) 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) + 2n2 M 1 ( G1 )( 2m2 + n2 )
(
+ 2n1m1 ( 2 M 1 ( G2 ) + 4m2 ) − 4 M 1 ( G1 ) 2 ( n2 − 1) m2 + 2m2 − M 1 ( G2 )
+ 4n2 M 1 ( G1 ) m2 + 4n1m1 M 1 ( G2 ) − 4 M 1 ( G1 ) M 1 ( G2 )
)
+ 2  n2 ( 2 M 1 ( G1 ) + 4m1 ) ( 2m2 + n2 ) + n1 ( 2 M 1 ( G2 ) + 4m2 ) ( 2m1 + n1 )
)(
(
)
− 4 2m1 ( n1 − 1) + 2m1 − M 1 ( G1 ) 2m2 ( n2 − 1) + 2m2 − M 1 ( G2 ) 


− 2 4n22 m1 + 4n12 m2 − 16m1m2  

(
)
n1n2 ( n1n2 −1)
Lemma 6.2.
DD* [ K m ⊕ K1 ] = ( 2m − 2 )
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R. Muruganandam et al.
Proof: Clearly the graph K m ⊕ K1 is the complete graph K m
DD* [ K m ⊕ K1 ] = DD* K m = ( 2m − 2 )
m( m−1)
2
(7)
Remark 6.3. Let G1 = K m and G2 = K1 . We get
=
n1 m,=
n2 1,=
m1
m ( m − 1)
2
,=
m2 0,=
m1 0, =
m2 0
M 1 ( G1 ) =
M1 ( K m ) =
m ( m − 1) , M 1 ( G2 ) =
M 1 ( K1 ) =
0
2
( )
( )
M
=
M=
M=
0, M=
0
1 G1
1 ( Km )
1 G2
1 ( K1 )
0, M
0
M
=
M=
=
M
=
1 ( G1 )
1 ( Km )
1 ( G2 )
1 ( K1 )
∴ In Theorem 6.1, put G1 = K m and G2 = K1 , we get
DD* [ K m ⊕ K1 ] ≤ ( 2m − 2 )
m( m−1)
2
(8)
From (7) and (8) our bound is tight.
References
[1]
Dobrynin, A.A. and Kochetova, A.A. (1994) Degree Distance of a Graph: A Degree
Analogue of the Wiener Index. Journal of Chemical Information and Computer
Sciences, 34, 1082-1086. https://doi.org/10.1021/ci00021a008
[2]
Gutman, I. (1994) Selected Properties of the Schultz Molecular Topological Index.
Journal of Chemical Information and Computer Sciences, 34, 1087-1089.
[3]
Gutman, I. and Trinajstic, N. (1972) Graph Theory and Moleclar Orbitals. Total
pi-Electron Energy of Alternant Hydrocarbons. Chemical Physics Letters, 17, 535538.
[4]
Devillers, J. and Balaban, A.T. (1999) Topological Indices and Related Descriptors
in QSAR and QSPR. Gordon and Breach, Amsterdam, The Netherlands.
[5]
Doslic, T. (2008) Vertex-Weighted Wiener Polynomials for Composite Graphs. Ars
Mathematica Contemporanea, 1, 66-80.
[6]
Ashrafi, A.R., Doslic, T. and Hamzeha, A. (2010) The Zagreb Coindices of Graph
Operations. Discrete Applied Mathematics, 158, 1571-1578.
https://doi.org/10.1016/j.dam.2010.05.017
[7]
Ashrafi, A.R., Doslic, T. and Hamzeha, A. (2011) Extremal Graphs with Respect to
The Zagreb Coindices, MATCH Commun. Math. Comput. Chem., 65, 85-92.
[8]
Hamzeha, A., Iranmanesh, A., Hossein-Zadeh, S. and Diudea, M.V. (2012) Generalized Degree Distance of Trees, Unicyclic and Bicyclic Graphs. Studia Ubb Chemia,
LVII, 4, 73-85.
[9]
Hamzeha, A., Iranmanesh, A. and Hossein-Zadeh, S. (2013) Some Results on Generalized Degree Distance. Open Journal of Discrete Mathematics, 3, 143-150.
https://doi.org/10.4236/ojdm.2013.33026
[10] Imrich, W. and Klavzar, S. (2000) Product Graphs: Structure and Recognition. John
Wiley and Sons, New York.
[11] Kahlifeh, M.H., Yousefi-Azari, H. and Ashrafi, A.R. (2008) The Hyper-Wiener Index of Graph Operations. Computers and Mathematics with Applications, 56, 14021407. https://doi.org/10.1016/j.camwa.2008.03.003
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