Suppose
G
is a simple graph. The irregularity of
G, irr
(
G
), is the summation of
imb
(
e
) over all edges
uv
=
e
∈
G
, where
imb
(
e
) = 
deg
(
u
) ∈
deg
(
v
). In this paper, we investigate the behavior of this graph parameter under some old and new graph operations.
AMS Mathematics Subject Classification : 05C12.
1. Introduction
The degree−based graph invariants are parameters defined by degrees of vertices. The first of such graph parameters was introduced by Gutman and Trinajstić
[10]
. Suppose
G
is a graph
e
=
uv
∈
E
(
G
).
The first Zagreb index
of
G
is defined as
M
_{1}
(
G
) = Σ
_{v∈V (G)}
deg
(
v
)
^{2}
. There are a lot of works on Zagreb group invariants and interested readers can be consulted
[3
,
8
,
11
,
16
,
17]
for more information on this topic. The
imbalance
of
e
is defined as
imb
(
e
) = 
deg
(
u
)−
deg
(
v
). The summation of imbalances over all edges of
G
is denoted by
irr
(
G
). Albertson
[2]
, named this parameter
“irregularity”
of the graph
G
. After this paper, there was a lot of research considering the irregularity index, see
[12
,
14
,
15]
for details. It is easy to see that
M
_{1}
(
G
) = Σ
_{e=uv∈E (G)}
[
deg
(
u
) +
deg
(
v
)]. FathTabar
[9]
, unaware from the seminal paper of Albertson and because of similarity between
M
_{1}
and
irr
used the term
“third Zagreb index”
for “irregularity”.
Albertson
[2]
computed the maximum irregularity of various classes of graphs. As a consequence, he proved that the irregularity of an arbitrary graph with
n
vertices is less than
and this bound is tight. Some of the present authors
[20]
, characterized the graphs with minimum and maximum values of irregularity. Luo and Zhou
[18]
determined the maximum irregularity of trees and unicyclic graphs with a given number of vertices and matching number. They also characterized the extremal graphs with the mentioned property. Zhou and Luo
[22]
, established an upper bound for
irr
(
G
) in terms of
n,m
, and
r
, where
n
is the order of
G
,
m
≥ 1 is its size, and
G
is assumed to have no complete subgraph of order
r
+ 1 where 2 ≤
r
≤
n
− 1. They also provided new upper bounds for the irregularity of trees and unicyclic graphs. These are both functions of the number of pendant vertices of the graph under consideration. For each of these three inequalities, the authors supplied a characterization of all graphs which attain the bound. Henning and Rautenbach
[15]
obtained the structure of bipartite graphs having maximum possible irregularity with given cardinalities of its bipartition and given number of edges. They derived a result for bipartite graphs with given cardinalities of its bipartition and presented an upper bound on the irregularity of these graphs. In particular, they shown that if
G
is a bipartite graph of order
n
with a bipartition of equal cardinalities, then
while if
G
is a bipartite graph with partite sets of cardinalities
n
_{1}
and
n
_{2}
, where
n
_{1}
≥ 2
_{n2}
, then
irr
(
G
) ≤
irr
(
K
_{n1}
,
_{n2}
).
Abdo and Dimitrov
[1]
introduced the concept of
total irregularity
of a graph
G
and obtain some exact formula for computing total irregularity of some old graph operations. The aim of this paper is to compute formulas for the regularity of graphs under some old and new graph invariants.
Throughout this paper the
path, complete
and
star
graphs of order
n
are denoted by by
P_{n}
,
K_{n}
and
S_{n}
, respectively. The degree of a vertex
v
is denoted by
deg_{G}
(
v
). We denote by Δ(
G
) the maximum degree of vertices of
G
.
Lemma 1.1.
Let G be a connected graph on n, n
> 2
vertices. If G has exactly k pendant vertices then irr
(
G
) ≽
k, with equality if and only if
Proof
. Suppose
u
is a pendant vertex of
G
and
v
∈
V
(
G
) is a vertex adjacent to
u
. Since 
V
(
G
) > 2,
deg
(
v
) ≽ 2 and so 
deg
(
u
)−
deg
(
v
) ≽ 1. But,
G
has exactly
k
pendant vertices, and so
irr
(
G
) ≽
k
. Notice that
irr
(
G
) =
k
if and only if
deg_{G}
(
u
) = 1 or 2, for each vertex
u
∈
V
(
G
). This implies that
Corollary 1.2.
Let T be a tree with
Δ(
T
) > 1.
Then irr
(
T
) ≽ Δ(
T
).
Proof
. By assumption,
T
has at least Δ(
T
) pendant vertices and so, by Lemma 1.1,
irr
(
T
) ≽ Δ(
T
).
Lemma 1.3.
Let T be a tree with n
> 2
vertices. Then
and the lower bound is attained if and only if
The upper bound is attained if and only if
Proof
. Since
T
is a tree with
n
> 2 vertices, it has at least two pendant vertices and so, by lemma 1.1, 2 ≼
irr
(
T
). On the other hand, it is not difficult to check that for each
uv
∈
E
(
T
), 
deg
(
u
) −
deg
(
v
) ≼
n
− 2 and
E
(
T
) =
n
− 1. So,
irr
(
T
) ≼ (
n
− 1)(
n
− 2). Notice that
S_{n}
is the unique graph with
n
− 2 as difference of degrees between adjacent vertices. But by Lemma 1.1,
P_{n}
is the unique tree that its third Zagreb index is equal to 2, as desired.
2. Main results
The
join
G
=
G
_{1}
+
G
_{2}
,
Figure 1
, of graphs
G
_{1}
and
G
_{2}
with disjoint vertex sets
V
_{1}
and
V
_{2}
and edge sets
E
_{1}
and
E
_{2}
is the graph union
G
_{1}
∪
G
_{2}
together with all the edges joining
V
_{1}
and
V
_{2}
. For example,
The
composition
G
=
G
_{1}
[
G
_{2}
] of graphs
G
_{1}
and
G
_{2}
with disjoint vertex sets
V
_{1}
and
V
_{2}
and edge sets
E
_{1}
and
E
_{2}
is the graph with vertex set
V
_{1}
×
V
_{2}
and
u
= (
u
_{1}
,
v
_{1}
) is adjacent with
v
= (
u
_{2}
,
v
_{2}
) whenever (
u
_{1}
is adjacent to
u
_{2}
) or (
u
_{1}
=
u
_{2}
and
v
_{1}
is adjacent to
v
_{2}
)
[13]
. For instance,
The Join of
The
Strong product
G
☒
H
,
Figure 2
, of graphs
G
and
H
has the vertex set
V
(
G
☒
H
) =
V
(
G
) ×
V
(
H
) and (
a, x
)(
b, y
) is an edge of
G
☒
H
if
a
=
b
and
xy
∈
E
(
H
), or
ab
∈
E
(
G
) and
x
=
y
, or
ab
∈
E
(
G
) and
xy
∈
E
(
H
). As an example,
C_{n}
☒
K
_{2}
is the closed fence. The
tensor product
G
⊗
H
,
Figure 3
, is defined as the graph with vertex set
V
(
G
) ×
V
(
H
) and
E
(
G
⊗
H
) = {(
u
_{1}
,
u
_{2}
)(
v
_{1}
,
v
_{2}
) 
u
_{1}
v
_{1}
∈
E
(
G
)
and
u
_{2}
v
_{2}
∈
E
(
H
)}. For example,
C_{n}
⊗
P
_{2}
=
C
_{2n}
. The
corona product
GoH
,
Figure 4
, is obtained by taking one copy of
G
and 
V
(
G
) copies of
H
; and by joining each vertex of the
i
th copy of
H
to the
i
th vertex of
G
, 1 ≤
i
≤ 
V
(
G
)
[19]
. For example,
Finally, for a connected graph
G, R
(
G
) is a graph obtained from
G
by adding a new vertex corresponding to each edge of
G
, then joining each new vertex to the end vertices of the corresponding edge
[21]
. For example, we can see
R
(
P
_{2}
) =
K
_{3}
.
The Strong Product of C_{5} and K_{2}.
The Tensor Product of Two Graphs.
The Corona Product of Two Graphs.
It is wellknown that the Cartesian product of graphs can be recognized efficiently, in time
O
(
mlogn
) for a graph with
n
vertices and
m
edges
[22]
. This operation is commutative and associative as an operation on isomorphism classes of graphs, but it is not commutative. The Cartesian product is not a product in the category of graphs, but the tensor product is the categorical product.
Suppose
G
and
H
are graphs with disjoint vertex sets. Following Došslić
[8]
, for given vertices
y
∈
V
(
G
) and
z
∈
V
(
H
) a splice of
G
and
H
by vertices
y
and
z
, (
G
·
H
)(
y; z
), is defined by identifying the vertices
y
and
z
in the union of
G
and
H
. Similarly, a link of
G
and
H
by vertices
y
and
z
is defined as the graph (
G
∼
H
)(
y; z
) obtained by joining
y
and
z
by an edge in the union of these graphs. Let
H
is a tree of progressive degree
p
and generation
r
that whose root vertex is
r
_{1}
. Also, let
DD_{p,r}
be the graph of the regular dicentric dendrimer,
Figure 6
. So, it is clear that
DD_{p,r}
= (
H
∼
H
)(
r
_{1}
;
r
_{1}
).
The Molecular Graph of Octanitrocubane.
Regular Dicentric (DD_{2.4}) Dendrimer.
Lemma 2.1.
Suppose G and H are rooted graphs with respect to the rooted vertices of a and b, respectively. Then
and the upper bound is attained if and only if for every vertex u
∈
V
(
G
)
that ua
∈
E
(
G
),
deg_{G}
(
a
) ≽
deg_{G}
(
u
)
and for every vertex v
∈
V
(
H
)
that vb
∈
E
(
H
),
deg_{H}
(
b
) ≽
deg_{H}
(
v
).
Moreover, the lower bound is attained if and only if for each edge au
∈
E
(
G
),
deg_{G}
(
u
) −
deg_{G}
(
a
) ≽
deg_{H}
(
b
)
and for each edge bv
∈
E
(
H
),
deg_{H}
(
v
) −
deg_{H}
(
b
) ≽
deg_{G}
(
a
).
Proof
. This follows immediately from the definition of splice of two graphs.
In a similar way, by definition of link of two graphs, we have:
Lemma 2.2.
Suppose G and H are rooted graphs with respect to the rooted vertices of a and b, respectively. Then
and the upper bound is attained if and only if for every vertex u
∈
V
(
G
)
that ua
∈
E
(
G
),
deg_{G}
(
a
) ≽
deg_{G}
(
u
)
and for every vertex v
∈
V
(
H
)
that vb
∈
E
(
H
),
deg_{H}
(
b
) ≽
deg_{H}
(
v
).
Moreover, the lower bound is attained if and only if for every vertex u
∈
V
(
G
)
that ua
∈
E
(
G
),
deg_{G}
(
a
) <
deg_{G}
(
u
)
and for every vertex v
∈
V
(
H
)
that vb
∈
E
(
H
),
deg_{H}
(
b
) <
deg_{H}
(
v
).
Lemma 2.3.
Let G be a connected graph. Then irr
(
G
) = 0
if and only if G is regular.
The line graph
L
(
G
) of a graph
G
is defined as follows: each vertex of
L
(
G
) represents an edge of
G
, and any two vertices of
L
(
G
) are adjacent if and only if their corresponding edges share a common endpoint in
G
[21]
.
Lemma 2.4.
Let G be a connected graph. Then irr
(
L
(
S
(
G
))) =
irr
(
G
).
Proof
. We assume that
uv
is an edge of
L
(
G
) that
u
and
v
are vertices corresponding to edges
wx
and
wy
of
G
, respectively. Then 
deg
_{L(G)}
(
u
)−
deg
_{L(G)}
(
v
) = 
deg_{G}
(
x
) −
deg_{G}
(
y
). To compute the third Zagreb index of
L
(
G
), it is enough to calculate the summation of all degree differences of vertices of distance 2 in
L
(
G
). Therefore,
irr
(
L
(
S
(
G
))) =
irr
(
G
).
In the next result, the relationship between strong and tensor products of two graphs under the third Zagreb index is investigated.
Lemma 2.5.
Let G and H be graphs. Then
Proof
. The summation of 
deg
_{G☒H}
(
u
) −
deg
_{G☒H}
(
v
) over all edges (
a, x
)(
b, y
) such that (
a
=
b
and
xy
∈
E
(
H
)) or (
x
=
y
and
ab
∈
E
(
G
)), is equal to:
On the other hand, for each edge (
a, x
)(
b, y
) of
G
☒
H
that
ab
∈
E
(
G
) and
xy
∈
E
(
H
), we have:
and hence
which completes the proof.
Lemma 2.6.
Let G and H be two connected graphs. Then
Proof
. Let
H_{i}
be the
i
th copy of
H
, 1 ≤
i
≤ 
V
(
G
), and let
G′
be the copy of
G
in
GoH
. We partition edges of
GoH
into the following three subsets:

A= {uv∈E(GoH) u, v∈V(Hi),i= 1, 2, ..., V(G)},

B= {uv∈E(GoH) u, v∈V(G′)},

C= {uv∈E(GoH) u∈V(G′),v∈V(Hi), i = 1, 2, ..., V(G)}.
If
u′
v′
∈
A
, then
deg_{GoH}
(
u′
) =
deg_{H}
(
u
) + 1 and
deg_{GoH}
(
v′
) =
deg_{H}
(
v
) + 1 that
u′
,
v′
∈
V
(
H_{i}
) are corresponding to
u, v
∈
V
(
H
), respectively. Thus 
deg_{GoH}
(
u′
) −
deg_{GoH}
(
v′
) = 
deg_{H}
(
u
) −
deg_{H}
(
v
). Since 
V
(
G
) edges
u′
v′
in
E
(
GoH
) are corresponding to each edge
uv
∈
E
(
H
),
irr
_{1}
= Σ
_{uv∈A}

deg_{GoH}
(
u
)−
deg_{GoH}
(
v
) = 
V
(
G
)
irr
(
H
).
It is clear that for each
u′
v′
∈
B, deg_{GoH}
(
u′
) =
deg_{G}
(
u
) + 
V
(
H
) and
deg_{GoH}
(
v′
) =
deg_{G}
(
v
) + 
V
(
H
), where
u′
,
v′
∈
V
(
G′
) are corresponding to
u, v
∈
V
(
G
), respectively. Hence
irr
_{2}
= Σ
_{uv∈B}

deg_{GoH}
(
u
) −
deg_{GoH}
(
v
) =
irr
(
G
).
Finally, if
u′
v′
∈
C
, then
deg_{GoH}
(
u′
) =
deg_{G}
(
u
) + 
V
(
H
) and
deg_{GoH}
(
v′
) =
deg_{H}
(
v
) + 1, where
u′
∈
V
(
G′
);
v′
∈
V
(
H_{i}
) are corresponding to
u
∈
V
(
G
),
v
∈
V
(
H
), respectively. Hence 
deg_{GoH}
(
u′
) −
deg_{GoH}
(
v′
) =
deg_{G}
(
u
) + 
V
(
H
) −
deg_{H}
(
v
) − 1. Consequently,
By summation of
irr
_{1}
,
irr
_{2}
and
irr
_{3}
, the result can be proved.
As in the proof of Lemma 2.6, 
deg_{GoH}
(
u′
)−
deg_{GoH}
(
v′
) =
deg_{G}
(
u
)+
V
(
H
)−
deg_{H}
(
v
) − 1, where for each edge
u′
v′
∈
E
(
GoH
), vertices
u′
∈
V
(
G′
) and
v′
∈
V
(
H_{i}
) are corresponding to
u
∈
V
(
G
) and
v
∈
V
(
H
), respectively. Thus,
Corollary 2.7.
Let G and H be two connected graphs. Then
and the upper bound is attained if and only if H is a tree and
Moreover, the lower bound is attained if and only if G is a tree and
Let
G_{i}
= (
V_{i}
,
E_{i}
) be
N
graphs with each vertex set
V_{i}
, 1 ≤
i
≤
N
, having a distinguished or root vertex, labeled 0. The hierarchical product
H
=
G_{N}
⊓ ... ⊓
G
_{2}
⊓
G
_{1}
is the graph with vertices the
N
−tuples
x_{N}
...
x
_{3}
x
_{2}
x
_{1}
,
x_{i}
∈
V_{i}
, and edges defined by the adjacencies:
We encourage the reader to consult
[5
,
6]
for the mathematical properties of this new graph operation.
Lemma 2.8.
Let G and H be connected rooted graphs and r is the root vertex of H. Then
The upper bound is attained if and only if for each ur
∈
E
(
H
),
deg_{H}
(
r
) ≽
deg_{H}
(
u
).
Moreover, the lower bound is attained if and only if for each ur
∈
E
(
H
)
and v
∈
V
(
G
),
deg_{H}
(
r
) +
deg_{G}
(
v
) ≼
deg_{H}
(
u
).
Proof
. Let
ur
∈
E
(
H
) and
v
∈
V
(
G
), then (
v, r
)(
v, u
) ∈
E
(
G
⊓
H
) and so
Consequently, if
deg_{H}
(
r
) ≽
deg_{H}
(
u
) then 
deg
_{G}
_{⊓}
_{H}
((
v, r
)) −
deg
_{G}
_{⊓}
_{H}
((
v, u
)) = (
deg_{H}
(
r
) −
deg_{H}
(
u
)) +
deg_{G}
(
v
) and if
deg_{H}
(
r
) +
deg_{G}
(
v
) ≼
deg_{H}
(
u
) then 
deg
_{G}
_{⊓}
_{H}
((
v, r
))−
deg
_{G}
_{⊓}
_{H}
((
v, u
)) = (
deg_{H}
(
u
) −
deg_{H}
(
r
))−
deg_{G}
(
v
). On the other hand for each edge (
u, r
)(
v, r
) of
G
⊓
H
that
uv
∈
E
(
G
), we have 
deg
_{G}
_{⊓}
_{H}
((
u, r
))−
deg
_{G}
_{⊓}
_{H}
((
v, r
)) = 
deg_{G}
(
u
) −
deg_{G}
(
v
) and for each edge (
w, u
)(
w, v
) of
G
⊓
H
that
u
≠
v
≠
r
, we have 
deg
_{G}
_{⊓}
_{H}
((
w, u
)) −
deg
_{G}
_{⊓}
_{H}
((
w, v
)) = 
deg_{H}
(
u
) −
deg_{H}
(
v
), which proves the result.
In what follows, let
for each
i, j
∈ {0, 1, 2, ...}, that
i
−
j
= 1. Let
G
_{1}
,
G
_{2}
,...,
G_{n}
be connected rooted graphs with root vertices
r
_{1}
,
r
_{2}
,...,
r_{n}
, respectively. We set
Also, if
G
=
G_{n}
⊓ ... ⊓
G
_{2}
⊓
G
_{1}
then we will use
deg_{G}
(
r
) to denote
deg
_{G1}
(
r
_{1}
) +
deg
_{G}
_{1}
(
r
_{2}
) + ... +
deg_{Gn}
(
r_{n}
).
Corollary 2.9.
Let
G
_{1}
,
G
_{2}
, ...,
G_{n} be connected rooted graphs with root vertices
r
_{1}
,
r
_{1}
, ...,
r_{n}, respectively. Then
The upper bound is attained if and only if for each ur_{i}
∈
E
(
G_{i}
),
deg_{Gi}
(
r_{i}
) ≽
deg_{Gi}
(
u
),
i
= 1, 2, ...,
n
− 1.
Moreover, the lower bound is attained if and only if for each ur_{i}
∈
E
(
G_{i}
)
and v
∈
V
(
G_{i}
),
i
= 1, 2, ...,
n
− 1,
j
=
i
+ 1,
i
+ 2, ...,
n
.
Proof
. Use induction on
n
. By Lemma 2.8, the result is valid for
n
= 2. Let
n
≽ 3 and assume the result holds for
n
. Set
G
=
G_{n}
⊓ ... ⊓
G
_{2}
⊓
G
_{1}
. Thus
G
_{n}
_{+1}
⊓ ... ⊓
G
_{2}
⊓
G
_{1}
=
G
_{n}
_{+1}
⊓
G
. Then by our assumption,
On the other hand, again by our assumption,
which completes our argument.
Example 2.10.
Octanitrocubane is the most powerful chemical explosive with formula
C
_{8}
(
NO
_{2}
)
_{8}
),
Figure 5
. Let Γ be the graph of this molecule. Then obviously Γ =
Q
_{3}
⊓
P
_{2}
. If
r
is the root vertex of
P
_{2}
, one can easily see that
and for each
and so, by Lemma 2.8, we have
Example 2.11.
Dendrimers are branched molecules have a high degree of molecular uniformity. The molecular graph of this molecules is constructed from a core and some branches connecting to the core. Let
DD_{p,r}
be the graph of the regular dicentric dendrimer, see
[7]
for more information. Then
DD_{p,r}
=
P
_{2}
⊓
H
, where
H
is a tree of progressive degree
p
and generation
r
,
Figure 6
. One can see that
irr
(
P
_{2}
) = 0,
irr
(
H
) =
p
^{r+1}
+
p
, and for each
ur
∈
E
(
H
) and
Therefore, by Lemma 2.8, we have:
Example 2.12.
Consider the graph
S
_{4}
whose root vertex is 0. If
then by Lemma 2.8 we have:
and if
then again by Lemma 2.8 we have:
Lemma 2.13.
Let G and H be connected graphs. Then

(1)If G is regular, then we have irr(G[H]) = E(G)irr(2H) + (V(G) − 2E(G))irr(H).

(2)If H is regular, then irr(G[H]) = V(H)3irr(G).
Proof
. 1). Let
G
be regular. Clearly, for each vertex (
u, v
) ∈
V
(
G
[
H
]),
deg_{G}
_{[H]}
((
u, v
)) = 
V
(
H
)
deg_{G}
(
u
) +
deg_{H}
(
v
). Since
G
is regular, for each edge (
u, x
)(
v, y
) ∈
G
[
H
] that
uv
∈
E
(
G
), we have 
deg
_{G[H]}
((
u, x
))−
deg
_{G[H]}
((
v, y
)) = 
deg_{H}
(
x
) −
deg_{H}
(
y
) and for every edge (
u, x
)(
u, y
) that
xy
∈
E
(
H
), we have 
deg
_{G[H]}
((
u, x
))−
deg
_{G[H]}
((
u, y
)) = 
deg_{H}
(
x
)−
deg_{H}
(
y
), which proves the result.
2). Let
H
be regular. Then for every edge (
u, x
)(
u, y
) that
xy
∈
E
(
H
), we have 
deg
_{G[H]}
((
u, x
))−
deg
_{G[H]}
((
u, y
)) = 0 and for every edge (
u, x
)(
v, y
) of
G
[
H
] that
uv
∈
E
(
G
), we have 
deg
_{G[H]}
((
u, x
))−
deg
_{G[H]}
((
v, y
)) = 
V
(
H
)
deg_{G}
(
u
)−
deg_{G}
(
v
). On the other hand, to each edge
uv
∈
E
(
G
), there correspond 
V
(
H
)
^{2}
edges (
u, x
)(
v, y
) in
E
(
GoH
), so
irr
(
G
[
H
]) = 
V
(
H
)
^{3}
irr
(
G
).
Acknowledgements
The authors are indebted to anonymous referees for their suggestions and helpful remarks.
BIO
Mostafa Tavakoli was born in Isfahan, Iran in 1986. He received his M.Sc. from University of Tehran under the direction of Hassan YousefiAzari. He is currently a Ph.D candidate at Ferdowsi University of Mashhad. His research interests focus on metric graph theory and mathematical chemistry.
Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
email:M.Tavakoly@ut.ac.ir,Mostafa Tavakoli@ymail.com
Fereydon Rahbarnia received M.Sc. from Boston College, and Ph.D. from Ferdowsi University of Mashhad. He is currently an assistant professor at Ferdowsi University of Mashhad.
Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
email:Rahbarnia@ferdowsi.um.ac.ir
Ali Reza Ashrafi was born in Tehran, Iran in 1964. He received his BSc from Teacher Training University of Tehran, MSc from Shahid Beheshti University and Ph.D at the University of Tehran under the direction of Mohammad Reza Darafsheh. Since 1996 he has been at the University of Kashan. He is now a professor of mathematics and supervised more than fifty MSc students in the field of computational group theory, graph theory and mathematical chemistry. In 2010, he elected as one of the best Iranian scientists in basic sciences. His research interests focus on computational group theory, metric graph theory and mathematical chemistry.
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731751167, I. R. Iran.
email:ashrafi@kashanu.ac.ir
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