Chemistry Reference
In-Depth Information
Suppose that
v
is a vertex of a graph
G
, and let
G
i
=
G
and
v
i
=
v
for all
i
∈
{
1,2,
...
,
d
}
.
Using Theorem 7.2.2, we easily arrive at:
Corollary 7.2.3
The first Zagreb index of the bridge graph
B
1
,(
d
≥
2 times), is
given by:
M
1
(
B
1
)
=
dM
1
(
G
)
+
4
υ
(
d
−
1)
+
4
d
−
6,
where
υ
=
deg
G
(
v
)
.
Theorem 7.2.4
If d
=
2, the second Zagreb index of the bridge graph
B
1
,isgiven
by:
M
2
(
B
1
)
=
M
2
(
G
1
)
+
M
2
(
G
2
)
+
α
G
1
(
v
1
)
+
α
G
2
(
v
2
)
+
(
υ
1
+
1)(
υ
2
+
1),
and for
d
≥
3,
d
d
−
1
d
−
1
=
+
+
+
+
M
2
(
B
1
)
M
2
(
G
i
)
α
G
1
(
v
1
)
α
G
d
(
v
d
)
2
α
G
i
(
v
i
)
υ
i
υ
i
+
1
i
=
1
i
=
2
i
=
1
d
−
1
+
2(
υ
1
+
υ
d
)
−
(
υ
2
+
υ
d
−
1
)
+
4
υ
i
+
4(
d
−
2),
i
=
2
where
υ
i
=
deg
G
i
(
v
i
), for 1
≤
i
≤
d
.
Proof
For the case d
=
2, see the proof of Lemma 2.4 in (Ashrafi et al.
2011
).
Now let
d
≥
3.
By definition of the second Zagreb index,
M
2
(
B
1
)is
equal
to
the
sum
of
deg
B
1
(
a
)deg
B
1
(
b
),
where
summation
is
taken
over
all
edges
ab
∈
E
(
B
1
).
From the definition of the bridge graph
B
1
,
E
(
B
1
)
=
E
(
G
1
)
E
(
G
2
)
...
E
(
G
d
)
{
v
i
v
i
+
1
|
1
≤
i
≤
d
−
1
}
. In order to compute
M
2
(
B
1
), we partition our sum into the four sums as follows:
The first sum
S
1
is taken over all edges
ab
∈
E
(
G
1
). Using Lemma 7.2.1,
S
1
=
deg
B
1
(
a
)deg
B
1
(
b
)
ab
∈
E
(
G
1
)
=
deg
G
1
(
a
)deg
G
1
(
b
)
+
deg
G
1
(
a
)[deg
G
1
(
v
1
)
+
1]
ab
∈
E
(
G
1
);
a
,
b
=
v
1
ab
∈
E
(
G
1
);
a
∈
V
(
G
1
),
b
=
v
1
α
G
1
(
v
1
)
.
The second sum
S
2
is taken over all edges
ab
=
M
2
(
G
1
)
+
∈
E
(
G
d
). Using Lemma 7.2.1, we
obtain:
S
2
=
deg
B
1
(
a
)deg
B
1
(
b
)
ab
∈
E
(
G
d
)
=
deg
G
d
(
a
)deg
G
d
(
b
)
+
deg
G
d
(
a
)[deg
G
d
(
v
d
)
+
1]
ab
∈
E
(
G
d
);
a
,
b
=
v
d
ab
∈
E
(
G
d
);
a
∈
V
(
G
d
),
b
=
v
d
=
M
2
(
G
d
)
+
α
G
d
(
v
d
)
.
