chemia - Studia
chemia - Studia
chemia - Studia
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MAHBOUBEH SAHELI, MODJTABA GHORBANI , MONICA L. POP, MIRCEA V. DIUDEA<br />
244<br />
Table 1. Omega polynomial and CI index of the Net A: Examples<br />
a Omega Polynomial CI<br />
1 1 2 6<br />
4x + 8x<br />
+ 2x<br />
916<br />
2<br />
1 2 16 18 32<br />
24x + 46x<br />
+ 2x<br />
+ 2x<br />
+ 1x<br />
44264<br />
3<br />
1<br />
2 30 36 88<br />
60x + 122x<br />
+ 3x<br />
+ 3x<br />
+ 2x<br />
437060<br />
4<br />
1<br />
2 48 60 168<br />
112x + 248x<br />
+ 4x<br />
+ 4x<br />
+ 3x<br />
2274544<br />
5<br />
1<br />
2 70 90 272<br />
180x + 436x<br />
+ 5x<br />
+ 5x<br />
+ 4x<br />
8280740<br />
6<br />
1<br />
2 96 126 400<br />
264x + 698x<br />
+ 6x<br />
+ 6x<br />
+ 5x<br />
23966456<br />
7 1<br />
2 126 168 552<br />
364x + 1046x<br />
+ 7x<br />
+ 7x<br />
+ 6x<br />
59104804<br />
8 1<br />
2 160 216 728<br />
480x + 1492x<br />
+ 8x<br />
+ 8x<br />
+ 7x<br />
129524240<br />
Net B. The unit of this net is as for the case A but the edges sharing triangles were<br />
deleted. Moreover, the net is constructed not translated (Figure 2). Note, these<br />
networks and only double periodic, as can be seen from bottom rows of figures.<br />
The computed data for the Omega polynomial of this net were<br />
rationalized as in the formulas presented below and Table 2.<br />
a =<br />
+<br />
6 8<br />
1 ⇒ Ω(<br />
G,<br />
x)<br />
= 2x<br />
2x<br />
a−1<br />
3<br />
( 10+<br />
4( a−2)<br />
) i 2a(2a+<br />
1) 16a<br />
∑ x + 2x<br />
1x<br />
(9)<br />
Ω ( G , x)<br />
= 4<br />
+<br />
i=<br />
1<br />
3 2<br />
| E ( G) | = Ω′ ( G,1)<br />
= 24a<br />
+ 4a<br />
2<br />
= 4a<br />
(6a<br />
+ 1)<br />
(10)<br />
2<br />
| V ( G) | = 8a<br />
( a + 1)<br />
(11)<br />
6 64 5 64 4 3 20 2 8<br />
Ω ′ ( G,1)<br />
= 256a<br />
+ a + a −8a<br />
+ a + a<br />
3 3<br />
3 3<br />
(12)<br />
6 512 5 16 4 3 32 2 8<br />
CI ( G) = 320a + a − a − 16a + a − a =<br />
3 3 3 3<br />
⎛ 5 64 4 2 3 2 4 1 ⎞<br />
8a ⎜ 40a + a − a − 2a − a − ⎟<br />
⎝ 3 3 3 3 ⎠<br />
(13)<br />
Table 2. Omega polynomial and CI index of the Net B: Examples<br />
a Omega Polynomial CI<br />
1 6 8<br />
2x + 2x<br />
584<br />
2<br />
10 20 128<br />
4x + 2x<br />
+ 1x<br />
25680<br />
3<br />
14 28 42 432<br />
4x + 4x<br />
+ 2x<br />
+ 1x<br />
273784<br />
4<br />
18 36 54 72 1024<br />
4x + 4x<br />
+ 4x<br />
+ 2x<br />
+ 1x<br />
1482912<br />
5<br />
22 44 66 88 110 2000<br />
4x + 4x<br />
+ 4x<br />
+ 4x<br />
+ 2x<br />
+ 1x<br />
5527720<br />
6<br />
26 52 78 104 130 156 3456<br />
4x + 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 2x<br />
+ 1x<br />
16246256<br />
7 30 60 90 120 150 180 210 5488<br />
4x + 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 2x<br />
+ 1x<br />
40497240<br />
8 34 68 102 136 170 204 238 272 8192<br />
4x + 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 4x<br />
+ 2x<br />
+ 1x<br />
89447744
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