Chemistry Reference
In-Depth Information
Fig. 7.25
The dendrimer tree
T
4,3
adding branching blocks around a central core (thus obtaining a new, larger orbit or
generation-the “divergent growth” approach) or by building large branched blocks
starting from the periphery and then attaching them to the core (Diudea
1995
). De-
tails on dendrimers, an important and recently much studied class of nano-materials,
and especially on their topological properties can be found in (Diudea
2006
; Diudea
and Nagy
2007
; Iranmanesh and Gholami
2009
; Iranmanesh and Dorosti
2011
) and
the references quoted therein.
A dendrimer tree
T
d
,
k
is a rooted tree such that the degree of its non-pendent
vertices is equal to
d
and the distance between the rooted (central) vertex and the
pendent vertices is equal to
k
(Dobrynin et al.
2001
), see Fig.
7.25
.So
T
d
,
k
can be
considered as a generalized Bethe tree with
k
+
1 levels, such that whose non-pendent
vertices have equal degrees. Note that
T
2,
k
=
P
2
k
+
1
and
T
d
,1
=
S
d
+
1
. Using Corollary
7.5.18, we get the following relations for Zagreb indices of the dendrimer tree
T
d
,
k
.
Corollary 7.5.22
The first and second Zagreb indices of the dendrimer tree
T
d
,
k
are
given by:
s
=
0
(
d
d
3
k
−
2
d
2
1)
k
−
1
1)
s
,
1.
M
1
(
T
d
,
k
)
=
+
d
(
d
−
+
−
+
d
3
k
−
2
s
=
0
=
d
2
(
d
−
1)
k
−
1
1)
s
.
2.
M
2
(
T
d
,
k
)
(
d
−
Now we introduce a class of dendrimers for which Corollary 7.5.4 is applicable. This
molecular structure can be seen in some of the dendrimer graphs such as tertiary
phosphine dendrimers. Let
D
0
be the graph of Fig.
7.26
.
For positive integers d and k, suppose
D
d
,
k
be a series of dendrimer graphs obtained
by attaching d pendent vertices to each pendent vertex of
D
d
,
k
−
1
and set
D
d
,0
=
D
0
.
Some examples of this class of dendrimer graphs are shown in Figs.
7.27
and
7.28
.
It is easy to check that,
M
1
(
D
0
)
45. Choose a numbering for
vertices of
D
0
such that its pendant vertices have numbers 1,2, 3 and its non-pendant
=
42 and
M
2
(
D
0
)
=
