chemia - Studia

ON OMEGA POLYNOMIAL OF ((4,7)3) NETWORK
Figure 1. The 2-dimensional ((4,7)3) net (3×3 units) designed by the sequence of
map operations Trs(Ca(4,4)): non-optimized (left) and optimized (right) structure.
Figure 2. The ops strips of length s=1 and s=2 in the 2-dimensional co-net
(the repeat unit (left) and 2×2 units (right) of ((4,7)3) pattern
Structure
Net
Table 1. Omega and Sadhana polynomials in the ((4,7)3) modified graphene
Formulas
2 2 2 2
Ω ( ) = 2[ + ( −1) ] ⋅ + (10 + 14 −4)
⋅
x k k x k k x
2
Ω ′(1) = 18x + 6k = 6 k(3k + 1) = e( G)
CI G k k k k
4 3 2
( ) = 2(162 + 108 + 5 + − 2)
2 2
2 2 18k + 6k− 2 2 18k + 6k−1
Sd( x) = 2[ k + ( k −1) ] ⋅ x + (10k + 14k −4)
⋅ x
2 2
Sd′ (1) = 6 k(3k + 1)(14k + 10k − 3) = e⋅ (14k + 10k
− 3)
′
4 3 2
Sd( G) = Sd (1) = 252x + 264x + 6k − 18k
vG ( ) = 4 k(5+ 3( k−
1))
Co-Net
2 2
Ω ( x) = 4 kk ( −1) ⋅ x + (10k + 14k−1)
⋅ x
2
Ω ′(1) = 18x + 6k− 1 = e( G)
4 3 2
CI( G) = 2(162k + 108k −13k − 5k
+ 1)
2 2
18k + 6k− 3 2 18k + 6k−2
Sd( x) = 4 k( k −1) ⋅ x + (10k + 14k −1)
⋅ x
2 2 2
Sd′ (1) = 2(7k + 5k − 1)(18k + 6k − 1) = 2(7k + 5k −1)
⋅ e
Sd G Sd′
x x k k
4 3 2
( ) = (1) = 252 + 264 + 10 − 22 + 2
vG ( ) = 4 k(5+ 3( k−
1))
Table 2. Examples for the formulas in Table 1.
k Omega polynomial v(G) e(G) CI(G) Sd(G)
Net
2 64X+10X 2 64 84 6952 6132
3 128X+26X 2 132 180 32168 27540
199