chemia - Studia

COMPUTING WIENER AND BALABAN INDICES OF DENDRIMERS BY AN ALGEBRAIC APPROACH
these H
k
by defining ( ak
). g = akg
∈ Hkg
where g ∈ K and ak
∈ Hk
. So
H ~ K = ⊕ k ∈K
K ∝ H .
Proposition. In the graph G , if Aut (G)
acts on V (G)
and the orbits of this
k
1
action are V 0
, V
1
, … , V k
then W ( G)
= ∑|
Vi | d(
x i
) where xi
∈ Vi
. If
2 i=
1
Aut (G) acts on E (G)
and the orbits of this action are E 1
, E
2
, … , E
k
k
m | Ei
|
then J ( G)
= ∑
where xi
−1
xi
∈ Ei
.
μ + 1 d(
x ) d(
x
i= 1 i−1 i)
Proof. It is sufficient to show that if α ∈ Aut(G)
then d( u)
= d(
α(
u))
that is
evident.
Define D[k] as the dendrimer molecule depicted in Figure 2. We
label the vertices of D[k] by 0, 1, ..., 2 × (3 k – 1). If an edge ij ( i < j ) is
shown by j then the edges of D[k] can be labelled by 1, 2, ..., 2 × (3 k – 1). So,
the number of vertices and edges of D[k] are 1 + 2 × (3 k – 1) and 2×(3 k – 1),
respectively.
Figure 2. The Dendrimer Molecule D[4].
Theorem. The automorphism group of D[k] is isomorphic to the wreath
k
product S
3
~ S4
where S
4
act on Ω = { 1,2,...,2×
(3 −1)}
.
Proof. Fix a vertex x 0
as root and assume that α ∈ Aut( D[
k])
. Then for
vertex v in level i , Figure 2, α (v)
is also in level i , since v and α (v)
have the same eccentricity. Consider the action of S 4
on { 1,2,...,2× (3
k −1)}
.
Therefore Aut ( D[
k])
is isomorphic to the wreath product of group S 3
via
the permutation group S
4
. █
Corollary 1. The orbits of Aut ( D[
k])
under its natural action on V ( D[
k])
are V
0
= {0}
, V = {1,2,3,4
1
} , … , { 1 2 (3 k−1
1),...,2 (3 k
V = + × − × − 1)}
.
k
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