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Let G = (V,E) be a connected graph, and u, v E V. The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices. a. Show that if A(G) > 4, then A(G) < 2. b. Show that if G has a cut vertex and A(G) = 2, then G has a vertex without neighbours.

Let G = (V,E) be a connected graph, and u, v E V. The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices. a. Show that if A(G) > 4, then A(G) < 2. b. Show that if G has a cut vertex and A(G) = 2, then G has a vertex without neighbours.

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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Let G = (V,E)  be a connected graph, and  u, v in V The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices.

 

  1. Show that if A(G) ≥ 4, then A(Ḡ) ≤ 2.
  2. Show that if G has a cut vertex and A(G) = 2, then Ḡ has a vertex without neighbors.

 

Let G = (V,E) be a connected graph, and u, v E V. The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices. a. Show that if A(G) 2 4, then A(G) < 2. b. Show that if G has a cut vertex and A(G) = 2, then G has a vertex without neighbours.
Transcribed Image Text:Let G = (V,E) be a connected graph, and u, v E V. The distance between u and v, d(u,v) is the length of the shortest route between u and v, while the width of G, W(G), is the greatest distance between two of its vertices. a. Show that if A(G) 2 4, then A(G) < 2. b. Show that if G has a cut vertex and A(G) = 2, then G has a vertex without neighbours.
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