Chemistry Reference
In-Depth Information
Fig. 7.10
The chain graph
C
=
C
(
G
1
,
G
2
,
...
,
G
d
;
v
1
,
w
1
,
v
2
,
w
2
,
...
,
v
d
,
w
d
)
1.
M
1
(
D
n
)
=
106
n
−
10,
2.
M
2
(
D
n
)
=
128
n
−
17
.
7.4
Zagreb Indices of Chain Graphs
In this section, we compute the first and second Zagreb indices for chain graphs. All
of the results of this section have been reported in (Azari et al.
2013
). We start this
section by definition of this class of composite graphs.
Let
d
i
=
{
G
i
}
1
be a set of finite pairwise disjoint graphs with distinct vertices
v
i
,
w
i
∈
V
(
G
i
). The chain graph
C
=
C
(
G
1
,
G
2
,
...
,
G
d
;
v
1
,
w
1
,
v
2
,
w
2
,
...
,
v
d
,
w
d
)
d
i
=
d
i
=
of
1
is the graph obtained from the graphs
G
1
,
G
2
,
...
,
G
d
by identifying the vertices
w
i
and
v
i
+
1
for all
i
{
G
i
}
1
with respect to the vertices
{
v
i
,
w
i
}
∈{
1,2,
...
,
d
−
1
}
,
as shown in Fig.
7.10
.
We first state a simple lemma which immediately follows from the definition of
C
.
Lemma 7.4.1
The degree of an arbitrary vertex u of the chain graph C,
d
≥
2, is
given by:
⎧
⎨
deg
G
1
(
u
)
if u
∈
V
(
G
1
)
−{
w
1
}
deg
G
d
(
u
)
if u
∈
V
(
G
d
)
−{
v
d
}
deg
C
(
u
)
=
⎩
deg
G
i
(
u
)
if u
∈
V
(
G
i
)
−{
v
i
,
w
i
}
,2
≤
i
≤
d
−
1,
ω
i
+
=
w
i
=
≤
≤
−
υ
i
+
1
if u
v
i
+
1
,1
i
d
1
where
υ
i
=
deg
G
i
(
v
i
),
ω
i
=
deg
G
i
(
w
i
), for 1
≤
i
≤
d
.
Theorem 7.4.2
The first Zagreb index of the chain graph C,
d
≥
2, is given by:
d
d
−
1
M
1
(
C
)
=
M
1
(
G
i
)
+
2
ω
i
υ
i
+
1
,
i
=
1
i
=
1
where
υ
i
=
deg
G
i
(
v
i
),
ω
i
=
deg
G
i
(
w
i
), for 1
≤
i
≤
d
.
Proof
Similar to the proof of Theorem 7.3.2 and by definition of the chain graph,
we have:
