Chemistry Reference
In-Depth Information
Fig. 7.22
The ordinary Bethe
tree
B
2,4
The ordinary Bethe tree
B
d
,
k
is a rooted tree of
k
levels whose root vertex has degree
d
, the vertices from levels 2 to
k
−
1 have degree
d
+
1, and the vertices at level
k
have degree 1, see Fig.
7.22
.
Note that
B
1,
k
=
P
k
and
B
d
,2
=
S
d
+
1
. Using Corollary 7.5.18, we get the following
relations for computing the first and second Zagreb indices of ordinary Bethe tree
B
d
,
k
.
Corollary 7.5.19
The first and second Zagreb indices of the ordinary Bethe tree
B
d
,
k
are given by:
s
=
0
d
s
,
1)
2
k
−
3
d
2
d
k
−
1
1.
M
1
(
B
d
,
k
)
=
+
+
d
(
d
+
⎧
⎨
d
2
=
if k
2
2.
M
2
(
B
d
,
k
)
=
3
.
s
=
0
d
s
k
−
3
⎩
2
d
2
(
d
+
1)
if k
≥
Denote by
C
(
d
,
k
,
n
), the unicyclic graph obtained by attaching the root vertex of
B
d
,
k
to the vertices of
n
-vertex cycle
C
n
, see Fig.
7.23
. For more information about
this graph, see (Rojo
2007
).
It is easy to see that
C
(
d
,
k
,
n
) is the cluster of
C
n
and
B
d
,
k
. So we can apply
Corollary 7.5.2 and then Corollary 7.5.19, to get the formulae for the first and second
Zagreb indices of
C
(
d
,
k
,
n
).
Corollary 7.5.20
The first and second Zagreb indices of
C
(
d
,
k
,
n
) are given by:
s
=
0
1)
2
k
−
3
nd
2
nd
k
−
1
d
s
,
1.
M
1
(
C
(
d
,
k
,
n
))
=
4
n
(
d
+
1)
+
+
+
nd
(
d
+
⎧
⎨
2
n
(
d
2
+
3
d
+
2)
if k
=
2
2.
M
2
(
C
(
d
,
k
,
n
))
=
.
k
−
s
=
0
3
⎩
n
(3
d
2
2
nd
2
(
d
d
s
+
6
d
+
4)
+
+
1)
if k
≥
3
Denote by
P
(
d
,
k
,
n
), the tree obtained by attaching the root vertex of
B
d
,
k
to the
vertices of
n
-vertex path
P
n
, see Fig.
7.24
. For more information about this class
of trees, see (Robbiano et al.
2008
). The graph
P
(
d
,
k
,
n
) can be considered as the
bridge graph
B
1
(
B
d
,
k
,
B
d
,
k
,
...
,
B
d
,
k
;
w
,
w
,
...
,
w
), where
w
denotes the root vertex
