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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.

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. Hence for r ∈ R(g) we have (1 − Wr ([p])) = Pg (−1) i=1 a(E ′ )2i−1 a(E ′ )2i +(J−r)a(E ′ ) X(E ′ ), where the sum is over all even subsets E ′ of G. Let E ′ be an arbitrary even subset of G. 5, since we can replace r by J − r in the summation.

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.

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