For n ≥ 3, let Cn denote the cycle of length n.
(a) What is P(C3, λ)?
(b) If n > 4, show that
P(Cn,λ) = P(Pn-1, λ) - P(Cn-1, λ),
where Pn-1 denotes the path of length n - 1.
(c) Verify that P(Pn-1, λ) = λ(λ - l)n-1, for all n ≥ 2.
(d) Establish the relations
P(Cn, λ) - (λ - 1)n = (λ - l)n-1 - P(Cn-1 λ), n ≥ 4,
P(Cn, λ) - (λ - l)n = P(Cn-2, λ) - (λ - l)n-2, n ≥ 5.
(e) Prove that for all n ≥3,
P(Cn,λ) = (λ-l)n + (-l)n(λ-l).