Chemistry Reference
In-Depth Information
Fig. 7.14
The thorn graph
G
∗
(
p
1
,
p
2
,
...
,
p
n
)
Let
G
be a labeled graph on
n
vertices and let
p
1
,
p
2
,
...
,
p
n
be non-negative integers.
The thorn graph
G
∗
(
p
1
,
p
2
,
...
,
p
n
) of the graph G is obtained from G by attaching
p
i
pendant vertices to the i-th vertex of
G
,
i
, see Fig.
7.14
.
The concept of thorny graphs was introduced in (Gutman
1998
) and eventu-
ally found a variety of chemical applications (Bytautas et al.
2001
; Vukicevic and
Graovac
2004
; Walikar et al.
2006
; Heydari and Gutman
2010
). The thorn graph
G
∗
(
p
1
,
p
2
,
...
,
p
n
) can be considered as the rooted product of
G
by the sequence
∈{
1,2,
...
,
n
}
S
p
1
+
1
,
S
p
2
+
1
,
...
,
S
p
n
+
1
, where the root vertex of
S
p
i
+
1
is assumed to be in the
vertex of degree
p
i
,
i
. So we can apply Theorem 7.5.1 to compute
the first and second Zagreb indices of the thorn graph
G
∗
(
p
1
,
p
2
,
...
,
p
n
).
Corollary
∈{
1,2,
...
,
n
}
7.5.5
The
first
and
second
Zagreb
indices
of
the
thorn
graph
G
∗
(
p
1
,
p
2
,
...
,
p
n
) are given by:
1.
M
1
(
G
∗
(
p
1
,
p
2
,
...
,
p
n
))
+
i
=
1
p
i
(
p
i
+
2
i
=
1
p
i
deg
G
(
i
),
=
M
1
(
G
)
1)
+
2. If
G
P
2
then
M
2
(
G
∗
(
p
1
,
p
2
,
...
,
p
n
))
=
i
=
1
p
i
2
i
=
1
p
i
deg
G
(
i
)
n
n
=
M
2
(
G
)
+
+
+
[
p
j
deg
G
(
i
)
+
p
i
deg
G
(
j
)
+
p
i
p
j
],
ij
∈
E
(
G
)
and M
2
(
P
2
∗
(
p
1
,
p
2
))
=
p
1
(
p
1
+
1)
+
p
2
(
p
2
+
1)
+
(
p
1
+
1)(
p
2
+
1)
.
n
i
=
Remark 7.5.6
Let
{
G
i
}
1
be a set of finite pairwise disjoint graphs with distinct
vertices
w
i
∈
B
1
(
G
1
,
G
2
,
...
,
G
n
;
w
1
,
w
2
,
...
,
w
n
) can be considered as the rooted product of the
n
-vertex path
P
n
by the sequence
V
(
G
i
),
i
∈{
1,2,
...
,
n
}
. The bridge graph
B
1
=
{
G
1
,
G
2
,
...
,
G
n
}
, where the root vertex of the graph
G
i
is assumed
to be in the vertex
w
i
,
i
. So using Theorem 7.5.1, we can reproduce
the results of the Theorem 7.2.2 and Theorem 7.2.4.
Now we consider several classes of molecular graphs constructed from rooted
product and determine their Zagreb indices.
Our first example is about caterpillar trees. A caterpillar or caterpillar tree is a
tree in which all the vertices are within distance one of a central path. If we delete
all pendent vertices of a caterpillar tree, we reach to a path. So caterpillars are thorn
graphs whose parent graph is a path, see Fig.
7.15
.
∈{
1,2,
...
,
n
}
