Chemistry Reference
In-Depth Information
Fig. 7.19
The starlike tree
S
(4,3, 3,1, 4,2, 5,3) with
vertex v of degree
n
=
8
and
I
={
2. Suppose
I
={
i
|
1
≤
i
≤
n
,
k
i
=
2
}
i
|
1
≤
i
≤
n
,
k
i
≥
3
}
and
let
|
I
| =
r
. The second Zagreb index of the sunlike graph
G
(
k
1
,
k
2
,
...
,
k
n
)is
given by:
4
i
∈
I
M
2
(
G
(
k
1
,
k
2
,
...
,
k
n
))
=
M
1
(
G
)
+
M
2
(
G
)
−
deg
G
(
i
)
+
k
i
−
8
n
+
9
r
+
5
m.
i
∈
I
If in particular for all 1
≤
i
≤
n
,
k
i
≥
3, then
n
M
2
(
G
(
k
1
,
k
2
,
...
,
k
n
))
=
M
1
(
G
)
+
M
2
(
G
)
+
4
k
i
+
5
m
−
8
n.
i
=
1
A starlike tree is a tree with exactly one vertex having degree greater than two. We
denote by
S
(
k
1
,
k
2
,
...
,
k
n
), the starlike tree which has a vertex
v
of degree
n
≥
3
P
k
1
P
k
2
...
P
k
n
, where
and has the property that
S
(
k
1
,
k
2
,
...
,
k
n
)
−
v
=
k
1
≥
1. Clearly,
k
1
,
k
2
,
...
,
k
n
determine the starlike tree up to
isomorphism and
S
(
k
1
,
k
2
,
...
,
k
n
) has exactly
k
1
+
k
2
≥
...
≥
k
n
≥
k
2
+
...
+
k
n
+
1 vertices. For
1
n
, denote by
v
i
, the root vertex of
P
k
i
attached to the vertex
v
, see Fig.
7.19
.
Clearly, for
k
i
≥
≤
i
≤
2, deg
P
k
i
(
v
i
)
=
1. Now consider the subtree T of
S
(
k
1
,
k
2
,
...
,
k
n
)
with the vertex set
V
(
T
)
={
v
,
v
1
,
v
2
,
...
,
v
n
}
. Clearly T is isomorphic to
n
+
1-vertex
≤
≤
star,
S
n
+
1
. Choose a numbering for vertices of T such that the vertex
v
i
,1
i
n
,
+
has number
i
and the vertex
v
has number
n
1. So we can consider the starlike
tree
S
(
k
1
,
k
2
,
...
,
k
n
) as the sunlike graph
T
(
k
1
,
k
2
,
...
,
k
n
+
1
) where
k
n
+
1
=
1 and
we can apply Corollary 7.5.11 to compute the first and second Zagreb indices of the
starlike tree
S
(
k
1
,
k
2
,
...
,
k
n
).
