For n 3, let Cn denote the cycle of length n. (a) What is P(C3, )?
Question:
(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).
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a 1 2 b Follows from Theorem 1110 c Follows by the rule of product d P C n PP n1 PC n1 ...View the full answer
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Related Book For
Discrete and Combinatorial Mathematics An Applied Introduction
ISBN: 978-0201726343
5th edition
Authors: Ralph P. Grimaldi
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