chemia - Studia

MAHBOUBEH SAHELI, MIRCEA V. DIUDEA
230
Omega Polynomial
The Omega polynomial (calculated at R max [8]) for the graphs transformed
from the trivalent Platonics is as follows:
4
2
( )
k+
− ×
×
( ) k
−
( )
k
k k 2 k 1 3 2
Ω ( Le(( P (T)) ), x) = 3(2 − 1) + 6 x + 4(2 − 1) x (10)
4
1 k 2
k k 3 2 + k 1 2
+
Ω ( Le(( P (C)) ), x) = 4(2 − 1) + 6 x + 6(2 − 1) + 3 x (11)
4
1 3
k k 5× 2 k+ k− 1 5×
2 k 2
k+
Ω ( Le(( P (Do)) ), x, R[8]) = 6(2 − 1) x + 12(2 − 1) x + 15 x (12)
Generalizing, we have:
⎛
⎜
⎝
1 + ( −1)
2
s0
k
k
k −1
s0×
2
Ω ( Le(( P4 (G)) ), x) = f0(2 − 1) + 3 x +
⎛ ⎢ s0
+ 1⎥
⎞ k+
1
s0 + − 1 × 2
1
0
+ 1⎥
⎞
⎜
k
k
⎢
6
⎥ ⎟ e
+
⎝ ⎣ ⎦ ⎠
0 ( s0
− 1) × 2
⎛ ⎢ s
⎜ s0
+ ⎢ (2 1) x x
6 ⎥ ⎟ − +
⎝ ⎣ ⎦ ⎠
⎡ s0
⎤
⎢ 3
⎥
And for CI we have:
⎞
⎟
⎠
( ((
k 2k k k k
4(T)) )) 324 4 6 4 (11 2 1) 18 4
( ((
k 2k k k k
4(C)) )) 1296 4 12 4 (16 2 1) 36 4
( ((
k 2k k k k
4(Do)) )) 8100 4 30 4 (25 2 1) 90 4
(13)
CI Le P = ⋅ − ⋅ ⋅ − − ⋅ (14)
CI Le P = ⋅ − ⋅ ⋅ − − ⋅ (15)
CI Le P = ⋅ − ⋅ ⋅ − − ⋅ (16)
The Omega polynomial, calculated at R max =10, in case M=Do, is as follows.
2
k
k 2 5( k − p) 2 + 3
Ω ( Le(( P4
(Do)) ), x, R[10]) = 6( k − p − 2) ⋅ x + 15 ⋅ x +
(17)
2
2 10( k − p)
6( k − p −1)
⋅ x
k
2 k
2 4 2
Ω ′( Le(( P (Do)) ),1, R[10]) = 120 p − 180k p + 120⋅ 2 − 120k + 90k + 90 p (18)
4
k
4 2 4 6
CI( Le(( P4
(Do)) ), R[10]) = 2250k p − 1800k p + 900k − 750k
+
(19)
900 p + 750 p − 2250k p − 960⋅ 2 + ( Ω′ ( Le(( P (Do)) ),1, R[10]))
2 3 2 2 2k
k
2
4
Table 4 lists some examples for the formulas derived within this
paper. Computations were made by Nano Studio software [34].
Table 4. Examples for the herein derived formulas
Le((P 4 (M)) k ) V Omega polynomial CI Ring polynomial
M ; k ; R max
T; k=3 ; R[8] 768 12x 24 +27x 32 1292544 192x 4 +8x 6 +210x 8
C; k=3 ; R[8] 1536 21x 32 +34x 48 5208576 384x^4+8x 6 +402x 8
Do; k=3;R[8] 3840 30x 8 +30x 24 +36x 40 +42x 80 32832000 960x 4 +20x 6 +930x 8 +12x 10
Do; k=3;R[10] 36x 40 +15x 64 +42x 80 32789760
T; k=4; R[8] 3072 28x 48 +51x 64 20960256 768x 4 +8x 6 +786x 8
C; k=4; R[8] 6144 45x 64 +66x 96 84142080 1536x 4 +8x 6 +1554x 8
Do; k=4; R[8] 15360 30x 16 +30x 48 +84x 80 +90x 160 527923200 3840x 4 +20x 6 +3810x 8 +12x 10
Do; k=4; R[10] 84x 80 +15x 128 +90x 160 527754240