Chemistry Reference
In-Depth Information
Further,
S
2
=
M
2
(
G
d
)
+
ω
d
−
1
α
G
d
(
v
d
),
where
S
2
is the sum over all edges
ab
E
(
G
d
).
The third sum
S
3
is taken over all edges
ab
∈
∈
E
(
G
i
) for all
i
∈
I
. Using Lemma
7.4.1, we get:
S
3
=
[
M
2
(
G
i
)
+
ω
i
−
1
α
G
i
(
v
i
)
+
υ
i
+
1
α
G
i
(
w
i
)
+
ω
i
−
1
υ
i
+
1
]
.
i
∈
I
Similarly,
S
4
=
[
M
2
(
G
i
)
+
ω
i
−
1
α
G
i
(
v
i
)
+
υ
i
+
1
α
G
i
(
w
i
)],
i
∈
I
where
S
4
is the sum over all edges
ab
∈
E
(
G
i
) for all
i
∈
I
.
Adding the quantities
S
1
,
S
2
,
S
3
,
S
4
, we arrive at the expression for
M
2
(
C
)given
in Theorem 7.4.4.
Suppose that
v
and
w
are two vertices of a graph
G
, and let
G
i
=
G
,
v
i
=
v
, and
w
i
=
w
for all
i
∈
1, 2,
...
,
d
. Then from Theorem 7.4.4 follows:
Corollary 7.4.5
Let
υ
=
deg
G
(
v
) and
ω
=
deg
G
(
w
). If
v
and
w
are adjacent in
G
,
then
M
2
(
C
)
=
dM
2
(
G
)
+
(
d
−
1)[
ωα
G
(
v
)
+
υα
G
(
w
)]
+
(
d
−
2)
υω
,
whereas otherwise;
M
2
(
C
)
=
dM
2
(
G
)
+
(
d
−
1)[
ωα
G
(
v
)
+
υα
G
(
w
)]
.
Now as an application of our results, we compute the first and second Zagreb indices
of the spiro-chain of the cycle
C
n
for arbitrary
n
3. Choose a numbering for
vertices of
C
n
such that the vertex
v
has number 1, the number
i
of the vertex
w
has
to be in {2, 3,
...
,
n
}. However, due to the symmetry
k
≥
↔
n
−
k
+
2, one can restrict
i
to
2, 3, 4
...
,
2
+
1
. Denote the graph
C
n
by
C
n
(
k
,
l
), where
k
and
l
are the
numbers of the vertices
v
and
w
, respectively. The spiro-chain of the graph
C
n
(
k
,
l
)
can be considered as the chain graph
C
(
G
,
G
,
...
,
G
;
v
,
w
,
v
,
w
,
...
,
v
,
w
), where
G
C
n
(
k
,
l
). The spiro-chains of
C
3
,
C
4
,
C
6
are shown in Fig.
7.11
. We denote the
spiro-chain containing
d
times the component
C
n
(
k
,
l
)
,
by
S
d
(
C
n
(
k
,
l
)).
Since all vertices of
C
n
(
k
,
l
) are of degree two, so
M
1
(
C
n
(
k
,
l
))
=
=
M
2
(
C
n
(
k
,
l
))
=
4
n
,
υ
=
ω
=
2, and
α
C
n
(
k
,
l
)
(
v
)
=
α
C
n
(
k
,
l
)
(
w
)
=
4. So using Corollaries 7.4.3 and
7.4.5, we easily arrive at:
Corollary 7.4.6
The first and second Zagreb indices of
S
d
(
C
n
(
k
,
l
)) are given by:
1.
M
1
(
S
d
(
C
n
(
k
,
l
)))
=
4
nd
+
8
d
−
8,
⎧
⎨
4
nd
+
20
d
−
24
if l
∈{
2,
n
}
2.
M
2
(
S
d
(
C
n
(1,
l
)))
=
.
⎩
4
nd
+
16
d
−
16
if l
∈{
3,
...
,
n
−
1
}
