Chemistry Reference
In-Depth Information
Fig. 7.6
Ortho-, meta- and
para-positions of vertices in
hexagon
Fig. 7.7
Ortho-, meta-, and
para-polyphenyl chains with
seven hexagons
Our first example is about polyphenyl chains. Two vertices
v
and
w
of a hexagon
H are said to be in ortho-position if they are adjacent in H. If two vertices
v
and
w
are at distance two, then they are said to be in meta-position, and if two vertices
v
and
w
are at distance three, then they are said to be in para-position. Examples of
vertices in the above three types of positions are illustrated in Fig.
7.6
.
An internal hexagon H in a polyphenyl chain is said to be an ortho-hexagon,
meta-hexagon, or para-hexagon, respectively if two vertices of H incident with two
edges which connect other two hexagons are in ortho-position, meta-position, para-
position, respectively. A polyphenyl chain of h hexagons is ortho-PPC
h
and is denoted
by O
h
, if all its internal hexagons are ortho-hexagons. In a fully analogous manner,
we can define meta-PPC
h
(denoted by M
h
) and para-PPC
h
(denoted by L
h
). See
Fig.
7.7
.
We may view the polyphenyl chains
O
h
,
M
h
, and
L
h
as the bridge graph
B
2
(
C
6
,
...
,
C
6
;
v
,
w
,
v
,
w
,
...
,
v
,
w
), (
h
times), where
C
6
is the cycle with six vertices
and
v
and
w
are the vertices shown in Fig.
7.6
. Since all vertices of
C
6
are of degree
two, so
M
1
(
C
6
)
4. Note that
v
and
w
are adjacent in
O
h
but are not adjacent in
M
h
and
L
h
.Using Corollaries 7.3.3
and 7.3.5, we obtain the following result.
=
M
2
(
C
6
)
=
24,
υ
=
ω
=
2, and
α
C
6
(
v
)
=
α
C
6
(
w
)
=
Corollary 7.3.6
The first and second Zagreb indices of
O
h
,
M
h
, and
L
h
are given by:
1.
M
1
(
O
h
)
=
M
1
(
M
h
)
=
M
1
(
L
h
)
=
34
h
−
10,
2.
M
2
(
O
h
)
=
42
h
−
19,
M
2
(
M
h
)
=
M
2
(
L
h
)
=
41
h
−
17
.
