Chemistry Reference
In-Depth Information
Fig. 7.13
The cluster of H
and K,
H
{
K
}
topological indices of clusters see (Yeh and Gutman
1994
; Došlic
2008
; Azari and
Iranmanesh
2013b
).
In what follows, we denote the root vertex of
G
i
by
w
i
and the degree of
w
i
in
G
i
by
ω
i
,
i
∈{
1,2,
...
,
n
}
.
Theorem 7.5.1
The first and second Zagreb indices of the rooted product
H
(
G
) are
given by:
+
i
=
1
M
1
(
G
i
)
2
i
=
1
ω
i
deg
H
(
i
),
1.
M
1
(
H
(
G
))
=
M
1
(
H
)
+
+
i
=
1
M
2
(
G
i
)
+
i
=
1
deg
H
(
i
)
α
G
i
(
w
i
)
2.
M
2
(
H
(
G
))
=
M
2
(
H
)
+
ij
∈
E
(
H
)
[
ω
j
deg
H
(
i
)
+
ω
i
deg
H
(
j
)
+
ω
i
ω
j
].
Proof
1. Using definition of the first Zagreb index, we have:
[deg
H
(
i
)
deg
G
i
(
u
)
2
n
ω
i
]
2
=
+
+
M
1
(
H
(
G
))
i
=
1
u
∈
V
(
G
i
)
−{
w
i
}
ω
i
2
deg
G
i
(
u
)
2
n
n
n
deg
H
(
i
)
2
=
+
+
+
2
ω
i
deg
H
(
i
)
i
=
1
i
=
1
u
∈
V
(
G
i
)
−{
w
i
}
i
=
1
n
n
=
M
1
(
H
)
+
M
1
(
G
i
)
+
2
ω
i
deg
H
(
i
)
.
i
=
1
i
=
1
2. Using definition of the second Zagreb index, we have:
n
=
+
+
+
M
2
(
H
(
G
))
[deg
H
(
i
)
ω
i
][deg
H
(
j
)
ω
j
]
ij
∈
E
(
H
)
i
=
1
uv
∈
E
(
G
i
);
u
,
v
=
w
i
deg
G
i
(
u
)deg
G
i
(
v
)
+
deg
G
i
(
u
)[deg
H
(
i
)
+
ω
i
]
}
uv
∈
E
(
G
i
);
u
∈
V
(
G
i
),
v
=
w
i
=
deg
H
(
i
)deg
H
(
j
)
+
[
ω
j
deg
H
(
i
)
+
ω
i
deg
H
(
j
)
+
ω
i
ω
j
]
ij
∈
E
(
H
)
ij
∈
E
(
H
)
