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Let G be a graph with n 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G? (b) Prove that x(Kn) = n, where Kn is the complete graph on n vertices. (c) Explain what the function x(G, k) counts, where k € N.

Let G be a graph with n 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G? (b) Prove that x(Kn) = n, where Kn is the complete graph on n vertices. (c) Explain what the function x(G, k) counts, where k € N.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a,b,c

5. Let G be a graph with n ≥ 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G? (b) Prove that x(Kn) = n, where K₁ is the complete graph on n vertices. (c) Explain what the function x(G, k) counts, where k € N. (d) If G has no edges, prove that x(G, k) = k". (e) Using the equation x(G, k) = x(G - e, k) — x(G/e, k), where e is an edge of G, show - that x(G, k) is a polynomial in k of degree n. (Hint: Strong induction on E(G) and use part (d).)
Transcribed Image Text:5. Let G be a graph with n ≥ 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G? (b) Prove that x(Kn) = n, where K₁ is the complete graph on n vertices. (c) Explain what the function x(G, k) counts, where k € N. (d) If G has no edges, prove that x(G, k) = k". (e) Using the equation x(G, k) = x(G - e, k) — x(G/e, k), where e is an edge of G, show - that x(G, k) is a polynomial in k of degree n. (Hint: Strong induction on E(G) and use part (d).)
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