Transactions on Combinatorics
ISSN (print): 2251-8657, ISSN (on-line): 2251-8665
Vol. 4 No. 4 (2015), pp. 5-14.
c 2015 University of Isfahan
⃝
www.combinatorics.ir
www.ui.ac.ir
ON THE HARMONIC INDEX OF GRAPH OPERATIONS
B. SHWETHA SHETTY, V. LOKESHA∗ AND P. S. RANJINI
Communicated by Ivan Gutman
Abstract. The harmonic index of a connected graph G, denoted by H(G), is defined as H(G) =
∑
2
uv∈E(G) du +dv where dv is the degree of a vertex v in G. In this paper, expressions for the Harary
indices of the join, corona product, Cartesian product, composition and symmetric difference of graphs
are derived.
1. Introduction and Preliminaries
Throughout this paper we consider only simple connected graphs, i.e. connected graphs without
loops and multiple edges. For a graph G, V (G) and E(G) denote the set of all vertices and edges,
respectively. For a graph G, the degree of a vertex v is the number of edges incident to v and denoted
by dv .
The Harmonic index H(G) is vertex-degree-based topological index. This index first appeared in [5],
and was defined as[14], [16]
H(G) =
∑
uv∈E(G)
2
du + dv
The connectivity index introduced in 1975 by Milan Randic [12], who has shown this index to reflect
molecular branching. Randic index was defined as follows;
∑
1
√
χ(G) =
du dv
uv∈E(G)
MSC(2010): Primary: 05C20; Secondary: 05C05.
Keywords: Harmonic index, Graph operations.
Received: 29 October 2014, Accepted: 1 December 2014.
∗Corresponding author.
5
6
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
Ernesto Estrada et al [4], introduced atom-bond connectivity (ABC) index, which it has been applied
up until now to study the stability of alkanes and the strain energy of cycloalkanes. This index is
defined as follows:
∑
ABC(G) =
e=uv∈E(G)
√
du + dv − 2
du dv
For a vertex v in a connected nontrivial graph G, The eccentricity eccG (v) of a vertex v is the
greatest geodesic distance between v and any other vertex. The Diameter d(G) of G is defined as
d(G) = max{eccG (v)|v ∈ V (G)}.
The Composition (also called lexicographic product [7]) G = G1 [G2 ] of graphs G1 and G2 with
disjoint vertex sets V (G1 ) and V (G2 ) and edge sets E(G1 ) and E(G2 ) is the graph with vertex set
V (G1 ) × V (G2 ) and (ui , vj ) is adjacent with (uk , vl ) whenever ui is adjacent with uk , or ui = uk and
vj is adjacent with vl .
The Cartesian product G1 × G2 of graphs G1 and G2 has the vertex set V (G1 × G2 ) = V (G1 ) × V (G2 )
and (ui , vj )(uk , vl ) is an edge of G1 × G2 if ui = uk and vj vl ∈ E(G2 ), or ui uk ∈ E(G1 ) and vj = vl
[2].
For given graphs G1 and G2 we define their Corona product G1 oG2 as the graph obtained by taking
|V (G1 )| copies of G2 and joining each vertex of the i-th copy with vertex vi ∈ V (G1 ). Obviously,
|V (G1 oG2 )| = |V (G1 )|(1 + |V (G2 )|) and |E(G1 oG2 )| = |E(G1 )| + |V (G1 )|(|V (G2 )| + |E(G2 )|)[6].
A sum G1 + G2 of two graphs G1 and G2 with disjoint vertex sets V (G1 ) and V (G2 ) is the graph on
the vertex set V (G1 ) ∪ V (G2 ) and the edge set
E(G1 )∪E(G2 )∪{uv | u ∈ V (G1 ), v ∈ V (G2 )}. Hence, the sum of two graphs is obtained by connecting
each vertex of one graph to each vertex of the other graph, while keeping all edges of both graphs[11].
The symmetric difference G1 ⊕ G2 of two graphs G1 and G2 is the graph with vertex set V (G1 ) ×
V (G2 ) and E(G1 ⊕ G2 ) = {(u1 , u2 )(v1 , v2 )|u1 v1 ∈ E(G1 )
or
u2 v2 ∈ E(G2 )
but
not
both}.
Obviously,|E(G1 ⊕ G2 )| = |E(G1 )||V (G2 )|2 + |E(G2 )||V (G1 )|2 − 4|E(G1 )||E(G2 )|
dG1 ⊕G2 (u, v) = |V (G2 )|dG1 (u) + |V (G1 )|dG2 (v) − 2dG1 (u)dG2 (v) [8].
In this paper, expressions for the Harmonic indices of the join, corona product, Cartesian product,
composition and symmetric difference of graphs are derived[1],[3],[10],[13],[15],[9].
2. Harmonic index of graph operations
Theorem 2.1. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively.
Then
{n
1
1
ABC(G2 ) + n1 R(G2 ) + n1 n2 m2
(1 + n2 )2 2
}
(n2 )3
+
ABC(G1 ) + (n2 )3 R(G1 ) + (n2 )2 m1 .
2
H(G1 [G2 ]) ≤
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
7
Proof. Let V (G1 ) = {u1 , u2 , . . . , un1 } and V (G2 ) = {v1 , v2 , . . . , vn2 } be a set of vertex for G1 and G2
respectively. By the definition of the composition of two graph one can see that,
|E(G1 [G2 ])| = |E(G1 )||V (G2 )|2 + |E(G2 )||V (G1 )|
dG1 [G2 ] (u, v) = |V (G2 )|dG1 (u) + dG2 (v)
∑
H(G1 [G2 ]) =
(ui ,vj ),(uk ,vl )∈E(G1 [G2 ]),(ui ,vj )̸=(uk ,vl )
∑
=
(ui ,vj ),(ui ,vl )∈E(G1 [G2 ]),j̸=l
2
dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (ui , vl )
∑
+
(ui ,vj ),(uk ,vl )∈E(G1 [G2 ]),i̸=k
=
∑
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
+
∑
∑
2
dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (uk , vl )
2
|V (G2 )|dG1 (ui ) + dG2 (vj ) + |V (G2 )|dG1 (ui ) + dG2 (vl )
∑
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
=
∑
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
+
∑
∑
2
|V (G2 )|dG1 (ui ) + dG2 (vj ) + |V (G2 )|dG1 (uk ) + dG2 (vl )
2
dG2 (vj ) + dG2 (vl ) + 2n2 dG1 (ui )
∑
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
(1)
2
dG1 [G2 ] (ui , vj ) + dG1 [G2 ] (uk , vl )
2
n2 dG1 (ui ) + n2 dG1 (uk ) + dG2 (vj ) + dG2 (vl )
= A1 + A2
where A1 , A2 are the sums of the above terms, in order.
For any vertex u, v, w of a graph G and for any positive integer n1 , n2 , n3
1
1
n1 n2
n3
≤
( +
+
)
(n
+
n
+
n
)
d
d
d
+ n2 dv + n3 dw )
1
2
3
u
v
w
1
(n1 +n2 +n3 ) (n1 du
with equality if and only if du = dv = dw .
Therefore
A1 =
∑
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
≤
2
2 + 2n2
∑
1
1
1 + n2 2 + 2n2
1
+
1 + n2
+
n2
1 + n2
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
{
=
2
dG2 (vj ) + dG2 (vl ) + 2n2 dG1 (ui )
∑
∑
∑
(
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
1 ( 1
1
2n2 )
+
+
2 + 2n2 dG2 (vj ) dG2 (vl ) dG1 (ui )
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
)
1
1
2
+
−
dG2 (vj ) dG2 (vl ) dG2 (vj )dG2 (vl )
1
dG2 (vj )dG2 (vl )
1 }
dG1 (ui )
8
Trans. Comb. 4 no. 4 (2015) 5-14
1 { 1
≤
1 + n2 2 + 2n2
+
1
1 + n2
n2
+
1 + n2
(2)
=
∑
B. S. Shwetha, V. Lokesha and P. S. Ranjini
∑
∑
√
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
1
dG2 (vj )dG2 (vl )
}
√
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
dG2 (vl ) + dG2 (vj ) − 2
dG2 (vl )dG2 (vj )
∑
1
ui ∈V (G1 ) vj ,vl ∈E(G2 )
1 { n1
n1
n1 n2 m2 }
ABC(G2 ) +
R(G2 ) +
1 + n2 2 + 2n2
1 + n2
1 + n2
Similarly
A2 =
∑
∑
∑
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
≤
2
2 + 2n2
∑
+
+
≤
(3)
n2
1
1 + n2 2 + 2n2
n2
1 + n2
∑
∑
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
{
=
2
n2 dG1 (ui ) + n2 dG1 (uk ) + dG2 (vj ) + dG2 (vl )
∑
∑
∑
∑
∑
∑
+
n2
1 + n2
+
1
2 + 2n2
∑ (
∑
∑
∑
)
1
1
2
+
−
dG1 (ui ) dG1 (uk ) dG1 (ui )dG1 (uk )
1
dG1 (ui )dG1 (uk )
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
1 { n2
1 + n2 2 + 2n2
∑ (
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
1
2 + 2n2
1 ( n2
n2
1
1 )
+
+
+
2 + 2n2 dG1 (ui ) dG1 (uk ) dG2 (vj ) dG2 (vj )
∑
1 )}
1
+
dG2 (vj ) dG2 (vj )
√
∑
dG1 (ui ) + dG1 (uk ) − 2
dG1 (ui )dG1 (uk )
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
∑
∑
1
√
dG1 (ui )dG1 (uk )
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
}
∑
∑
∑
(1 + 1)
ui ,uk ∈E(G1 ) vj ∈V (G2 ) vl ∈V (G2 )
1 { (n2 )3
(n2 )3
(n2 )2 m1 }
=
ABC(G1 ) +
R(G1 ) +
1 + n2 2 + 2n2
1 + n2
1 + n2
From equation (1),(2)and (3) we get
{n
1
1
ABC(G2 ) + n1 R(G2 ) + n1 n2 m2
(1 + n2 )2 2
}
(n2 )3
+
ABC(G1 ) + (n2 )3 R(G1 ) + (n2 )2 m1 .
2
H(G1 [G2 ]) ≤
□
Theorem 2.2. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively.
Then
H(G1 × G2 ) ≤
1{
n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2
8
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
9
}
+ n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1 .
Proof. Let V (G1 ) = {u1 , u2 , . . . , un1 } and V (G2 ) = {v1 , v2 , . . . , vn2 } be a set of vertex for G1 and G2
respectively. By the Definition of the cartesian product of two graph one can see that,
|E(G1 × G2 )| = |E(G1 )||V (G2 )| + |E(G2 )||V (G1 )|
dG1 ×G2 (u, v) = dG1 (u) + dG2 (v)
∑
H(G1 × G2 ) =
(ui ,vj ),(uk ,vl )∈E(G1 ×G2 ),(ui ,vj )̸=(uk ,vl )
∑
=
(ui ,vj ),(ui ,vl )∈E(G1 ×G2 ),vj vl ∈E(G2 )
2
dG1 ×G2 (ui , vj ) + dG1 ×G2 (ui , vl )
∑
+
(ui ,vj ),(uk ,vj )∈E(G1 ×G2 ),ui uk ∈E(G1 )
∑
=
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
+
∑
vj ∈V (G2 ) ui ,uk ∈E(G1 )
(4)
2
dG1 ×G2 (ui , vj ) + dG1 ×G2 (uk , vl )
2
dG1 ×G2 (ui , vj ) + dG1 ×G2 (uk , vj )
2
dG2 (vj ) + dG2 (vl ) + 2dG1 (ui )
2
dG1 (ui ) + dG1 (uk ) + 2dG2 (vj )
= B1 + B2
where B1 , B2 are the sums of the above terms, in order.
∑
B1 =
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
≤
=
1
8
∑
+2
∑
(
ui ∈V (G1 ) vj ,vl ∈E(G2 )
1{
8
2
dG2 (vj ) + dG2 (vl ) + 2dG1 (ui )
∑
∑
1
2 )
1
+
+
dG2 (vj ) dG2 (vl ) dG1 (ui )
(
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
)
1
2
1
+
−
dG2 (vj ) dG2 (vl ) dG2 (vj )dG2 (vl )
1
dG2 (vj )dG2 (vl )
1 }
dG1 (ui )
ui ∈V (G1 ) vj ,vl ∈E(G2 )
√
∑
dG2 (vl ) + dG2 (vj ) − 2
1{ ∑
≤
8
dG2 (vl )dG2 (vj )
+2
∑
∑
ui ∈V (G1 ) vj ,vl ∈E(G2 )
∑
∑
1
dG2 (vj )dG2 (vl )
ui ∈V (G1 ) vj ,vl ∈E(G2 )
}
∑
∑
+2
1
+2
ui ∈V (G1 ) vj ,vl ∈E(G2 )
√
10
Trans. Comb. 4 no. 4 (2015) 5-14
(5)
=
B. S. Shwetha, V. Lokesha and P. S. Ranjini
}
1{
n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2
8
Similarly we get,
}
1{
n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1
8
From equation (4),(5)and (6) we get
1{
H(G1 × G2 ) ≤
n1 ABC(G2 ) + 2n1 R(G2 ) + 2n1 m2
8
}
+ n2 ABC(G1 ) + 2n2 R(G1 ) + 2n2 m1 .
B1 ≤
(6)
□
Theorem 2.3. For i ∈ {1, 2}, let Gi be a graph of minimum degree δi , maximum degree ∆i ,order ni
and size mi . Then
m1
m2 n1
2n1 n2
+
+
∆1 + n2 ∆2 + 1 ∆1 + ∆2 + n2 + 1
m1
m2 n1
2n1 n2
H(G1 oG2 ) ≤
+
+
.
δ1 + n2 δ2 + 1 δ1 + δ2 + n2 + 1
H(G1 oG2 ) ≥
Proof. The edges of G1 oG2 are partitioned into three subsets E1 , E2 and E3 as follows
E1 = {e ∈ E(G1 oG2 ), e ∈ E(G1 )}
E2 = {e ∈ E(G1 oG2 ), e ∈ E(G2i ), i = 1, 2, . . . , |V (G1 )|}
E3 = {e ∈ E(G1 oG2 ), e = uv, u ∈ V (G2i ), i = 1, 2, . . . , |V (G1 )| and v ∈ V (G1 )}
and if u is a vertex of G1 oG2 , then
{
dG1 oG2 (u) =
dG1 (u) + |V (G2 )| if u ∈ V (G1 )
if u ∈ V (G2 )
dG2 (u) + 1
Let G1 = (Vi , Ei ), i ∈ {1, 2} and let G1 oG2 = (V, E)
we have
H(G1 oG2 ) =
∑
uv∈E(G1 oG2 )
(7)
2
dG1 oG2 (u) + dG1 oG2 (v)
= Q1 + Q2 + Q3
where
Q1 =
∑
uv∈E1
(8)
2
dG1 (u) + n2 + dG1 (v) + n2
2m1
m1
=
∆1 + n2 + ∆1 + n2
∆1 + n2
∑
2
Q2 = n1
dG2 (u) + 1 + dG2 (v) + 1
≥
uv∈E2
(9)
≥
n1 m2
2n1 m2
=
∆2 + 1 + ∆2 + 1
∆2 + 1
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
∑
Q3 =
uv∈E3 ,u∈V1 andv∈V2
≥
(10)
11
2
dG1 (u) + n2 + dG2 (v) + 1
2n1 n2
∆1 + ∆2 + n2 + 1
From equation (7),(8), (9)and (10) we get
H(G1 oG2 ) ≥
m1
m2 n1
2n1 n2
+
+
∆1 + n2 ∆2 + 1 ∆1 + ∆2 + n2 + 1
Similarly we deduce the lower bound
H(G1 oG2 ) ≤
m1
m2 n1
2n1 n2
+
+
.
δ1 + n2 δ2 + 1 δ1 + δ2 + n2 + 1
□
Theorem 2.4. Let G1 and G2 be two connected graphs with order n1 , n2 and size m1 , m2 respectively.
Then
H(G1 + G2 ) ≥
2
2
n1 n2
R(G1 ) +
R(G2 ) +
.
m1 + 2n2 + 1
m2 + 2n1 + 1
n1 + n2 − 1
Proof. Let V (G1 ) = {u1 , u2 , . . . , un1 } and V (G2 ) = {v1 , v2 , . . . , vn2 } be a set of vertex for G1 and G2
respectively. By the Definition of the join of two graph one can see that, if u is a vertex of G1 + G2 ,
then
{
dG1 +G2 (u) =
dG1 (u) + |V (G2 )| if u ∈ V (G1 )
dG2 (u) + |V (G1 )| if u ∈ V (G2 )
Therefore,
H(G1 + G2 ) =
∑
uv∈E(G1 +G2 )
=
∑
uv∈E(G1 )
+
2
dG1 +G2 (u) + dG1 +G2 (v)
2
+
dG1 (u) + n2 + dG1 (v) + n2
∑
u∈V (G1 ),v∈V (G2 )
(11)
∑
uv∈E(G2 )
2
dG2 (u) + n1 + dG2 (v) + n1
2
dG1 (u) + n2 + dG2 (v) + n1
= A1 + A2 + A3
where
A1 =
∑
uv∈E(G1 )
2
dG1 (u) + dG1 (v) + 2n2
Since for each edge uv of G, we have dG (u) + dG (v) ≤ |E(G)| + 1 with equality iff every other edge of
√
G is adjacent to the edge uv. And 1 ≤ du dv the equality hold iff du = dv = 1. Hence
∑
2
1
A1 ≥
×√
|E(G1 )| + 1 + 2n2
du dv
uv∈E(G1 )
(12)
=
2
R(G1 )
m1 + 1 + 2n2
12
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
Similarly
2
R(G2 )
m2 + 1 + 2n1
Since for any graph G with n vertices du ≤ n − 1, therefore
∑
2
A3 =
dG1 (u) + n2 + dG2 (v) + n1
A2 ≥
(13)
u∈V (G1 ),v∈V (G2 )
∑
≥
u∈V (G1 ),v∈V (G2 )
=
(14)
2
n1 − 1 + n2 + n2 − 1 + n1
n1 n2
n1 + n2 − 1
From equation (11),(12), (13)and (14) we get
H(G1 + G2 ) ≥
2
2
n1 n2
R(G1 ) +
R(G2 ) +
.
m1 + 2n2 + 1
m2 + 2n1 + 1
n1 + n2 − 1
□
Theorem 2.5. For i ∈ {1, 2}, let Gi be a graph of diameter d(Gi ), order ni and size mi . Then
H(G1 ⊕ G2 ) ≥
n22 m1 + n21 m2 − 4m1 m2
.
n2 (n1 − d(G1 )) + n1 (n2 − d(G2 )) − 2(n1 − d(G1 ))(n2 − d(G2 ))
Proof. Let V (G1 ) = {u1 , u2 , . . . , un1 } and V (G2 ) = {v1 , v2 , . . . , vn2 } be a set of vertex for G1 and G2
respectively.
∑
H(G1 ⊕ G2 ) =
(ui ,vj ),(uk ,vl )∈E(G1 ⊕G2 )
=
∑
∑
2
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )
∑
vj ∈V (G2 ) vl ∈V (G2 ) ui ,uk ∈E(G1 )
+
∑
∑
∑
ui ∈V (G1 ) uk ∈V (G1 ) vj ,vl ∈E(G2 )
(15)
−
∑
∑
ui ,uk ∈E(G1 ) vj ,vl ∈E(G1 )
2
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )
2
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )
2
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl )
As per the definition of the Symmetric difference of a graph
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) = n2 [dG1 (ui ) + dG1 (uk )] + n1 [dG2 (vj ) + dG2 (vl )]
− 2[dG1 (ui )dG2 (vj ) + dG1 (uk )dG2 (vl )]
Since for each vertex u of a graph G with n vertices, dG (u) ≤ n − eccG (u) and the diameter d(G) ≥
eccG (u).
Therefore,
dG1 ⊕G2 (ui , vj ) + dG1 ⊕G2 (uk , vl ) ≤ n2 [n1 − eccG1 (ui ) + n1 − eccG1 (uk )]
+ n1 [n2 − eccG2 (vj ) + n2 − eccG2 (vl )]
− 2[(n1 − eccG1 (ui ))(n2 − eccG2 (vj ))
Trans. Comb. 4 no. 4 (2015) 5-14
B. S. Shwetha, V. Lokesha and P. S. Ranjini
13
+ (n1 − eccG1 (uk ))(n2 − eccG2 (vl ))]
≤ n2 [2n1 − 2d(G1 )] + n1 [2n2 − 2d(G2 )]
− 2[(n1 − d(G1 ))(n2 − d(G2 )) + (n1 − d(G1 ))(n2 − d(G2 ))]
= 2{n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]}
(16)
From equation (15)and (16) we get
H(G1 ⊕ G2 ) ≥
n22 m1
n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]
n21 m2
n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]
4m1 m2
−
n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]
+
=
n22 m1 + n21 m2 − 4m1 m2
.
n2 [n1 − d(G1 )] + n1 [n2 − d(G2 )] − 2[(n1 − d(G1 ))(n2 − d(G2 ))]
□
Acknowledgments
The authors wish to record their sincere thanks to the anonymous referees for carefully reading the
manuscript and making suggestions that improve the content and presentation of the paper.
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B. Shwetha Shetty
Don Bosco Institute of Technology, Bangalore-78, India
Email:[email protected]
V. Lokesha
Dept. of Mathematics, V S K University, Bellary - 583105, India
Email:[email protected]
P. S. Ranjini
Department of Mathematics, Don Bosco Institute of Technology, Bangalore-78, India
Email:ranjini p [email protected]