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.

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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Question

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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