5. Let G = (V, E) be a graph with n > 3 vertices.(a) Define a relation on V by "ry if and only if there is a walk between x and y".Prove that is an equivalence relation on V.(b) Give an example of a graph G with 2 connected components.(c) Suppose that dege(x) + degc(y) > n + 4 for all non-adjacent vertices x, y € V. Showthat G has two Hamilton cycles that have no edge in common.(d) Using Kruskal's Greedy Algorithm, find a minimum weight spanning tree of the edgeweighted graph below.I731YW1100132NU 215

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Answered: 5. Let G=(V, E) be a graph with n > 3…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...

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In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.
Draw a directed graph diagram for relation S on {0, 1, 2, 3,4}, where TSy iff x + y = 4
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True or False? For every nonempty set A, there exists a relation R which is both an equivalence relationand a partial ordering on A.

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5. Let G=(V, E) be a graph with n > 3 vertices. (a) Define a relation on V by "ry if and only if there is a walk between r and y". Prove that is an equivalence relation on V.