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Fig. 7.11
The spiro-chains of
C
3
,
C
4
,
C
6
Fig. 7.12
The rooted product
of
H
by
G
,
H
(
G
)
=
H
(
G
1
,
G
2
,
...
,
G
n
)
7.5
Zagreb Indices of Rooted Product of Graphs
In this section, we compute the first and second Zagreb indices of rooted product of
graphs. All of the results of this section have been reported in (Azari and Iranmanesh
2013a
). We start this section by definition of the graph of rooted product.
Let
H
be a labeled graph on
n
vertices with the vertex set
V
(
H
)
}
and let G be a sequence of
n
rooted graphs
G
1
,
G
2
,
...
,
G
n
. According to (God-
sil and McKay
1978
),
={
1, 2,
...
,
n
=
H
(
G
1
,
G
2
,
...
,
G
n
) is the graph obtained by identifying the root vertex of
G
i
with
the
i
-th vertex of
H
for all
i
the rooted product of
H
by G, denoted by
H
(
G
)
, see Fig.
7.12
.
In the special case when the components
G
i
,
i
∈{
∈{
1, 2,
...
,
n
}
, are mutually
isomorphic to a graph K, the rooted product of
H
by G is denoted by
H
{
K
}
1, 2,
...
,
n
}
and is
called the cluster of
H
and K, see Fig.
7.13
. For more information on computing
