chemia - Studia

THE OMEGA POLYNOMIAL OF THE CORCOR DOMAIN OF GRAPHENE
2) ε
1 10 / 3 ⎡ / 6⎤ 1+
10 / 3+
⎡(
−3)
/ 6⎤ 2 = x
− + n + n
+ x
n n ,
3) ε 4 / 3 2 3 1 ⎡ / 2⎤ ⎡(3
3) / 9⎤ 3 = ∑ n − x
n − + j + j − j −n−
,
j = n / 3 + 2
4) 7 / 3 2 7 / 3
4 ∑ − + + ′ + ′
ε = n j n S S
x
j j .
j = 4 n / 3 − 1
(
Therefore, ⎡ ⎤)
∑ − + ′ + ′′ + + − +
ε + ε + ε + ε = 2n
2+⎡n
/ 3 1 S S 2n
j 2 n / 3
1 2 3 4
⎤x
j j
.
j = 1
To simplify these quantities, two cases that n is odd or even are
considered. If n is even then
7n
⎛
3 6 2 1 4 5
⎞
⎜x + x + x ( 2x + x + x ) −
⎟
7
Ω ( Gx , ) = [6/(1 − x)]
×⎜
14n
⎟
1
3
⎛ −2 −1 1 3 4 1 5 n −2 −1 5 6 ⎞
x x x 2 x x x x ( x 2x x 2 x )
⎜ ⎜ + + + + + + − − + + ⎟
2 2 3
⎟
⎝ ⎝
⎠⎠
if n is odd, then
7n
1 3 9
⎛
3 6
⎛ ⎞
⎞
2 2 2 2
⎜x + x + x ⎜x + x + 2x
⎟−
⎟
7
Ω ( Gx , ) = [6/(1 − x)]
×
⎜ ⎝ ⎠
⎟
⎜ 14n
⎟
1
3
⎛ −2 −1 1 3 4 1 5 n −2 −1 5 6 ⎞
x x x 2 x x x x ( x 2x x 2 x )
⎜ ⎜ + + + + + + − − + + ⎟
2 2 3
⎟
⎝ ⎝
⎠⎠
Using a similar argument as above, if n ≡ 1 (mod 3) then for even n,
7n
⎛
3 6 2
4 5
⎞
⎜x + x + x ( 2x+ x + x ) −
⎟
7
Ω ( Gx , ) = [6/(1 − x)]
×⎜ ⎟
14n
−5 −2 1 4 10 16 19 19 16 −5 −2
⎜ ⎛2 5 1 1 n
⎞
3 3 3 3 3 3 3 3 3 3 3 3 ⎟
⎜
x ⎜ x + x + x + x + 2 x + x + x + ( x + 2x −2 x −x
) ⎟
3 6 3 6 3
⎟
⎝ ⎝
⎠⎠
and for odd n,
7n
1 3 9
⎛
3 6
⎛
⎞
⎞
2 2 2 2
⎜ −x − x + x ⎜x + x + 2x
⎟−
⎟
7 ⎜
⎝
⎠
⎟
Ω ( G, x) = [6/(1 − x )] × ⎜ 14n
−5 −2 1 4 10 16 19 19 16 −5 −2
⎟
⎜
⎛ 2 5 1 1 n
⎞
3 3 3 3 3 3 3 3 3 3 3 3
x x + x + x + x + 2 x + x + x + ( −x − 2x + 2 x + x ) ⎟
⎜ ⎜
⎟
3 6 3 6 3
⎟
⎝ ⎝
⎠ ⎠
Finally, if n ≡ 2 (mod 3) then for even n,
7n
⎛
7 10 2 5 8 9
⎞
⎜x + x + x ( 2x + x + x ) −
⎟
4 7
Ω ( Gx , ) = [6/ x(1 − x)]
× ⎜
⎟,
14n
8 14 17 20 23 29 8 29
⎜ ⎛3 1
⎞
3 3 3 3 3 3 3 3 3 ⎟
⎜
x ⎜ x + x + x + x + x + x + n( − x + x ) ⎟
2 2
⎟
⎝ ⎝
⎠⎠
and for odd n,
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