chemia - Studia

OMEGA POLYNOMIAL FOR NANOSTRUCTURES DESIGNED BY (P 4 ) k Le OPERATIONS
parameters are given in Table 3. Observe, in the dual pair, the face of
parent becomes the vertex of transform and this interchanging operates
also on the corresponding parameters: s 0 f 0 becomes d 0 v 0 , while the number
of edges remains unchanged.
Table 2. Platonic solid graph parameters
Graph Vertices |v 0 | Degree d 0 Edges |e 0 | Ring folding s 0 Faces |f 0 |
T 4 3 6 3 4
C 8 3 12 4 6
Oct 6 4 12 3 8
Do 20 3 30 5 12
Ico 12 5 30 3 20
Table 3.Transforms of the Platonic solid graphs by Le(P 4 (M)) k )
M Vertices |v 0 | Edges |e 0 | Faces |f 0 |
T 12× 4 k
18× 4 k
k
6× 4 + 2
C 24× 4 k
36× 4 k
k
12× 4 + 2
Do 60× 4 k
90× 4 k
k
30× 4 + 2
Formula | v| = 4 k × s0f
| | 3 4 k
k
e = × × e
0
0 | f | = 4 × e0
+ 2
Oct 24× 4 k
36× 4 k
k
12× 4 + 2
Ico 60× 4 k
90× 4 k
k
30× 4 + 2
Formula | v| = 4 k × d0v0
| e| 3 4 k
k
= × × e
| f | = 4 × e0
+ 2
Ring polynomial
The ring polynomial for the graphs originating in trivalent Platonics
is as follows:
4
0
( )
4 6
( )
8
( )
4 6
( )
8
( ) ( )
k a a
R( Le(( P (T)) ), x) = 3× 4 x + 8x + 3× 4 − 6 x
(5)
k a a
R( Le(( P (C)) ), x) = 6× 4 x + 8x + 6× 4 − 6 x (6)
4
4
k a 4 6 a 8 10
R( Le(( P (Do)) ), x) = 15× 4 x + 20 x + 15× 4 − 30 x + 12 x (7)
Generalizing, for the graphs transformed from the trivalent Platonics,
the formula for ring polynomial is of the form:
2( k−1) 4 6
( )
k−1 k−1 k 8 2s0
( s0f0× 2 (2 − 1) + e0(2 − 1) ) x + f0
x
RLe ( (( P( Gd ( :3))) ), x) = sf× 2 x + vx +
k
4 0 0 0 0
Now, considering the relation between the dual pairs, for the trigonal
Platonics we have:
2( k−1) 4 6
( )
k−1 k−1 k 8 2d0
( dv
0 0× 2 (2 − 1) + e0(2 − 1) ) x + v0x
RLe ( (( P( G( f :3))) ), x) = dv× 2 x + fx +
k
4 0 0 0 0
(8)
(9)
229