By Martin Loebl
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Example text
We define (−1)rot(p) by ′ (−1)rot(p ) = (−1)Ja(p) (−1)rot(p) . Let R(g) denote the set of all 0, 1-vectors of length 2g. For r ∈ R(g) we let Wr ([p]) = (−1)ra(p) W ([p]). Hence W ([p′ ]) = WJ ([p]). Theorem 22. If G = (V, E) is a g-graph where each degree is even and at most four, then (−1)s(J−r) E(G, x) = 2−g (1 − Wr ([p])), r∈R(g) where is the formal infinite product over all equivalence classes of prime reduced cycles of G. Proof. We proceed as in the proof of Theorem 21. , over all even subsets of G.
30 MARTIN LOEBL special and note that each special vertex has degree 4 in G′ . Each nonperiodic closed walk p of G corresponds to the prime reduced cycle p′ of G′ which satisfies the crossover condition at each special vertex. Let J denote the vector (1, . . , 1) of all 1’s. We define (−1)rot(p) by ′ (−1)rot(p ) = (−1)Ja(p) (−1)rot(p) . Let R(g) denote the set of all 0, 1-vectors of length 2g. For r ∈ R(g) we let Wr ([p]) = (−1)ra(p) W ([p]). Hence W ([p′ ]) = WJ ([p]). Theorem 22. If G = (V, E) is a g-graph where each degree is even and at most four, then (−1)s(J−r) E(G, x) = 2−g (1 − Wr ([p])), r∈R(g) where is the formal infinite product over all equivalence classes of prime reduced cycles of G.
5, since we can replace r by J − r in the summation. Dept. of Applied Mathematics and, Institute of Theoretical Computer Science (ITI), Charles University, Malostranske n. 25, 118 00 Praha 1, Czech Republic.