chemia - Studia

TUTTE POLYNOMIAL OF AN INFINITE CLASS OF NANOSTAR DENDRIMERS
Figure 2. Denderimer D[2].
For example, let G be a tree with n vertices, then T(G, x, y) = x n -1 ,
because all the edges in a tree are bridges. The dendrimer D[n] in Figure 2
is a tree with 2×3 n+1 n 1
2× 3 −2
-1 vertices, thusT ( D[
n],
x,
y)
= x
+
.
The Figure 1 has been constructed by joining six Ns[0] units to the
hexagons in the outer layers, as detailed in Figures 3 and 4.
Figure 3. Ns[0] and Ns[0]-H 1 +C 5 .
Figure 4. Ns[1].
6
⎛ x −x
⎞
Lemma 1. Let H be a hexagon. Then T(D[H], x, y) =
⎜ + y
⎟.
⎝ x−1
⎠
Proof. By using the formula of Tutte polynomial, we have:
5
T(D[H], x, y) = x + T(D[C
5], x, y)
= + +
5 4
x x T(D[C
4
], x, y)
= + + +
5 4 3
x x x T(D[C
3], x, y)
6
x − x
= + y.
x −1
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