Chemistry Reference
In-Depth Information
n
2
Fig. 7.4
The van Hove comb lattice graphs with
N
=
vertices
This graph can be represented as the bridge graph
B
1
(
P
1
,
P
2
,
...
,
P
n
−
1
,
P
n
,
P
n
−
1
,
...
,
P
2
,
P
1
;
v
1,1
,
v
1,2
,
...
,
v
1,
n
−
1
,
v
1,
n
,
v
1,
n
−
1
,
...
,
v
1,2
,
v
1,1
),
where for 2
n
,
v
1,
i
is the first vertex (vertex of degree one) of the
i
-vertex
path
P
i
and
v
1,1
is the single vertex (vertex of degree zero) of the singleton graph
P
1
.
So by Theorems 7.2.2 and 7.2.4, the formulae for Zagreb indices of
CvH
(
N
) are
obtained at once.
Corollary 7.2.8
The first and second Zagreb indices of
CvH
(
N
) are given by:
≤
i
≤
4
n
2
1.
M
1
(
CvH
(
N
))
=
+
4
n
−
12,
⎧
⎨
9
if n
=
2
=
2.
M
2
(
CvH
(
N
))
.
⎩
4
n
2
+
10
n
−
28
if n
≥
3
7.3
Zagreb Indices of the Bridge Graph B
2
In this section, we compute the first and second Zagreb indices for the bridge graph
B
2
. 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
i
1
be a set of finite pairwise disjoint graphs with distinct vertices
v
i
,
w
i
∈
V
(
G
i
). The bridge graph
B
2
=
{
G
i
}
=
B
2
(
G
1
,
G
2
,
...
,
G
d
;
v
1
,
w
1
,
v
2
,
w
2
,
...
,
v
d
,
w
d
)of
i
=
1
is the graph obtained from the
graphs
G
1
,
G
2
,
...
,
G
d
by connecting the vertices
w
i
and
v
i
+
1
by an edge for all
i
∈{
i
=
1
{
G
i
}
{
v
i
,
w
i
}
with respect to the vertices
1,2,
...
,
d
−
1
}
, as shown in Fig.
7.5
.
