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OMEGA POLYNOMIAL IN TITANIUM OXIDE NANOTUBES
APPENDIX
Examples for calculating Omega polynomial.
1. Case of infinite 2-dimensional graph K.
We have the Omega polynomial:
qx + 2( x + x + ... + x ) + (2p − q + 1) x
1.1. Case: 2 p > q>
p,2| q,
If q = 6, p = 5 then, the graph is:
2 p+ 1 3 5 2 q− 1 2q+
1
and
Ω ( Gx , ) = 6x + 2( x+ x + x + x + x ) + 5x
1.2. Case: 2 p > q > p,2 / | q,
Now if p = 4, q = 7 then, the graph is:
11 3 5 7 9 11 13
and
9 3 5 7 9 13 15
Ω ( Gx , ) = 7x + 2( x+ x + x + x + x ) + 2x
2p+ 1 3 5 4p-1 4p+
1
We have also
qx + 2( x + x + ... + x ) + ( q - 2 p + 1) x
1.3. Case: q≥ 2 p. If p = 4, q = 9 then, the graph is:
and
Ω ( Gx , ) = 9x + 2( x+ x + x + x + x + x + x ) + 2x
9 3 5 7 9 11 13 15 17
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