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1. Given an undirected graph G = (V, E), and define a distance function d: V x V → R20 which assigns a non-negative real number to each pair of vertices satisfying d(u, v) = d(v, u). (a) Write down the rigorous mathematical definition for the diameter of G. (b) Write down the rigorous mathematical definition for the radius of G. (c) If d(u, v) is a metric (that is, it further satisfies the triangle inequality: d(x, z) ≤d(x, y)+d(y, z)), prove that the diameter of G is at most two times the radius of G.

1. Given an undirected graph G = (V, E), and define a distance function d: V x V → R20 which assigns a non-negative real number to each pair of vertices satisfying d(u, v) = d(v, u). (a) Write down the rigorous mathematical definition for the diameter of G. (b) Write down the rigorous mathematical definition for the radius of G. (c) If d(u, v) is a metric (that is, it further satisfies the triangle inequality: d(x, z) ≤d(x, y)+d(y, z)), prove that the diameter of G is at most two times the radius of G.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 42E
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1. Given an undirected graph G = (V, E), and define a distance function d: V x V → R20 which assigns a non-negative real number to each pair of vertices satisfying d(u, v) = d(v, u). (a) Write down the rigorous mathematical definition for the diameter of G. (b) Write down the rigorous mathematical definition for the radius of G. (c) If d(u, v) is a metric (that is, it further satisfies the triangle inequality: d(x, z) ≤d(x, y)+d(y, z)), prove that the diameter of G is at most two times the radius of G.
Transcribed Image Text:1. Given an undirected graph G = (V, E), and define a distance function d: V x V → R20 which assigns a non-negative real number to each pair of vertices satisfying d(u, v) = d(v, u). (a) Write down the rigorous mathematical definition for the diameter of G. (b) Write down the rigorous mathematical definition for the radius of G. (c) If d(u, v) is a metric (that is, it further satisfies the triangle inequality: d(x, z) ≤d(x, y)+d(y, z)), prove that the diameter of G is at most two times the radius of G.
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